1.1 Definitions

We have opted, not without thought, for the following definition, which is the opposite of the majority view.

This gives the basic data for a bicategory, $ {\mathscr{P}}{\kern.5pt}\textit {rof}$ , of profunctors. Composition is given by “matrix multiplication,” which takes the form of a coend. For and , the composite $ Q \otimes P$ is defined by

\begin{equation*} Q \otimes P (A, C) = \int ^{B \in \textbf {B}} Q (B, C) \times P (A, B) \rlap {\,.} \end{equation*}

The identity is the hom functor

\begin{equation*} {\textrm {Id}}_{\textbf {A}} (A, A') = \textbf {A} (A, A') \rlap {\,.} \end{equation*}

The reader is referred to the standard texts (see, e.g., Borceaux (Reference Borceaux1994b)) for a proof that we do get a bicategory.

For explicit computations involving profunctors, the following notation is useful. An element $ x \in P (A, B)$ is denoted by a pointed arrow, sometimes called a heteromorphism, if it’s necessary to keep track of the profunctor. The functoriality of $ P$ manifests itself as a composition

which is associative (left, right, and middle) and unitary.

It is in dealing with composition that this is most useful. An element of $ Q \otimes P (A, C)$ is an equivalence class of pairs

where the equivalence relation is generated by identifying and if we have

so they are equivalent iff there exists a path of pairs

We write the equivalence class as

\begin{equation*} y \otimes _B x \ \ \mbox{or simply} \ \ y \otimes x \rlap {\,.} \end{equation*}

The equivalence relation is generated by

\begin{equation*} y b \otimes x = y \otimes b x \rlap {\,.} \end{equation*}

Every functor $ F \colon \textbf {A} \longrightarrow \textbf {B}$ induces two profunctors

and

\begin{equation*}F_* (A, B) = \textbf {B} (FA, B) \quad \quad F^* (B, A) = \textbf {B} (B, FA) . \end{equation*}

$ F^*$ is right adjoint to $ F_*$ in $ {\mathscr{P}}{\kern.5pt}\textit {rof}$ .

1.2 Biclosedness

The bicategory $ {\mathscr{P}}{\kern.5pt}\textit {rof}$ is biclosed, that is, $ \otimes$ admits right adjoints in each variable giving two hom profunctors $ \oslash$ and characterized by natural bijections

for profunctors

We use Lambek’s notation for the internal homs. Inasmuch as $ \otimes$ is a product, the right adjoints are quotients of a sort.

An element of $ (Q\ \varnothing_{C}\ R) (A, B)$ is a $ \textbf {C}$ -natural transformation

\begin{equation*} t \colon Q (B, -) \longrightarrow R (A, -) \end{equation*}

and an element of $(R$ $_{\textbf {A}}\, P) (B, C)$ is an $ \textbf {A}$ -natural transformation

\begin{equation*} u \colon P (-, B) \longrightarrow R (-, C)\rlap {\,.} \end{equation*}

1.3 Cocontinuous functors

Our interest is in functors between functor categories and a profunctor will produce an adjoint pair of them. A profunctor: is a functor

\begin{equation*} \textbf {1}^{op} \times \textbf {A} \longrightarrow \textbf {Set} \end{equation*}

which we identify with a functor $ \Phi \colon \textbf {A} \longrightarrow \textbf {Set}$ . A profunctor will then produce, by composition, a functor

\begin{equation*} P \otimes _{\textbf {A}} (\ \ ) \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}} \end{equation*}

with a right adjoint

It follows that $ P \otimes _{\textbf {A}} (\ \ )$ is cocontinuous and is considered to be the linear functor corresponding to the matrix $ P$ .

As is well known, we have

Given a cocontinuous functor $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ , the corresponding profunctor is given by

\begin{equation*} P (A, B) = F (\textbf {A} (A, -)) (B) \rlap {\,.} \end{equation*}

Note that this doesn’t use cocontinuity of $ F$ , which leads to the following.

Definition 1.2. The core of a functor $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ is the profunctor defined by

\begin{equation*} \mbox{Cor} (F) (A, B) = F (\textbf {A} (A, -)) (B) \rlap {\,.} \end{equation*}

The functor

\begin{equation*} \mbox{Cor} (F) \otimes (\ \ ) \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}} \end{equation*}

is the “linear core” of $ F$ .

Proof. A profunctor can be viewed, by exponential adjointness, as a functor $ \textbf {A}^{op} \longrightarrow \textbf {Set}^{\textbf {B}}$ . Then, $ \mbox{Cor}$ is just restriction along the Yoneda embedding

\begin{equation*} F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}\quad \longmapsto \quad \textbf {A}^{op} \longrightarrow^Y \textbf {Set}^{\textbf {A}} \longrightarrow^F \textbf {Set}^{\textbf {B}} \end{equation*}

and $ P \otimes _{\textbf {A}} (\ \ )$ is left Kan extension

Thus for $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ , $ \mbox{Cor} (F) \otimes _A (\ \ )$ is the best approximation to $ F$ by a cocontinuous functor. As a matter of interest, the counit of the adjunction

\begin{equation*} \epsilon (F) \colon \mbox{Cor} (F) \otimes (\ \ ) \longrightarrow F \end{equation*}

is given as follows. An element of $ (\mbox{Cor} (F) \otimes \Phi )( B)$ is an equivalence class

so

\begin{equation*} [\textbf {A} (A, -) \xrightarrow{\hspace{6pt}{\bar {x}}\hspace{6pt}} \Phi , \ y \in F (\textbf {A} (A, -))(B)] \end{equation*}

giving

\begin{equation*} F (\textbf {A} (A, -))(B) \xrightarrow{F({\bar {x}})(B)} F (\Phi ) (B) \end{equation*}

\begin{equation*} y \longmapsto F ({\bar {x}})(B) (y) \rlap {\,.} \end{equation*}

Example 1.1. If $ \textbf {A}$ and $ \textbf {B}$ are discrete categories, i.e., sets $ A$ and $ B$ , then a profunctor is just a $ A \times B$ -matrix of sets $ [P_{ab}]$ and a morphism of profunctors $ P \longrightarrow P'$ a $ A \times B$ -matrix of functions. The identity $ {\textrm {Id}}_{\textbf {A}}$ is the matrix with $ 1$ ’s on the diagonal and $ 0$ elsewhere. If $ \textbf {C}$ is another discrete category and a profunctor, then $ Q \otimes _{\textbf {B}} P$ is the $ B \times C$ -matrix

\begin{equation*} \Bigg[ \sum _{b \in B} Q_{bc} \times P_{ab} \Bigg]_{\rlap {\,.}} \end{equation*}

If then

\begin{equation*} R \oslash _{\textbf {A}} P = \Bigg[\prod _{a \in A} R_{ac}^{P_{ab}} \Bigg] \end{equation*}

and

A profunctor is a $ 1 \times A$ matrix of sets, i.e., a vector $[X_a]$ and $ P \otimes _{\textbf {A}} X$ is the vector

\begin{equation*} \Bigg[ \sum _{a \in A} P_{ab} \times X_a\Bigg]_{b \rlap {\,.}} \end{equation*}

On the other hand for a $ B$ -vector,

\begin{equation*} \Bigg[\prod _b Y_b^{P_{ab}} \Bigg]_{a\rlap {\,.}} \end{equation*}

So, $ P \otimes _{\textbf {A}} (\ \ )$ is a “linear” functor and a “monomial” functor.

2.1 Complemented subobjects

In this section, we collect some useful facts about complemented subobjects in functor categories $ \textbf {Set}^{\textbf {A}}$ , most of which are well known from topos theory. We first list some general topos theory results, which will be useful for us. Proofs can be found in any of the standard topos theory books (see Borceaux (Reference Borceaux1994a) for an easily accessible account).

We will use the hooked arrow as a reminder that $ \Psi$ is complemented,

Recall that every subobject $ \Psi \, \succ\!\xrightarrow{\hspace{.3cm}} \, \Phi$ has a pseudo-complement $ \neg \Psi \, \succ\!\xrightarrow{\hspace{.3cm}} \, \Phi$ , the largest subobject of $ \Phi$ whose intersection with $ \Psi$ is $ 0$ . It can be calculated as the pullback of the element $ \mbox{false} \colon 1 \, \succ\!\xrightarrow{\hspace{.3cm}} \, \Omega$ along the characteristic morphism of $ \Psi$ .

Proposition 2.1. 1. A subobject $ \Psi \, \succ\!\xrightarrow{\hspace{.3cm}} \, \Phi$ is complemented iff its characteristic morphism factors through $ 1 + 1$

2. Complemented subobjects are closed under composition.

3. Complemented objects are stable under pullback: if is complemented and $ f \colon \Theta \longrightarrow \Phi$ , then $ \neg f^{-1} (\Psi ) = f^{-1} (\neg \Psi )$ , and we have isomorphisms

4. If is complemented, its complement is $ \neg \Psi$ , so complements are unique when they exist.

5. Given an inverse image diagram (pullback)

$ f$ restricts to

and the resulting square is also a pullback.

Complemented subobjects in functor categories $ \textbf {Set}^{\textbf {A}}$ are better behaved than in general toposes. For example, $ \neg \Psi \, \succ\!\xrightarrow{\hspace{.3cm}} \, \Phi$ is always complemented for any subobject $ \Psi \, \succ\!\xrightarrow{\hspace{.3cm}} \, \Phi$ .

Proposition 2.2. For $ \textbf {Set}^{\textbf {A}}$ , we have

(1) $ \Psi \, \succ\!\xrightarrow{\hspace{.3cm}} \, \Phi$ is complemented iff for all $ f \colon A \longrightarrow A'$ and $ x \in \Phi A$ , we have

\begin{equation*} x \in \Psi A \quad \Longleftrightarrow \quad \Phi (f) (x) \in \Psi (A) \rlap {\,.} \end{equation*}

This is equivalent to saying that for all $ f \colon A \longrightarrow A'$

is a pullback diagram. This in turn is equivalent to saying that for all $ f \colon A \longrightarrow A'$ , every commutative square

has a unique fill-in making the bottom triangle commute, i.e., $ \Psi \longrightarrow \Phi$ is orthogonal to every representable transformation.

(2) For $ \Psi \, \succ\!\xrightarrow{\hspace{.3cm}} \, \Phi$ ,

\begin{equation*} \neg \Psi (A) = \{a \in \Phi A\ |\ (\forall f \colon A \longrightarrow A') (\Phi (f) (a) \notin \Psi (A')\} \end{equation*}

and $ \neg \neg \Psi (A)$ consists of all elements, $ x$ of $ \Phi (A)$ connected to an element $ x'$ of $ \Psi$ by a zigzag of elements of $ \Phi$

\begin{equation*} A \longleftarrow A_1 \longrightarrow A_2 \longleftarrow \cdots \longrightarrow A_n =\kern-1pt= A' \end{equation*}

\begin{equation*} \Phi A \xleftarrow{\hspace{15pt}} \Phi A_1 \xrightarrow{\hspace{15pt}} \Phi A_2 \xleftarrow{\hspace{15pt}} \cdots \Phi A_n \xleftarrow{\hspace{8pt}}\!\prec \ \Psi A' \end{equation*}

\begin{equation*} x \,\longleftarrow\!\shortmid\, x_1 \longmapsto x_2 \longleftarrow\!\shortmid\, \cdots \longmapsto x_n = x' \end{equation*}

(3) For any $ \Psi \, \succ\!\xrightarrow{\hspace{.3cm}} \, \Phi$ , $ \neg \Psi$ is complemented and its complement is $ \neg \neg \Psi$ , which is the smallest complemented subobject of $ \Phi$ containing $ \Psi$ .

Thus, the class of complemented subobjects consists of all transformations right orthogonal to the representable transformations $ \textbf {A} (f, -)$ , suggesting that it may be the $ {\mathscr{M}}$ part of a factorization system on $ \textbf {Set}^{\textbf {A}}$ , which is indeed the case.

For $ \Phi$ in $ \textbf {Set}^{\textbf {A}}$ , let $ \sim$ be the equivalence relation on the set of all elements of $ \Phi$ generated by identifying $ x \in \Phi A$ with $ \Phi f (x) \in \Phi A'$ for all $ f \colon A \longrightarrow A'$ . Thus, $ x \in \Phi A \sim x' \in \Phi A'$ if there exists a zigzag path as in (2) above. The set of equivalence classes is the set of components of $ \Phi$ , $ \pi _0 \Phi = \varinjlim _A \Phi A$ , and two elements are equivalent if and only if they are in the same component.

Thus, $ t$ is $ \pi _0$ -surjective iff every element of $ \Phi$ is connected by a zigzag path to an element in the image of $ t$ .

Proof. (1) and (2) are obvious from the definition. For (3), let $ t \colon \Psi \longrightarrow \Phi$ be any transformation. Let $ \Phi _0 A \subseteq \Phi A$ be the set of all $ x \in \Phi A$ connected to an element in the image of $ t$ . $ \Phi _0$ is easily seen to be a complemented subfunctor of $ \Phi$ and is in fact the union of all of the components of $ \Phi$ that contain an element in the image of $ t$ . Then, $ t$ factors as

and $ t_0$ is $ \pi _0\mbox{-surjective}$ by construction. This is our factorization. The uniqueness part will follow from (4).

Consider a commutative square in $ \textbf {Set}^{\textbf {A}}$

where $ t$ is $ \pi _0\mbox{-surjective}$ and $ m$ is a complemented mono. Any $ x \in \Phi A$ is connected to some $ t (A') (y)$ for $ y \in \Psi A'$ , so $ s (A) (x)$ is connected to $ s (A') t(A') (y) = m (A') r (A') (y)$ . As $ m$ is complemented, this implies that $ s (A) (x)$ is in $ \Gamma (A)$ . This gives the diagonal fill-in $ \delta \colon \Phi \longrightarrow \Gamma$ such that $ m\ \delta = s$ and $ \delta \ t = r$ . $ \delta$ is unique as $ m$ is monic.

These results tell us that we have a factorization system on $ \textbf {Set}^{\textbf {A}}$ with $ {\mathscr{E}}$ the class of $ \pi _0$ -surjections and $ {\mathscr{M}}$ the class of complemented monos. We call it the Boolean factorization. Note that the class of $ \pi _0$ -surjections is not stable under pullback however. Consider morphisms $ f_i \colon A_0 \longrightarrow A_i , i = 1, 2$ in $ \textbf {A}$ and consider the pullback

$ \Sigma (A)$ consists of pairs of morphisms $ (g_1, g_2)$ such that

commutes, which well may be empty for all $ A$ . In that case, taking $ \pi _0$ of the above pullback gives

showing that $ \textbf {A} (g_1, -)$ is $ \pi _0$ -surjective but its pullback is not.

Nevertheless, it will be useful for us in Section 5 where we will be particularly interested in transformations defined on sums of representables. We record here the following facts for use later.

A natural transformation

\begin{equation*} t \colon \sum _{j \in J} \textbf {A} (C_j, -) \longrightarrow \sum _{i \in I} \textbf {A} (A_i, -) \end{equation*}

is determined by a function on the indices $ \alpha \colon J \longrightarrow I$ and a $ J$ -family of functions $ \langle\, f_j\rangle$ ,

\begin{equation*} f_j \colon A_{\alpha (j)} \longrightarrow C_j \rlap {\ .} \end{equation*}

Write $ t = \displaystyle {\sum _\alpha } \textbf {A} (f_j, -)$ .

It’s implicit in (1), but may be worth mentioning explicitly, that the complemented subobjects of $ \sum _{i \in I} \textbf {A} (A_i, -)$ are the subsums, i.e., of the form $ \sum _{k \in K} \textbf {A} (A_k, -)$ for $ K \subseteq I$ . It is also clear from the fact that each hom functor $ \textbf {A} (A_i, -)$ is connected and complemented, so is one of the components of $ \sum _{i \in I} \textbf {A} (A_i, -)$ , and any complemented subfunctor is a union of components.

The following is well known (see Borceaux (Reference Borceaux1994a, Example 7.2.4)).

We end this subsection with the following, which says that limits and confluent colimits of complemented subobjects are again complemented.

Proof. (1) is complemented iff for every $ f \colon A \longrightarrow A'$ ,

is a pullback (Proposition 2.2 (1)). Limits of pullback diagrams are pullbacks, and the result follows.

(2) Recall from Paré (Reference Paré2024) that a category $ \textbf {I}$ is confluent if any span can be completed to a commutative square and that confluent colimits commute with inverse image diagrams in $ \textbf {Set}$ . This gives (2) immediately.

Proof. Let be a family of complemented subobjects. Without loss of generality, we can assume that the total subobject is contained in it so that the indexing poset $ \textbf {I}$ is connected. Then, by the previous proposition

\begin{equation*} \varprojlim \Psi _i \longrightarrow \varprojlim \Phi \end{equation*}

is a complemented mono. Because $ \textbf {I}$ is connected the limit of the constant diagram $ \varprojlim \Phi \cong \Phi$ , and the $ \varprojlim \Psi _i$ is . The lattice of complemented subobjects of $ \Phi$ is self-dual, which implies the result for unions.

Note that this result does not hold in an arbitrary Grothendieck topos.

2.2 Tense functors

As mentioned above, the functors $ P \otimes (\ \ ) \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ arising from profunctors are not generally taut. In fact, they don’t even preserve monos in general. This may not be surprising if we consider the tensor product of modules, but one might have hoped that things would be better in the simpler $ \textbf {Set}$ case.

Example 2.1. For any epimorphism $ e \colon A \longrightarrow A'$ in $ \textbf {A}$ , the natural transformation $ \textbf {A} (e, -)\colon$ $ \textbf {A} (A', -)\longrightarrow \textbf {A} (A, -)$ is a monomorphism. If, for a profunctor $ P \otimes (\ \ ) \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ were to preserve monos, we would need that $ P \otimes \textbf {A} (e, -)$ be a mono, but $ P \otimes \textbf {A} (e, -)$ is

\begin{equation*} P (e, -) \colon P (A', -) \longrightarrow P (A, -) \rlap {\ .} \end{equation*}

So $ P (e, B) \colon P (A', B) \longrightarrow P (A, B)$ would have to be one-to-one for all $ B$ , but that’s hardly always the case. The simplest example is when $ \textbf {A} = \textbf {2}$ and $ \textbf {B} = \textbf {1}$ . Then, $ P (e, 0)$ is an arbitrary function in $ \textbf {Set}$ ( $ e$ is the unique morphism $ 0 \longrightarrow 1$ , which is of course epi).

Now, the functors $ P \otimes (\ \ )$ are “linear functors,” and any theory of functorial differences that doesn’t apply to them is seriously flawed. This leads to the main definition of the section.

Tense functors are closely related to, though incomparable with, taut functors. For this reason, we chose the word “tense” as an approximate synonym and homonym of “taut”.

Any functor preserving binary coproducts is tense, in particular $ P \otimes (\ \ )$ , which preserves all colimits, is tense. So, Example2.1 shows that tense does not imply taut. On the other hand, the functor

\begin{equation*} \textbf {Set} \longrightarrow \textbf {Set}^{\textbf {2}} \end{equation*}

\begin{equation*} A \longmapsto (A \longrightarrow 1) \end{equation*}

is taut (a right adjoint, so preserves all limits) but not tense: any proper subset $ A \subsetneq B$ gives a noncomplemented subobject

The following is obvious but worth stating explicitly.

Proof. (1) and (2) are obvious as is the “only if” part of (3), so assume $ F$ is taut and $ F(j)$ complemented. If is complemented, its characteristic morphism factors through $ 1 + 1 \, \succ\!\xrightarrow{\hspace{.3cm}} \, \Omega$ giving a pullback

$ F$ of which is also a pullback, so is complemented.

Evaluation functors preserve tenseness but, contrary to tautness, they don’t jointly create it. However, if we consider “evaluating at a morphism,” they do.

Proof. $ ev_B \colon \textbf {Set}^{\textbf {B}} \longrightarrow \textbf {Set}$ preserves coproducts so is tense and thus $ ev_B F$ will be tense if $ F$ is. To say that $ ev_g \colon ev_B \longrightarrow ev_{B'}$ is tense is to say that for every complemented subobject , we have a pullback

which Proposition2.2 (1) says is indeed the case. So, $ ev_g F$ will be tense when $ F$ is.

In fact, this says that being complemented is equivalent to every $ g$ giving a pullback as above. So our condition (2) implies that $ F$ preserves complemented subobjects. And the evaluation functors $ ev_B$ jointly create pullbacks. So (1) and (2) together imply that $ F$ is tense.

The second part is clear as the functors $ ev_B$ jointly create pullbacks and tenseness of natural transformations is a purely pullback condition.

Corollary 2.2. The following are equivalent.

1. (1) $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ is tense.

2. (2) (a) For every complemented subobject and every morphism $ g \colon B \longrightarrow B'$ ,

   

   is a pullback diagram, and

3. (b) For every pullback diagram of complemented subobjects

   

   and every $ B$ in $ \textbf {B}$ ,

   

   is a pullback.

4. (3) For every pullback diagram of complemented subobjects

   

   and every $g \colon B \longrightarrow B'$ ,

   

   is a pullback.

Furthermore, $ t \colon F \longrightarrow \ G$ is tense if and only if for every complemented subobject and every object $ B$ in $ \textbf {B}$ ,

is a pullback.

2.3 Limits and colimits of tense functors

Proposition 2.10. Let $ \Gamma \colon \textbf {I} \longrightarrow {{\mathscr{C}}{\kern.5pt}\textit {at}} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}})$ be a diagram such that for every $ I$ in $ \textbf {I}$ , $ \Gamma (I)$ is tense. Then,

(1) $ \varprojlim \Gamma$ is tense.

If $ t \colon \Gamma \longrightarrow \Theta$ is a natural transformation such that for every $ I$ in $ \textbf {I}$ , $ t I \colon \Gamma I \longrightarrow \Theta I$ is tense, then

(2) the induced transformation

\begin{equation*} \varprojlim t \colon \varprojlim \Gamma \longrightarrow \varprojlim \Theta \end{equation*}

is tense.

If $ \textbf {I}$ is confluent, then under the same conditions as above we have

(3) $ \varinjlim \Gamma$ is tense, and

(4) $ \varinjlim t$ is tense.

This is a result about limits and colimits of tense functors taken in $ {{\mathscr{C}}{\kern.5pt}\textit {at}} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}})$ . It is not assumed that the transition transformations $ \Gamma (I) \longrightarrow \Gamma (J)$ are tense, and unsurprisingly, we don’t get a universal property for tense cones or cocones. Given a tense cone or cocone, the uniquely induced natural transformation is tense but this doesn’t establish the required bijection because neither the projections in the limit case nor the injections in the colimit case are tense.

It’s more natural to consider diagrams where the transitions are tense, i.e., $ \Gamma \colon \textbf {I} \longrightarrow {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}})$ . For such diagrams, things are better. We lose products as the projections are not tense but that’s the only obstruction. Limits of connected tense diagrams are created by the inclusion

\begin{equation*} {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}}) \, \succ\!\xrightarrow{\hspace{12pt}} \, \ {{\mathscr{C}}{\kern.5pt}\textit {at}} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}}) \end{equation*}

as are all colimits, not just confluent ones.

First, we analyze diagrams $ \Gamma \colon \textbf {I} \longrightarrow {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}})$ .

Proof. Functors $ \Gamma \colon \textbf {I} \longrightarrow {{\mathscr{C}}{\kern.5pt}\textit {at}} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}})$ correspond bijectively to functors $ \overline {\Gamma } \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B} \times \textbf {I}}$ by exponential adjointness:

\begin{equation*} \overline {\Gamma } (\Phi ) (B, I) = \Gamma (I) (\Phi ) (B) \rlap {\,.} \end{equation*}

If $ \Gamma$ factors through $ {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}})$ , then we want to show that $ \overline {\Gamma }$ is tense.

First of all, $ \overline {\Gamma } (\Psi ) \longrightarrow \overline {\Gamma } (\Phi )$ must be a complemented subobject for complemented, i.e.,

for $ g \colon B \longrightarrow B'$ and $ \alpha \colon I \longrightarrow I'$ should be a pullback of monos. If we rewrite this in terms of $ \Gamma$ and use functoriality on the vertical arrows, we see that it is

(1) is a pullback of monos because $ \Gamma (I)$ is tense, and (2) is a pullback of monos because $ \Gamma (\alpha )$ is a tense transformation (the mono part because $ \Gamma (I')$ is tense).

This shows that if $ \Gamma (I)$ preserves complemented subobjects and $ \Gamma (\alpha )$ is tense, then $ \overline {\Gamma }$ preserves complemented subobjects. The converse is also true as can be seen by taking $ \alpha = {\textrm {id}}_I$ for $ \Gamma (I)$ and $ g = 1_B$ for $ \Gamma (\alpha )$ .

Preservation of inverse images by $ \overline {\Gamma }$ is equivalent to that of $ \Gamma (I)$ as can be seen immediately upon writing it down. Likewise for the tenseness of $ \overline {t}$ .

Proof. Given a diagram $ \Gamma \colon \textbf {I} \longrightarrow {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}})$ , its colimit is given by the composite

\begin{equation*} \textbf {Set}^{\textbf {A}} \xrightarrow{\overline {\Gamma }} \textbf {Set}^{\textbf {B} \times \textbf {I}} \xrightarrow{\varinjlim_{\textbf {I}}} \textbf {Set}^{\textbf {B}} \end{equation*}

$ \varinjlim _{\textbf {I}}$ is left adjoint to the diagonal functor $ D \colon \textbf {Set}^{\textbf {B}} \longrightarrow \textbf {Set}^{\textbf {B} \times \textbf {I}}$ , so it preserves coproducts and a fortiori is tense. And $ \overline {\Gamma }$ is tense by the previous proposition, so $ \varinjlim _I \Gamma (I)$ is tense.

$ D$ itself preserves coproducts being left adjoint to $ \varprojlim$ , the limit functor. So $ D$ is tense. Natural transformations between coproduct preserving functors are automatically tense, so the adjunction $ \varinjlim _{\textbf {I}} \dashv D$ is an adjunction in the bicategory $ {\mathscr{T}}\ \textit {ense}$ , and this gives the universal property of $ \varinjlim _I \Gamma (I)$ :

\begin{equation*} {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}})^{\textbf {I}} \xrightarrow{\cong} {\mathscr{T}}\ \textit {ense}(\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B} \times \textbf {I}}) \xrightarrow{{\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \ \varinjlim )} {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}}) \end{equation*}

is left adjoint to

\begin{equation*} {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}}) \xrightarrow{{\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \ D)} {\mathscr{T}}\ \textit {ense}(\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B} \times \textbf {I}}) \xrightarrow{\cong } {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}})^{\textbf {I}} \end{equation*}

which is itself the diagonal functor.

If $ \textbf {I}$ is nonempty and connected, then $ \varprojlim _{\textbf {I}} \colon \textbf {Set}^{\textbf {B} \times \textbf {I}} \longrightarrow \textbf {Set}^{\textbf {B}}$ preserves coproducts, so the same argument as above shows that $ \textbf {I}$ -limits are created in this case.

2.4 Internal homs

Part of the motivation for introducing tense functors was that the functors $ P \otimes (\ \ )$ , thought of as linear, were not in general taut but preserved coproducts, so were tense. The other side of the story is that the right adjoint to $ P \otimes (\ \ )$ , namely , is taut but not always tense. As Example1.1 suggests is a functorial version of a monomial with the $ P$ acting as the powers, and perhaps we shouldn’t expect them to be nice for all $ P$ . After all, even for real valued functions, fractional powers can be problematic, and for rings, the powers are taken to be integers, not elements of the ring.

Proposition 2.12. For a profunctor the internal hom functor is tense if and only if for every $ f \colon A \longrightarrow A'$ , the function

\begin{equation*} \pi _0 P (A', -) \longrightarrow \pi _0 P (A, -) \end{equation*}

is onto.

Proof.

preserves limits and so is taut. Thus, by Proposition2.8 (3), it is only necessary to check that

is complemented, and it’s also sufficient. This is equivalent to the condition that for every $ f \colon A \longrightarrow A'$

be a pullback. This says that every natural transformation $ t$ for which (the outside of)

commutes, factors through the injection $ j$ . This is in $ \textbf {Set}^{\textbf {B}}$ . Using the adjunction $ \pi _0 \dashv Const \colon \textbf {Set} \longrightarrow \textbf {Set}^{\textbf {B}}$ , we have, equivalently, that every function $ \overline {t}$ for which

commutes, factors through $ j$ (in $ \textbf {Set}$ ). This is equivalent to

\begin{equation*} \pi _0 P (A', -) \longrightarrow \pi _0 P (A, -) \end{equation*}

being onto.

The condition on $ P$ making tense is a kind of lifting condition. For every element of  $ P$ , and morphism $ f \colon A \longrightarrow A'$ there exist a $ B'$ and a $ P$ -element for which $ p' f$ is connected to $ P$ by a path of $ P$ -elements

Or more fancifully and more memorably, it’s a kind of homotopy pushout condition: for every $ f$ and $ p$ as below there exist a lifting to a $ p'$ with a fill in “fan”

2.5 Multivariable analytic functors

Following Fiore et al. (Reference Fiore, Gambino, Hyland and Winskel2008), we define analytic functors of several variables $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ as follows. First, for a category $ \textbf {A}$ , its exponential $ ! \textbf {A}$ (from linear logic) is the free symmetric strict monoidal category generated by $ \textbf {A}$ . In concrete terms, $ ! \textbf {A}$ is the category with objects finite sequences $ \langle A_1 \ldots A_n\rangle$ of objects of $ \textbf {A}$ and morphisms finite sequences of morphisms of $ \textbf {A}$ controlled by a permutation. There are no morphisms between sequences unless they have the same length and then

\begin{equation*} \langle A_1 \ldots A_n\rangle \longrightarrow \langle A'_1 \ldots , A'_n\rangle \end{equation*}

is a permutation of the indices, $ \sigma \in S_n$ and a sequence of morphisms

\begin{equation*} f_i \colon A_{\sigma i} \longrightarrow A'_i \rlap {\,.} \end{equation*}

Composition is as expected

\begin{equation*} (\tau , \langle g_i \rangle ) (\sigma , \langle\, f_i \rangle ) = (\sigma \tau , \langle g_i f_{\tau i} \rangle ). \end{equation*}

An $ \textbf {A}$ – $ \textbf {B}$ symmetric sequence is a profunctor which for us is a functor $ (! \textbf {A})^{op} \times \textbf {B} \longrightarrow \textbf {Set}$ . (Warning: Our definition of profunctor is the opposite of theirs.) $ P$ encodes what are to be the coefficients of a $ \textbf {B}$ -family of multivariable power series.

The analytic functor determined by $ P$

\begin{equation*} \widetilde {P} \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}} \end{equation*}

is given by

\begin{equation*} \widetilde {P} (\Phi ) (B) = \int ^{\langle A_1 \ldots A_n\rangle \in \, ! \textbf {A}} P (A_1 \ldots A_n;\ B) \times \Phi A_1 \times \Phi A_2 \times \ldots \times \Phi A_n \rlap {\,.} \end{equation*}

We will show that $ \widetilde {P}$ is tense. Define a profunctor by

\begin{equation*} Q (A_1, \ldots , A_n ;\ A) = \textbf {A} (A_1, A) + \textbf {A} (A_2, A) + \ldots + \textbf {A} (A_n, A) \end{equation*}

with the obvious definition on morphisms. We may consider $ Q$ as a functor $ (!\textbf {A})^{op} \longrightarrow \textbf {Set}^{\textbf {A}}$ and $ \widetilde {P}$ is the left Kan extension of $ P$ , considered as a functor $ (!\textbf {A})^{op} \longrightarrow \textbf {Set}^{\textbf {B}}$ , along $ Q$

For our purposes, a different description of $ \widetilde {P}$ will be useful.

Proposition 2.13. 1. $ \widetilde {P}$ is the composite

2. $ Q$ satisfies the condition of Proposition 2.12 .

Proof. (1) Let $ \Phi \in \textbf {Set}^{\textbf {A}}$ . An element of is a natural transformation

\begin{equation*} \textbf {A} (A_1, -) + \cdots + \textbf {A} (A_n, -) \longrightarrow \Phi \end{equation*}

which by the universal property of coproduct and the Yoneda lemma corresponds to an element of

\begin{equation*} \Phi A_1 \times \Phi A_2 \times \cdots \times \Phi A_n \rlap {\,.} \end{equation*}

Now the result follows by the definition of $ P \otimes (\ \ )$ and $ \widetilde {P}$ .

(2) $ Q (A_1, \ldots , A_n ; -) = \textbf {A} (A_1, -) + \cdots + \textbf {A}(A_n, -)$ a sum of representables each of which is connected. So,

\begin{equation*} \pi _0 Q (A_1, \ldots , A_n; -) \cong n \end{equation*}

and, as $ ! \textbf {A}$ has only morphisms between sequences of the same length, we get

\begin{equation*}\pi _0 Q (A_1, \ldots , A_n ; -) \cong \pi _0 Q (A'_1 , \ldots , A'_n; -) \rlap {\,.} \end{equation*}

Corollary 2.4. For an $ \textbf {A}$ – $ \textbf {B}$ symmetric sequence and a profunctor, we have

\begin{equation*} \widetilde {R \otimes P} \cong R \otimes \widetilde {P} \rlap {\,.} \end{equation*}

Proof.

3.1 Partial difference

A functor $ \Phi \in \textbf {Set}^{\textbf {A}}$ is a multisorted algebra, the sorts being the objects of $ \textbf {A}$ , with unary operations corresponding to the morphisms of $ \textbf {A}$ . Freely adding a single element of sort $ A$ gives

\begin{equation*} \Phi \leadsto \Phi + \textbf {A} (A, -) \rlap {\,.} \end{equation*}

Definition 3.1. The A-shift functor for an object $ A$ in $ \textbf {A}$ is

\begin{equation*} S_A \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {A}} \end{equation*}

\begin{equation*} S_A (\Phi ) = \Phi + \textbf {A} (A, -)\rlap {\,.} \end{equation*}

$ S_A$ is clearly tense, in fact a tense monad. Although we won’t use it here, it may be of interest to note that an Eilenberg–Moore algebra for $ S_A$ consists of a functor $ \Phi \in \textbf {Set}^{\textbf {A}}$ together with an element $ x \in \Phi A$ . A Kleisli morphism is a partial natural transformation

defined on a complemented subobject $ \Phi _0$ together with a transformation on the complement $ \Phi '_0 \longrightarrow \textbf {A} (A, -)$ , perhaps quantifying the degree of undefinedness.

These monads commute with each other

\begin{equation*} S_{A_1} \circ S_{A_2} \cong S_{A_2} \circ S_{A_1} \end{equation*}

and for every $ f \colon A \longrightarrow A'$ there is a monad morphism $ S_A \longrightarrow S_{A'}$ which is tense.

The main definition of the paper is the following.

Definition 3.2. The partial difference with respect to A, or the A-partial difference, $ \Delta _A [F]$ , of a tense functor $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ is given by

\begin{equation*} \Delta _A [F] \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}} \end{equation*}

\begin{equation*} \Delta _A [F] (\Phi ) = F (\Phi + \textbf {A} (A, -)) \setminus F (\Phi )\,, \end{equation*}

the complement of .

Proposition 3.1. For a tense functor $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ , $ \Delta _A [F]$ is also a tense functor. A tense natural transformation $ t \colon F \longrightarrow G$ restricts to one, $ \Delta _A [t] \colon \Delta _A [F] \longrightarrow \Delta _A [G]$ , making $ \Delta _A$ a functor

\begin{equation*} \Delta _A \colon {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}}) \longrightarrow {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}}) \,.\end{equation*}

Proof. Let $ \phi \colon \Psi \longrightarrow \Phi$ be a natural transformation. We have the following pullbacks

From the second one, we get that $ F (\phi + \textbf {A} (A, -))$ restricts to the complements and gives another pullback

which gives functoriality and tenseness.

Suppose $ t \colon F \longrightarrow G$ is a tense transformation. Then, we get a pullback for any $ \Phi$

so $ t (\Phi + \textbf {A} (A,-))$ restricts to the complements, giving another pullback

It follows immediately that $ \Delta _A [t]$ is natural. Tenseness follows by comparing the following diagrams that we get for any complemented subobject .

The pasted rectangles are equal, and (2), (3), and (4) are pullbacks, so (1) is too.

Corollary 3.1. $ \Delta _A [F]$ is a complemented subobject of the shifted $ F$

\begin{equation*} F + \Delta _A [F] {\xrightarrow{\cong}} F \circ S_A \end{equation*}

where the first component is $ F$ of the unit $ \eta _A \colon {\textrm {id}} \longrightarrow S_A$ .

3.2 Limit and colimit rules

$ \Delta _A$ satisfies all the same commutation properties with respect to limits and colimits as the $ \Delta$ of Paré (Reference Paré2024). This may be proved directly with virtually the same proofs as in loc. cit. However, just as the usual properties of partial derivatives follow from their single variable versions by fixing all the variables but one, those of $ \Delta _A$ follow from their $ \Delta$ counterparts.

Proposition 3.2. Objects $ A$ in $ \textbf {A}$ and $ \Phi$ in $ \textbf {Set}^{\textbf {A}}$ give an affine functor $ \textbf {Set} \longrightarrow \textbf {Set}^{\textbf {A}}$

\begin{equation*} Aff_{A, \Phi } (X) = \textbf{A} (A,-) \cdot X + \Phi \rlap {\,.} \end{equation*}

For any tense functor $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ and object $ B$ in $ \textbf {B}$ , the translated functor

\begin{equation*} F^B_{A, \Phi } = (\textbf {Set} {\xrightarrow{ Aff_{A, \Phi }}} \textbf {Set}^{\textbf {A}} {\xrightarrow{F}} \textbf {Set}^{\textbf {B}} {\xrightarrow{e v_B}} \textbf {Set}) \end{equation*}

is taut and

\begin{equation*} \Delta _A [F] (\Phi ) (B) \cong \Delta [F^B_{A, \Phi }](0)\rlap {\,.} \end{equation*}

Proof. The evaluation functors are tense as is $ Aff_{A,\Phi }$ so the composite $ e\!v_B \circ F \circ Aff_{A,\Phi }$ is too, so taut.

\begin{equation*} \begin{array}{lll} \Delta [F^B_{A, \Phi }] (0) & = & F^B_{A,\Phi } (1) \setminus F^B_{A,\Phi } (0)\\ & = & F (\textbf {A}(A,-) \cdot 1 + \Phi ) (B) \setminus F(\textbf {A} (A,-) \cdot 0 + \Phi )(B)\\ & \cong & F(\Phi + \textbf {A}(A,-)) (B) \setminus F (\Phi ) (B)\\ & = & \Delta _A [F] (B) \rlap {\,.} \end{array} \end{equation*}

Precomposing by any functor, in particular $ Aff_{A, \Phi }$ , preserves all limits and colimits (of the $ F$ ’s), and precomposing by a functor that preserves complemented subobjects preserves tense transformations. The same holds for postcomposing by $ ev_B$ . Furthermore, the $ ev_B$ jointly create limits and colimits. These considerations give the following results.

We now look at a few special cases.

Proof. $ P \otimes (\ \ )$ is cocontinuous so preserves binary coproducts

\begin{equation*} \begin{array}{lll} P \otimes (\Phi + \textbf {A} (A,-)) & \cong & P \otimes \Phi + P \otimes \textbf {A} (A,-)\\ & \cong & P \otimes \Phi + P (A,-) \rlap {\,.} \end{array} \end{equation*}

All that was used in Proposition 3.3 was that $ P \otimes (\ \ )$ preserved binary coproducts, so we can improve it.

Proposition 3.4. If $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ preserves binary coproducts, then

\begin{equation*} \Delta _A [F] (\Phi ) = F (\textbf {A} (A,-)) \rlap {\,.} \end{equation*}

Note that $ \Delta _A [F]$ is independent of $ \Phi$ , so $ \Delta _A [F]$ is the constant functor $ \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ with value $ F (\textbf {A} (A,-))$ .

We can do better than (2) in the Corollary3.2.

Proposition 3.5. Let $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ be tense and a profunctor. Then

\begin{equation*} \Delta _A [P \otimes F] \cong P \otimes \Delta _A [F] \rlap {\,.} \end{equation*}

Proof. We have a coproduct diagram preserved by $ P \otimes (\ \ )$

from which the result follows.

The notation $ P \otimes F$ may need some explanation as it doesn’t type check. It is componentwise tensor, $ (P \otimes F) (\Phi ) = P \otimes _{\textbf {B}} F (\Phi )$ . We can interpret Proposition 3.5 as saying that multiplying $ F$ by a matrix of constants is preserved by differences. But we can generalize this result to the following, although the interpretation of “pulling constants out” may be lost.

Proposition 3.6. If $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ is tense and $ G \colon \textbf {Set}^{\textbf {B}} \longrightarrow \textbf {Set}^{\textbf {C}}$ preserves binary coproducts, then

\begin{equation*} \Delta _A [GF] = G \Delta _A [F] \rlap {\,.} \end{equation*}

3.3 Analytic functors

In this section, we prove that the generalized analytic functors of Fiore et al. (Reference Fiore, Gambino, Hyland and Winskel2008) are closed under taking differences and, in fact, derive an explicit formula for the symmetric sequences so obtained.

We start with an addition formula for analytic functors, which may look obvious but is frustratingly hard to make precise. The integral notation for coends conveniently hides the functoriality of the arguments, which in the case at hand is not trivial, involving permutations as it does.

We introduce some notation, without which we run the risk of drowning in a sea of subscripts, subsubscripts, ellipses, and so on.

In what follows $ \vec A$ represents an arbitrary object of $ ! \textbf {A}$ , $ \langle A_1, \ldots , A_n \rangle$ of length $ n$ . Recall that a morphism $ (\sigma , \langle\, f_1, \ldots , f_n \rangle ) \colon \langle A_1, \ldots , A_n \rangle \longrightarrow \langle A'_1, \ldots , A'_n \rangle$ is a permutation $ \sigma \in S_n$ and a sequence of morphisms

\begin{equation*} f_i \colon A_{\sigma i} \longrightarrow A'_i\rlap {\,.} \end{equation*}

We will denote that by $ (\sigma , \vec f\,) \colon \vec A \longrightarrow \vec A'$ . We also use objects $ \vec X = \langle X_1, \ldots , X_k \rangle$ and $ \vec Y = \langle Y_1, \ldots , Y_l \rangle$ whose lengths are $ k$ and $ l$ , respectively. By construction, $ ! \textbf {A}$ is a monoidal category whose tensor is concatenation

\begin{equation*} \vec X \otimes \vec Y = \langle X_1, \ldots , X_k \, , \,Y_1, \ldots , Y_l \rangle \end{equation*}

a notation that we use extensively. Of course, it also applies to morphisms

\begin{equation*} (\tau , \vec g\,) \otimes (g, \vec h\,) = (\tau + \rho , \vec g \otimes \vec h\,) \end{equation*}

where $ \tau + \rho \colon k + l \longrightarrow k + l$ is the ordinal sum, and $ \vec g \otimes \vec h$ is concatenation.

We also use the notation and obvious variants,

\begin{equation*} \prod \Phi \vec A \colon = \Phi A_1 \times \ldots \times \Phi A_n \end{equation*}

for $ \Phi$ in $ \textbf {Set}^{\textbf {A}}$ . An element $ \langle a_1, \ldots , a_n \rangle$ of $ \prod \Phi \vec A$ is denoted $ \vec a \in \prod \Phi \vec A$ .

The addition formula alluded to above is given in the following statement.

Theorem 3.2. Let be an $ \textbf {A}$ - $ \textbf {B}$ symmetric sequence and $ \widetilde {P} \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ the analytic functor it defines. Then, for $ \Phi _1$ and $ \Phi _2$ in $ \textbf {Set}^{\textbf {A}}$ and $ B$ in $ \textbf {B}$ , we have a natural isomorphism

\begin{equation*} \widetilde {P} (\Phi _1 + \Phi _2) (B) \cong \int ^{\vec X} \int ^{\vec Y} P (\vec X \otimes \vec Y ;\ B) \times \prod \Phi _1 \vec X \times \prod \Phi _2 \vec Y \rlap {\,.} \end{equation*}

The idea of the proof is simple:

\begin{eqnarray*} \widetilde {P} (\Phi _1 + \Phi _2) (B) & = & \int ^{\vec A} P(\vec A;\ B) \times \prod (\Phi _1 \vec A + \Phi _2 \vec A\,)\\[6pt] & \cong & \int ^{\vec A} P(\vec A ;\ B) \times \sum _{\alpha \colon n \rightarrow 2} \prod \Phi _\alpha \vec A\\[6pt] & \cong & \int ^{\vec X, \vec Y} P (\vec X \otimes \vec Y ;\ B) \times \prod \Phi _1 \vec X \times \prod \Phi _2 \vec Y\\[6pt] & \cong & \int ^{\vec X} \int ^{\vec Y} P (\vec X \otimes \vec Y;\ B) \times \prod \Phi _1 \vec X \times \prod \Phi _2 \vec Y \rlap {\,.} \end{eqnarray*}

The first line is just the definition of $ \widetilde {P}$ , the second line is distributivity of $ \prod$ over $ +$ , and the last line is Fubini for coends. It’s in going from the second to the third line that everything happens. The “reason” for the isomorphism is that for each summand with $ \Phi _1$ and $ \Phi _2$ interspersed “at random” in the product, there is an isomorphism in $ ! \textbf {A}$ which permutes them so that all the $ \Phi _1$ come first followed by the $ \Phi _2$ . And, indeed that’s the reason. The devil is in the details, as they say.

We step back and consider how we might show that two coends are isomorphic. Let $ \Gamma \colon \textbf {I}^{op}\times \textbf {I} \longrightarrow \textbf {Set}$ be a functor which we might think of as a profunctor . The coend $ \int ^I \Gamma (I, I)$ consists of equivalence classes of elements of $ \Gamma$ , , the equivalence relation generated by identifying with when there are $ f \colon I \longrightarrow I'$ and such that $ x = {\bar {x}} f$ and $ x' = f {\bar {x}}$ :

So $ x$ is equivalent to $ x'$ if there’s a zigzag of such diagrams joining them.

But in the case at hand the equivalence relation is simpler because both of the diagrams whose coends we’re considering are separable into a product of a contravariant functor times a covariant one.

Definition 3.3. A diagram $ \Gamma \colon \textbf {I}^{op} \times \textbf {I} \longrightarrow \textbf {Set}$ is separable if for every $ f \colon I \longrightarrow I'$ ,

is a pullback.

For example, if $ \Gamma (I, I') = \Gamma _0 I \times \Gamma _1 I'$ for $ \Gamma _0 \colon \textbf {I}^{op} \longrightarrow \textbf {Set}$ and $ \Gamma _1 \colon \textbf {I} \longrightarrow \textbf {Set}$ , then $ \Gamma$ is separable. Or, if $ \textbf {I}$ is a groupoid, every $ \Gamma$ is separable.

The point of this definition is that the equivalence relation is generated by identifying $ x$ with $ x'$ when there is an $ f \colon I \longrightarrow I'$ such that $ x' f = f x$ :

The $ {\bar {x}}$ is automatic. This is important because we can compose such squares.

Let us call an $ x \in \Gamma (I, I)$ a $ \Gamma$ -algebra and an $ f$ as above a homomorphism. Then, we get a category $ \textbf {Alg} (\Gamma )$ and $ \int ^I \Gamma (I, I) = \pi _0 \textbf {Alg} (\Gamma )$ , the set of connected components of $ \textbf {Alg} (\Gamma )$ .

Let $ \Theta \colon \textbf {J}^{op} \times \textbf {J} \longrightarrow \textbf {Set}$ be another bivariant diagram. A morphism $ (\Xi , \xi ) \colon \Gamma \longrightarrow \Theta$ is a functor $ \Xi \colon \textbf {I} \longrightarrow \textbf {J}$ and a natural transformation $ \xi \colon \Gamma \longrightarrow \Theta (\Xi (-), \Xi (-))$

Such a morphism induces a functor

\begin{equation*} \textbf {Alg} (\Xi , \xi ) \colon \textbf {Alg} (\Gamma ) \longrightarrow \textbf {Alg} (\Theta ) \end{equation*}

We are now ready to apply this to our addition formula. Let be given by

\begin{equation*} \Gamma (\vec X, \vec Y ; \vec X', \vec Y'\,) = P (\vec X \otimes \vec Y ;\ B) \times \prod \Phi _1 \vec X' \times \prod \Phi _2 \vec Y' \end{equation*}

and by

\begin{equation*} \Theta (\vec A; \vec A'\,) = P (\vec A ;\ B) \times \sum _{\alpha \colon n \longrightarrow 2} \prod \Phi _\alpha \vec A' \end{equation*}

with the obvious action on morphisms. Note that $ \Gamma$ and $ \Theta$ are both products of a covariant part (with the primes) and a contravariant part (without primes) so that they are separable. Thus, we will be able to compute the coends by taking connected components of their categories of elements.

Theorem 3.3. With the above notation, there is a morphism

such that the induced functor

\begin{equation*} \textbf {Alg} (\otimes , \xi ) \colon \textbf {Alg} (\Gamma ) \longrightarrow \textbf {Alg} (\Theta ) \end{equation*}

is an equivalence of categories.

Proof. Throughout, $ B$ is a fixed object of $ \textbf {B}$ .

An element of $ \Gamma (\vec X, \vec Y ; \vec X', \vec Y'\,)$ is a triple

\begin{equation*} \large \Big(p \in P (\vec X \otimes \vec Y ;\ B),\ \vec x \in \prod \Phi _1 \vec X',\ \vec y \in \prod \Phi _2 \vec Y '\large \Big)\rlap {\,,} \end{equation*}

and an element of $ \Theta (\vec A, \vec A'\,)$ is a triple

\begin{equation*} \Big(p \in P (\vec A ;\ B), \ \alpha \colon n \rightarrow 2, \ \vec a \in \prod \Phi _\alpha \vec A'\Big) \rlap {\,,} \end{equation*}

where $ \prod \Phi _\alpha \vec A'$ is $ \prod ^{n'}_{i = 1} \Phi _{\alpha i} A'_i$ , as expected.

\begin{equation*} \xi \colon \Gamma (\vec X, \vec Y ; \vec X', \vec Y') \longrightarrow \Theta (\vec X \otimes \vec Y, \vec X' \otimes \vec Y') \end{equation*}

is given by

\begin{equation*} \xi (p, \vec x, \vec y) = \Big(p \in P (\vec X \otimes \vec Y ;\ B),\ \alpha _{k', l'} \colon k' + l' \longrightarrow 2,\ \langle \vec x, \vec y \rangle \in \prod \Phi _{\alpha _{k', l'}} (\vec X' \otimes \vec Y')\Big) \rlap {\,.} \end{equation*}

Here, $ \alpha _{k', l'}$ is the indexing that consists of $ 1$ ’s followed by $ 2$ ’s,

\begin{equation*} \alpha _{k', l'} (i) = \left \{ \begin{array}{ll} 1 & \mbox{if}\quad l \leq i \leq k'\\ 2 & \mbox{if}\quad k' \lt i \leq k' + l' \rlap {\,,} \end{array} \right . \end{equation*}

and $ \langle \vec x, \vec y \rangle$ is concatenation

\begin{equation*} \langle \vec x, \vec y \rangle = \langle x_1, \ldots , x_{k'}, y_1, \ldots , y_{l'} \rangle \in \Phi _1 X'_1 \times \ldots \times \Phi _1 X'_{k'} \times \Phi _2 Y'_1 \times \ldots \times \Phi _2 Y'_{l'} \rlap {\,.} \end{equation*}

Naturality of $ \xi$ is a straightforward calculation.

The morphism $ ( \otimes , \xi )$ induces a functor

\begin{equation*} \Xi \colon \textbf {Alg} (\Gamma ) \longrightarrow \textbf {Alg} (\Theta ) \rlap {\,.} \end{equation*}

Explicitly, a $ \Gamma$ -algebra is a $ 5$ -tuple

\begin{equation*} \Big(\vec X,\ \vec Y,\ p \in P (\vec X \otimes \vec Y ;\ B),\ \vec x \in \prod \Phi _1 \vec X ,\ \vec y \in \prod \Phi _2 \vec Y\Big) \end{equation*}

and a $ \Theta$ -algebra is a quadruple

\begin{equation*} \Big(\vec A, \ p \in P (\vec A ;\ B), \ \alpha \colon n \longrightarrow 2,\ \vec a \in \prod \Phi _\alpha \vec A\,\Big) \rlap {\,.} \end{equation*}

$ \Xi$ assigns to $ (\vec X, \vec Y, p, \vec x , \vec y\,)$ the algebra $ (\vec X \otimes \vec Y, \ p, \ \alpha _{k, l},\ \langle \vec x, \vec y \rangle )$ .

A homomorphism $ (\vec X, \vec Y, p, \vec x, \vec y\,) \longrightarrow (\vec X', \vec Y', p', \vec x', \vec y\,')$ is a pair of morphisms in $ ! \textbf {A}$ ,

\begin{equation*} (\tau , \vec g) \colon \vec X \longrightarrow \vec X' \quad \mbox{ and }\quad (\rho , \vec h\,) \colon \vec Y \longrightarrow \vec Y' \end{equation*}

preserving everything. It is sent to $ (\tau , \vec g) \otimes (\rho , \vec h\,)$ by $ \Xi$ .

$ \otimes$ is faithful as it is just concatenation, so $ \Xi$ is also faithful.

If $ (\sigma , \vec f\,)$ is a homomorphism

\begin{equation*} (\vec X \otimes \vec Y, \ p, \ \alpha _{k, l},\ \langle \vec x, \vec y\,\rangle ) \longrightarrow ( \vec X' \otimes \vec Y' , \ p', \ \alpha _{k', l'},\ \langle \vec x', \vec y\,'\rangle ) \end{equation*}

we have

which implies that $ \sigma$ restricts to bijections $ \tau \colon k' \longrightarrow k$ and $ \rho \colon l' \longrightarrow l$ (by taking inverse images of $ \{1\}$ and $ \{2\}$ ) so $ k' = k$ and $ l' = l$ and $ \sigma = \tau + \rho$ . It follows that $ \vec f$ consists of morphisms $ (\tau , \vec g) \colon \vec X \longrightarrow \vec X'$ and $ (\rho , \vec h) \colon \vec Y \longrightarrow \vec Y'$ and the preservation of $ \langle \vec x, \vec y\,\rangle$ becomes preservation of $ \vec x$ and $ \vec y\,$ separately. That is, $ ( \sigma , \vec f\,)$ is $ \Xi ((\tau , \vec g), (\rho , \vec h))$ and so $ \Xi$ is full.

For any $ \Theta$ -algebra $ (\vec A, p, \alpha , \vec a)$ , there is a permutation of $ \sigma \in S_n$ such that

\begin{equation*} n \xrightarrow{\sigma} n \xrightarrow{\alpha} 2 \end{equation*}

is order -preserving, i.e., all the $ 1$ ’s come first and then the $ 2$ ’s, so that $ \alpha \sigma = \alpha _{k, l}$ , where $ k$ is the cardinality of $ \alpha ^{-1} \{1\}$ and $ l$ that of $ \alpha ^{-1}\{2\}$ . Associated to $ \sigma$ is an isomorphism

\begin{equation*} (\sigma , \vec 1\,) \colon \vec A \longrightarrow \vec {A_\sigma } \end{equation*}

where $ \vec {A_\sigma }$ is $ \langle A_{\sigma 1}, \ldots , A_{\sigma n} \rangle$ and $ \vec 1 = \langle 1_{A_{\sigma 1}}, \ldots , 1_{A_{\sigma n}} \rangle$ . We can transport the $ \Theta$ -algebra structure on $ \vec A$ to one on $ \vec {A_\sigma }$ giving an algebra isomorphism

\begin{equation*} (\sigma , \vec 1\,) \colon (\vec A, p, \alpha , \vec a) \longrightarrow (\vec {A_\sigma } , \ p \cdot (\sigma ^{-1}, \vec 1\,),\ \alpha _{k, l} ,\ \vec {a}_\sigma ) \end{equation*}

where $ p \cdot (\sigma ^{-1}, \vec 1\,) = P ((\sigma ^{-1}, \vec 1\,);B)(p)$ and $ \vec {a}_\sigma = \langle a_{\sigma 1}, \ldots , a_{\sigma n} \rangle$ in $ \prod \Phi _{\alpha \sigma } \vec {A_\sigma }$ . The $ \Theta$ -algebras with indexing of the form $ \alpha _{k, l}$ are precisely those in the image of $ \Xi$ . Indeed, the $ \vec X$ are the first $ k$ $ A$ ’s, $ \langle A_{\sigma 1}, \ldots , A_{\sigma k} \rangle$ in this case and $ \vec Y$ the last $ l$ of them $ \langle A_{\sigma (k + l)}, \ldots , A_{\sigma (n)} \rangle$ . Similarly, $ \vec x = \langle a_{\sigma 1} , \ldots , a_{\sigma k}\rangle$ and $ \vec y = \langle a_{\sigma (k+1)}, \ldots a_{\sigma n} \rangle$ . Then, $ \Xi (\vec X, \vec Y, p \cdot (\sigma ^{-1}, \vec 1\,), \vec x, \vec y\,)$ is $ (\vec {A_\sigma }, p \cdot (\sigma ^{-1}, \vec 1\,), \alpha _{k, l}, \vec a)$ , so $ \Xi$ is essentially surjective, which shows it’s an equivalence.

If we take connected components we get

\begin{equation*} \pi _0 \textbf {Alg}(\Gamma ) \cong \pi _0 \textbf {Alg} (\Theta ) \end{equation*}

so the coend of $ \Gamma$ is isomorphic to that of $ \Theta$ .

Corollary 3.4.

\begin{align*} \int ^{\vec X, \vec Y} P (\vec X \otimes \vec Y ;\ B) \times \prod \Phi _1 \vec X \times \prod \Phi _2 \vec Y \cong \int ^{\vec A} P (\vec A ;\ B) \times \sum _{\alpha \colon n \longrightarrow 2} \prod \Phi _\alpha \vec A \rlap. \end{align*}

Our addition formula, Theorem3.2, now follows by a simple application of the Fubini theorem for coends, which is what we wanted, but Theorem3.3 is a stronger result.

Our next step in the derivation of the formula for $ \Delta _A [\widetilde {P}]$ is to specialize our addition formula to the case $ \Phi _1 = \Phi$ and $ \Phi _2 = \textbf {A} (A, -)$ . This gives

\begin{equation*} \widetilde {P} (\Phi + \textbf {A} (A, -)) (B) = \int ^{\vec X} \int ^{\vec Y} P (\vec X \otimes \vec Y ;\ B) \times \prod \Phi \vec X \times \prod \textbf {A} (A, \vec Y) \end{equation*}

in which the expression

\begin{equation*} \prod \textbf {A} (A, \vec Y) = \textbf {A} (A, Y_1) \times \ldots \times \textbf {A}(A, Y_l) \end{equation*}

appears, not surprisingly, as it already appears in the definition of $ \widetilde {P}$ . It defines a functor

\begin{equation*} \prod \textbf {A}(A, -) \colon !\textbf {A} \longrightarrow \textbf {Set} \end{equation*}

closely related to the representable functor $ ! \textbf {A} (A^{\otimes n}, -)$ where $ A^{\otimes n} = \langle A, \ldots , A \rangle$ , the $ n$ -fold tensor of $ A$ .

Proposition 3.7. With the above notation, we have

\begin{equation*} \prod \textbf {A} (A, -) \cong \sum ^\infty _{n = 0} ! \textbf {A} (A^{\otimes n}, -)/S_n \rlap {\,.} \end{equation*}

Proof. If $ \vec Y = \langle Y_1, \ldots , Y_l \rangle$ , then $ ! \textbf {A} (A^{\otimes n} , \vec Y)$ is $ 0$ unless $ l = n$ in which case an element of $ ! \textbf {A} (A^{\otimes n}, \vec Y)$ is a morphism

\begin{equation*} (\sigma , \vec f\,) \colon A^{\otimes n} \longrightarrow \vec Y \end{equation*}

so that $ ! \textbf {A}(A^{\otimes n}, \vec Y) \cong S_n \times \textbf {A} (A_1 Y_1) \times \ldots \times \textbf {A} (A, Y_n)$ and if we quotient by $ S_n$ we get

\begin{equation*} ! \textbf {A} (A^{\otimes n}, \vec Y) / {S_n} \cong \prod \textbf {A} (A, \vec Y) \end{equation*}

easily seen to be natural in $ \vec Y$ . The result follows.

Lemma 3.1. Let $ W \colon !\textbf {A}^{op} \longrightarrow \textbf {Set}$ . Then

\begin{equation*} \int ^{\vec Y} W(\vec Y) \times \prod \textbf {A} (A, \vec Y) \cong \sum ^\infty _{n = 0} W (A^{\otimes n})/S_n \rlap {\,.} \end{equation*}

Proof.

\begin{eqnarray*} \kern-3pc\int ^{\vec Y} W({\vec Y}) \times \prod \textbf {A} (A, \vec Y) & \cong & \int ^{\vec Y} W (\vec Y) \times \sum ^\infty _{n = 0} ! \textbf {A}(A^{\otimes n}, \vec Y)/S_n\\ & \cong &\sum ^\infty _{n = 0} \int ^{\vec Y} W (\vec Y) \times (! \textbf {A} (A^{\otimes n}, \vec Y)/{S_n}\\\end{eqnarray*}

\begin{eqnarray*} && \kern85pt\cong \sum ^\infty _{n =0} \Bigl (\int ^{\vec Y} W (\vec Y) \times ! \textbf {A} (A^{\otimes n} , \vec Y) \Bigr )/S_n\\&& \kern85pt\cong \sum ^\infty _{n = 0} W (A^{\otimes n})/S_n \rlap {\,.} \end{eqnarray*}

The second isomorphism is commutation of coends and coproducts, the third commutation of coends with colimits (“modding out” by $ S_n$ is a colimit), and the last isomorphism comes from the fact that tensoring with a representable is substitution.

Corollary 3.5.

\begin{align*} \widetilde {P} (\Phi + \textbf {A} (A, -))(B) \cong \int ^{\vec X} \sum ^\infty _{n = 0} P (\vec X \otimes A^{\otimes n} ;\ B)/ (\{{\textrm {id}}_k\} \times S_n) \times \prod \Phi \vec X \end{align*}

Proof.

\begin{equation*} \widetilde {P} (\Phi + \textbf {A} (A, -)) (B) \cong \int ^{\vec X} \int ^{\vec Y} P (\vec X \otimes \vec Y ;\ B) \times \prod \Phi \vec X \times \prod \textbf {A} (A, \vec Y) \rlap {\,.} \end{equation*}

If we fix $ \vec X$ and consider the coend over $ \vec Y$ , we can apply the previous lemma with

\begin{equation*} W (\vec Y) = P (\vec X \otimes \vec Y ;\ B) \times \prod \Phi \vec X \end{equation*}

and the result follows immediately.

Corollary 3.6.

\begin{align*} \Delta _A [\widetilde {P}] (\Phi ) (B) \cong \int ^{\vec X} \sum ^\infty _{n = 1} P(\vec X \otimes A^{\otimes n} ;\ B)/ (\{{\textrm {id}}_k\} \times S_n) \times \prod \Phi \vec X \rlap. \end{align*}

For any $ \textbf {A}$ - $ \textbf {B}$ symmetric sequence and object $ A$ of $ \textbf {A}$ , we define a new symmetric sequence by the formula

\begin{equation*} \nabla _A P (\vec X ;\ B) = \sum ^\infty _{n = 1} P (\vec X \otimes A^{\otimes n} ;\ B) /(\{{\textrm {id}}_k\} \times S_n) \rlap {\,.} \end{equation*}

Now Corollary3.6 can be stated in its final form, giving the main theorem of the section.

Theorem 3.4. Analytic functors $ \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ are closed under taking differences. If is a symmetric sequence, then

\begin{equation*} \Delta _A [\widetilde {P}] \cong \widetilde {\nabla _A P} \rlap {\,.} \end{equation*}

The definition of $ \nabla _A P$ as a coproduct of quotients is clear but for formal manipulations a more abstract definition is useful. Let $ \textbf {S}_+$ be the category whose objects are positive finite cardinals, $ k \gt 0$ , and whose morphisms are bijections. So $ \textbf {S}_+$ is the coproduct

\begin{equation*} \textbf {S}_+ = \sum ^\infty _{k = 1} \textbf {S}_k \end{equation*}

where $ \textbf {S}_k$ is the symmetric group $ \textbf {S}_k$ considered as a one-object category.

Given an $ \textbf {A}$ - $ \textbf {B}$ symmetric sequence and an object $ A$ of $ \textbf {A}$ we get an $ \textbf {S}_+$ family of $ \textbf {A}$ - $ \textbf {B}$ symmetric sequences

\begin{equation*} P_A \colon \textbf {S}^{op}_+ \longrightarrow {\mathscr{P}}{\kern.5pt}\textit {rof} (! \textbf {A}, \textbf {B}) \end{equation*}

\begin{equation*} P_A (k) (A_1 \ldots A_n ;\, B) = P(A_1 \ldots A_n , A, A, \ldots , A ;\ B) \end{equation*}

where there are $ k$ $ A$ ’s. Functoriality and naturality are obvious. Now $ \nabla _A P = \varinjlim _k P_A (k)$ .

Proposition 3.8. For an $ \textbf {A}$ - $ \textbf {B}$ symmetric sequence and a profunctor, we have

\begin{equation*} \nabla _A (Q \otimes P) \cong Q \otimes \nabla _A P \rlap {\,.} \end{equation*}

Proof.

\begin{eqnarray*} \nabla _A (Q \otimes P) (A_1 \ldots A_n ;\ C) & \cong & \varinjlim _k \int ^B Q (B, C) \times P(A_1 \ldots A_n, A \ldots A ;\ B)\\ & \cong & \int ^B Q (B, C) \times \varinjlim _k P(A_1 \ldots A_n , A \ldots A ;\ B)\\ & \cong & \int ^B Q (B, C) \times \nabla _A P (A_1 \ldots A_n ;\ B)\\ & \cong & (Q \otimes \nabla _A P) (A_1 \ldots A_n ;\ C) \rlap {\,.} \end{eqnarray*}

Corollary 3.7. For any $ \textbf {A}$ - $ \textbf {B}$ symmetric sequence $ P$ we have

\begin{equation*} \nabla _A P \cong P \otimes \nabla _A {\textrm {Id}}_{!\textbf {A}} \rlap {\,.} \end{equation*}

Proof.

\begin{equation*} \nabla _A P \cong \nabla _A (P \otimes {\textrm {Id}}_{!\textbf {A}}) \cong P \otimes \nabla _A {\textrm {Id}}_{!\textbf {A}} \rlap {\,.} \end{equation*}

$ \nabla _A {\textrm {Id}}_{!\textbf {A}}$ is easy to describe:

\begin{equation*} \nabla _A {\textrm {Id}}_{!\textbf {A}} \colon \textbf {Set}^{!\textbf {A}} \longrightarrow \textbf {Set}^{!\textbf {A}} \end{equation*}

\begin{equation*} \nabla _A {\textrm {Id}}_{!\textbf {A}} (A_1 \ldots A_n ; A'_1 \ldots A'_m) \cong \sum ^\infty _{k = 1} ! \textbf {A} (A_1 \ldots A_n, A \ldots A ; A'_1 \ldots A'_m) / (\{{\textrm {id}}_n\} \times S_k) \end{equation*}

which is $ 0$ if $ m \leq n$ and

\begin{equation*} ! \textbf {A} (A_1 \ldots A_n , A, \ldots , A ; A'_1 \ldots A'_m)/ (\{{\textrm {id}}_n\} \times S_{m-n}) \end{equation*}

when $ m \gt n$ . There are $ m - n$ $ A$ ’s and the action we’re modding out by is $ S_{m - n}$ acting on those $ A$ ’s.

There is also a generic difference formula.

Corollary 3.8.

\begin{align*} \Delta _A [\widetilde {P}] \cong P \otimes \Delta _A [{\textrm {Id}}_{!\textbf {A}}] \rlap. \end{align*}

Proof.

\begin{equation*} \begin{array}{lll} \Delta _A \widetilde {P} & \cong \widetilde {\nabla _A P} &(\mbox{Thm. 3.4})\\[5pt] & \cong (P \otimes \nabla _A {\textrm {Id}}_{! \textbf {A}}){\widetilde {\ \ \ }} &(\mbox{Cor. 3.7})\\[5pt]& \cong P \otimes \widetilde {\nabla _A {\textrm {Id}}_{! \textbf {A}}} &(\mbox{Cor. 2.4})\\[5pt]& \cong P \otimes \Delta _A {[}{\textrm {Id}}_{! \textbf {A}}{]} & (\mbox{Thm. 3.4}) \end{array} \end{equation*}

3.4 Higher differences

As $ \Delta _A [F]$ is also tense, its difference can also be taken $ \Delta _{A'} [\Delta _A [F]] = \Delta _{A', A} [F]$ and so on, iteratively. For any sequence $ \langle A_1 \ldots A_n \rangle$ of length $ n$ of objects of $ \textbf {A}$ we define

\begin{equation*} \Delta _{\langle A_i \rangle } {[}F{]} = \left \{ \begin{array}{ll} F & \mbox{ if } n = 0\\[-2pt] \Delta _{A_1} {[}\Delta _{\langle A_2 \ldots A_{n}\rangle } {[}F{]}{]} & \mbox{ if } n \geq 1 \rlap {\,.} \end{array} \right . \end{equation*}

If an element is in $ F$ of a subsum, it’s in every bigger subsum, so it is sufficient to consider only those subsums with one less term. Thus the new elements are those in the set difference

\begin{equation*} F \Bigg(\Phi + \sum _{i = 1}^n \textbf {A} (A_i, -)\Bigg)(B) \setminus \bigcup _{j = 1}^n F \Bigg(\Phi + \sum _{i \neq j} \textbf {A} (A_i, -)\Bigg) (B) \rlap {\ .} \end{equation*}

Proof. We prove this by induction on $ n$ . For $ n = 0, 1$ the result holds by definition. Assume the result holds for sequences of length $ n - 1$ and take $ \langle A_i \rangle = \langle A_1, \ldots , A_n \rangle$ . Let $ \langle A_i \rangle ^+ = \langle A_2, \ldots , A_n \rangle$ .

An element of $ \Delta _{\langle A_i\rangle } [F](\Phi )(B)$ is an element of $ \Delta _{\langle A_i \rangle ^+} [F] (\Phi + \textbf {A} (A_1, -)) (B)$ which is not in $ \Delta _{\langle A_I \rangle ^+} [F] (\Phi ) (B)$ . An element of $ \Delta _{\langle A_i \rangle ^+} [F] (\Phi + \textbf {A} (A_1, -)) (B)$ is, by the induction hypothesis, an element of

(1) \begin{equation} F \Big(\Phi + \textbf {A} (A_1, -) + \sum ^n_{i = 2} \textbf {A} (A_i, -)\Big) (B) \cong F \Big(\Phi + \sum ^n_{i = 1} \textbf {A} (A_i, -)\Big) (B) \end{equation}

not in

(2) \begin{equation} F \Bigg(\Phi + \textbf {A} (A_1, -) + \sum ^n_{i = 2, i \neq j} \textbf {A} (A_i, -)\Bigg) (B) \cong F \Bigg(\Phi + \sum ^n_{i = 1, i \neq j} \textbf {A} (A_i, -)\Bigg) (B) \end{equation}

for any $ 2 \leq j \leq n$ . From this we must exclude the elements of $ \Delta _{\langle A_i\rangle ^+} [F] (\Phi )(B)$ and these, again by the induction hypothesis, are elements of

(3) \begin{equation} F \Bigg(\Phi + \sum ^n_{i = 2} \textbf {A} (A_i, -)\Bigg) (\Phi ) (B) \end{equation}

except for any in some

(4) \begin{equation} F \Bigg(\Phi + \sum ^n_{i = 2, i \neq j} \textbf {A} (A_i, -)\Bigg) (\Phi )(B) \end{equation}

for $ 2 \leq j \leq n$ .

To summarize,

\begin{equation*} \Delta _{\langle A_i\rangle } [F] (\Phi ) (B) = ((1) \setminus (2)) \setminus ((3) \setminus (4)) \rlap {\,,} \end{equation*}

but $ (4) \subseteq (2)$ so

\begin{equation*} \Delta _{\langle A_i\rangle } [F] (\Phi ) (B) = (1) \setminus ((2) \cup (3)) \rlap {\,.} \end{equation*}

Now $ (2) \cup (3)$ is the union of

\begin{equation*} F \Bigg(\Phi + \sum _{i = 1, i \neq j} \textbf {A} (A_i, -)\Bigg)(B) \end{equation*}

over all $ j,$ $1 \leq j \leq n$ , and the result follows.

We see from this formula that $ \Delta _{\langle A_i \rangle } [F]$ is independent of the order of the differences, a version of Clairaut’s theorem.

Corollary 3.9. Let $ \langle A_i \rangle$ be a sequence of length $ n$ of objects of $ \textbf {A}$ and $ \sigma \in S_n$ a permutation, then

\begin{equation*} \Delta _{\langle A_{\sigma i} \rangle } [F] \cong \Delta _{\langle A_i \rangle } [F] \rlap {\,.} \end{equation*}

4.1 Definitions and functoriality

Let $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ be a tense functor and let $ f \colon A \longrightarrow A'$ be a morphism of $ \textbf {A}$ . Then, as

is a pullback of complemented objects, so is

and it follows that $ F (\Phi + \textbf {A} (f,-))$ restricts to complements giving another pullback

This proves the following:

Proposition 4.1. For any $ \Phi$ in $ \textbf {Set}^{\textbf {A}}$ , $ \Delta _A[F](\Phi )$ is functorial in $ A$ , i.e., is the object part of a functor

\begin{equation*} \Delta [F] (\Phi ) \colon \textbf {A}^{op} \longrightarrow \textbf {Set}^{\textbf {B}} \rlap {\,.} \end{equation*}

By exponential adjointness we get a functor $ \textbf {A}^{op} \times \textbf {B} \longrightarrow \textbf {Set}$ , i.e., a profunctor .

Definition 4.1. The (discrete) Jacobian profunctor of $ F$ at $ \Phi$

is given by

\begin{equation*} \Delta [F](\Phi ) (A,B) = \Delta _A [F](\Phi ) (B) \rlap {\,.} \end{equation*}

It’s more or less clear that $ \Delta [F](\Phi )$ is functorial in $ \Phi$ , which we express in the following proposition.

Proposition 4.2. For any tense functor $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ , $ \Delta [F] (\Phi )$ is the object part of a tense functor

\begin{equation*} \Delta [F] \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}} = {\mathscr{P}}\,\textit{rof}\ (\textbf {A}, \textbf {B}) \rlap {\,.} \end{equation*}

Proof. For a natural transformation $ t \colon \Phi \longrightarrow \Psi$ and object $ A$ in $ \textbf {A}$ ,

is a pullback of a complemented subobject, so

is too. So $ F(t + \textbf {A}(A,-))$ restricts to the complements, giving another pullback

(*)

hence functoriality.

We still must prove that it is tense.

Proposition3.1 says that for a fixed $ A$ , $ \Delta _A[F] \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ is tense and $ \Delta _A [F]$ is the composite

\begin{equation*} \textbf {Set}^{\textbf {A}} \longrightarrow^{\Delta [F]} \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}} \longrightarrow^{ev_B} \textbf {Set}^{\textbf {B}} \rlap {\,.} \end{equation*}

The $ ev_B$ are the evaluation functors which preserve pullbacks and collectively reflect them, so that $ \Delta [F]$ will preserve pullbacks of complemented subobjects. However, the $ ev_B$ don’t reflect complemented subobjects, so we still must show that $ \Delta [F]$ preserves those.

Let be a complemented subobject. We want to show that $ \Delta [F] (\Phi _0) \succ\!\xrightarrow{\hspace{.3cm}} \Delta [F] (\Phi )$ is complemented, or equivalently, for every $ f \colon A'\to A$ and $ g \colon B \longrightarrow B'$

is a pullback. We can do this separately for $ f$ and $ g$ , fixing $ B$ and then $ A$ . We already know for fixed $ A$ it’s a pullback. So let’s fix $ B$ .

Let $ f \colon A' \longrightarrow A$ and consider

and

The second and third squares are pullbacks by $ (\!*\!)$ and the fourth because $ F$ is tense. As the composite of the first and second squares is equal to the composite of the third and fourth, we get that the first square is a pullback, which shows that $ \Delta {[}F{]} (\Phi _0) \succ\!\xrightarrow{\hspace{.3cm}} \Delta {[}F{]}(\Phi )$ is complemented.

To complete the discussion of functoriality of $ \Delta$ note that $ \Delta _A [F] (\Phi )$ is a subfunctor of $ F (\Phi + \textbf {A} (A,-))$ , which is not only functorial in $ \Phi$ and $ A$ but by Proposition3.1 also in $ F$ but only for tense transformations. Proposition2.9 says that the evaluation functors $ ev_B$ jointly reflect tenseness of transformations, so that $ \Delta _A [t]$ itself will be tense. Thus, we get a functor

\begin{equation*} \Delta \colon {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {B}}) \longrightarrow {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}}) \end{equation*}

the (discrete) Jacobian functor.

There are various ways of reformulating the Jacobian which are of independent interest.

Given a tense functor $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ , we get another tense functor analogous to the differential operator

\begin{equation*} D[F] \colon \textbf {Set}^{\textbf {A}} \times \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}} \end{equation*}

\begin{equation*} D[F] (\Phi , \Psi ) = \Delta {[}F{]} (\Phi ) \otimes _{\textbf {A}} \Psi \end{equation*}

where $ \Psi$ is considered as a profunctor

Paré (Reference Paré2024) we used the finite projection $ \textbf {Set} \times \textbf {Set} \longrightarrow \textbf {Set}$ as a tangent bundle and saw that this supported a definition of functorial differences where the lax chain-rule was actually a lax functor. This generalizes to the multivariable setting. We define

by $ T[F] (\Phi , \Psi ) = (F \Phi , \Delta {[}F{]}(\Phi ) \otimes _{\textbf {A}} \Psi )$ . We see that $ T[F]$ preserves colimits in the second variable.

Profunctors are in bijection with profunctors

\begin{equation*} \frac {P \colon \textbf {A}^{op} \times \textbf {B} \longrightarrow \textbf {Set}} {P^\top \colon (\textbf {B}^{op}) \times \textbf {A}^{op} \longrightarrow \textbf {Set}}, \end{equation*}

i.e., $ P^\top (B,A) = P(A,B)$ , the transpose as matrices. This gives the reverse difference operator

\begin{equation*} \Delta ^\top [F] \colon \textbf {Set}^A \longrightarrow {\mathscr{P}}{\kern.5pt}\textit {rof} (\textbf {B}^{op}, \textbf {A}^{op}) . \end{equation*}

This suggests that we take as the cotangent bundle the first projection $ \textbf {Set}^{\textbf {A}} \times \textbf {Set}^{\textbf { A}^{op}} \longrightarrow \textbf {Set}^{\textbf {A}}$ . As the Yoneda embedding $ Y \colon \textbf {A}^{op} \succ\!\xrightarrow{\hspace{.3cm}} \textbf {Set}^{\textbf {A}}$ is the cocompletion of $ \textbf {A}$ , the category of cocontinuous functors

\begin{equation*} \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set} \end{equation*}

is equivalent to the category of functors

\begin{equation*} \textbf {A}^{op} \longrightarrow \textbf {Set} \end{equation*}

i.e., $ \textbf {Set}^{\textbf {A}^{op}}$ . So $ \textbf {Set}^{\textbf {A}^{op}}$ has a legitimate claim to be the (linear) dual of $ \textbf {Set}^{\textbf {A}}$ . Now we can extend the reverse difference to the cotangent bundle. Given a tense functor $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ , we first pull back the cotangent bundle along $ F$

and then take the functor $ \textrm {coT} [F]$

\begin{equation*} (\Phi , \Theta ) \succ\!\xrightarrow{\hspace{.3cm}} (\Phi , \Delta ^\top [F](\Phi ) \otimes \Theta ) \end{equation*}

In this $ \Theta$ in $ \textbf {Set}^{\textbf {B}^{op}}$ is considered as a profunctor

A differential form is a global section of the cotangent bundle, which in our case amounts to a functor $ \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {A}^{op}}$ .

For a tense $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ , we get another tense functor $ \Delta {[}F{]} \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}}$ which, upon composing with the evaluation at $ B$ , $ ev_B \colon \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}} \longrightarrow \textbf {Set}^{\textbf {A}^{op}}$ gives another tense functor $ \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {A}^{op}}$ . This way the difference $ \Delta {[}F{]}$ may be viewed as a $ \textbf {B}$ -family of differential forms

\begin{equation*} \textbf {B} \longrightarrow {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf { A}^{op}})\rlap {\,.} \end{equation*}

It is tempting to write $ \Omega ^1 (\textbf {Set}^{\textbf {A}})$ for $ {\mathscr{T}}\ \textit {ense} (\textbf {Set}^{\textbf {A}}, \textbf {Set}^{\textbf {A}^{op}})$ .

4.2 Product and sum rules

The evaluation functors $ ev_A \colon \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}} \longrightarrow \textbf {Set}^{\textbf {B}}$ jointly create limits and colimits, and as the composites $ ev_A \circ \Delta {[}F{]} (\Phi )$ are $ \Delta _A [F] (\Phi )$ , the limit rules of Section 3.2 lift to $ \Delta {[}F{]}$ .

Theorem3.1 gives the following.

Note that on the right-hand side of (3), we have $ \Delta {[}F{]} \times G$ , for example. $ \Delta {[}F{]}$ is a functor $ \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}}$ , whereas $ G$ is a functor $ \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ . Looking at where this came from

\begin{equation*} \Delta _A [F \times G] \cong (\Delta _A [F] \times G) + (F \times \Delta _A [G]) + (\Delta _A [F] \times \Delta _A [G]), \end{equation*}

we see that the $ G$ is the same for all $ A$ , which means the $ G$ in (3) should be interpreted, as is often done, to be the functor

\begin{equation*} \textbf {Set}^{\textbf {A}} \longrightarrow^G \textbf {Set}^{\textbf {B}} \longrightarrow^{\textbf {Set}^{P_2}} \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}} \end{equation*}

for $ P_2 \colon \textbf {A}^{op} \times \textbf {B} \longrightarrow \textbf {B}$ the second projection, i.e., $ G$ followed by the inclusion of $ \textbf {Set}^{\textbf {B}}$ in $ \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}}$ given by functors $ \textbf {A}^{op} \times \textbf {B} \longrightarrow \textbf {Set}$ constant in the first variable.

Of course similar remarks go for the $ F$ in the second term of (3) and the $ F_i$ in (3) of Theorem4.1.

Proposition 4.3. For a profunctor, we have

\begin{equation*} \Delta [P \otimes (\ \ )] (\Phi ) \cong P \rlap. \end{equation*}

This is just a restatement of Proposition3.3.

We think of $ P \otimes (\ \ )$ as a linear functor with coefficients $ P$ , and its difference is the constant functor

\begin{equation*} \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}} \end{equation*}

with constant value $ P$ .

Corollary 4.2.

\begin{align*} \Delta [{\textrm {id}}_{\textbf {Set}^{\textbf {A}}}] (\Phi ) = {\textrm {Id}}_{\textbf {A}} \end{align*}

where $ {\textrm {id}}_{\textbf {Set}^{\textbf {A}}} \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {A}}$ is the identity functor and is the identity profunctor.

Like we did in Proposition3.4, we can generalize Proposition 4.3 to the following:

Proposition 4.4. If $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ preserves binary coproducts, then

\begin{equation*} \Delta {[}F{]} (A, B) = F (\textbf {A} (A,-)) (B), \end{equation*}

i.e., $ \Delta {[}F{]} = C\!or (F)$ , the core of $ F$ (see Definition 1.2 ).

We can improve (2) in Corollary4.1, replacing the set $ C$ by a profunctor . Given a tense functor $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ , we can compose it with $ P \otimes (\ \ )$ to get another tense functor

\begin{equation*} \textbf{Set}^{\textbf{A}} {\xrightarrow{F}} \textbf {Set}^{\textbf {B}} {\xrightarrow{P \otimes (\ \ )}} \textbf {Set}^{\textbf {C}} \end{equation*}

which will be called $ P \otimes F$ as its value at $ \Phi$ is $ P \otimes (F (\Phi ))$ although it might be hard to parse.

Proposition 4.5.

\begin{align*} \Delta [P \otimes F] \cong P \otimes \Delta {[}F{]} \rlap. \end{align*}

Proof. The $ P \otimes F$ is the composite

\begin{equation*} \textbf{Set}^{\textbf{A}} {\xrightarrow{F\hspace{4pt}}} \textbf {Set}^{\textbf {B}} {\xrightarrow{P \otimes (\ \ )}} \textbf {Set}^{\textbf {C}} \end{equation*}

so $ \Delta [P \otimes F]$ is the functor $ \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {A}^{op} \times \textbf {C}}$ with values

\begin{equation*} \Delta [P \otimes F](\Phi ) (A, C) = \Delta _A [P \otimes F](\Phi ) (C) \rlap . \end{equation*}

On the other hand, $ P \otimes \Delta {[}F{]}$ is the composite

\begin{equation*} \textbf {Set}^{\textbf {A}} \xrightarrow{\Delta {[}F{]}} \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}} \xrightarrow{P \otimes _{\textbf {B}} (\ \ )} \textbf {Set}^{\textbf {A}^{op} \times \textbf {C}} \end{equation*}

so has values

\begin{equation*} (P \otimes \Delta {[}F{]}) (\Phi ) (A, C) = (P \otimes _{\textbf {B}} (\Delta {[}F{]} (\Phi ))) (A, C) \rlap . \end{equation*}

By the definition of composition of profunctors, this is

\begin{equation*} \begin{array}{lll} & &\kern-2pc \int ^B P (B, C) \times \Delta {[}F{]} (\Phi ) (A, B)\\ & = & \int ^B P(B, C) \times \Delta _A [F] (\Phi ) (B)\\ & = & (P \otimes \Delta _A [F] (\Phi )) (C) \end{array} \end{equation*}

and by Proposition3.5 this is isomorphic to $ \Delta _A [P \otimes F] (\Phi ) (C)$ .

4.3 A natural reformulation

It will be conceptually clearer to reformulate the definition of $ \Delta$ in more categorical terms, that is, in terms of natural transformations, Yoneda style. This rids us of many of the element-based proofs, eliminating, as it does, membership and especially non-membership. The results are cleaner and clearer, especially in the next section where we see the chain rule reduced to composition. This is a vast improvement over the construction and proof of the one-variable chain rule given in Paré (Reference Paré2024) which is far from transparent.

So why not just start with this as a definition? The basic intuition of finite differences would be lost. It is hard to imagine why one would define a profunctor using (2) or (3) in the proposition below, or formulate the product and sum rules or the chain rule.

Proposition 4.6. Let $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ be a tense functor and $ \Phi$ an object of $ \textbf {Set}^{\textbf {A}}$ . Then there is a natural bijection between the following:

(1) Elements $ x \in \Delta {[}F{]} (\Phi ) (A, B)$

(2) Natural transformations $ t \colon \textbf {B} (B, -) \longrightarrow F (\Phi + \textbf {A}(A, -))$ giving a pullback

(3) Natural transformations $ u \colon F(\Phi ) + \textbf {B}(B,-) \longrightarrow F (\Phi + \textbf {A} (A,-))$ giving a pullback

Proof. An element of $ \Delta {[}F{]} (\Phi ) (A, B)$ is an element of $ F (\Phi + \textbf {A} (A,-)) (B)$ , which is not in $ F (\Phi ) (B)$ . By Yoneda, this corresponds bijectively to a natural transformation

\begin{equation*} t \colon \textbf {B} (B, -) \longrightarrow F (\Phi + \textbf {A}(A,-)) \end{equation*}

for which $ t (B)(1_B) \notin F (\Phi )(B)$ . As is complemented by tenseness, that’s equivalent to none of the values of $ t$ being in $ F (\Phi )$ , which means that

is a pullback. And this, in turn, is equivalent to

being a pullback, where $ u$ is the inclusion on the first summand and $ t$ on the second.

As mentioned in Definition 1.1, it is useful to think of the elements of a profunctor as some sort of morphism but between objects of different categories (sometimes called heteromorphisms). Because of the representables appearing in the natural transformations above, it’s not unreasonable to think of them as morphisms from $ A$ to $ B$ , as a kind of Kleisli morphism although $ F$ is not a monad. If $ F$ were the identity for example, $ t$ is equivalent to a natural transformation $ \textbf {B}(B,-) \longrightarrow \textbf {A} (A,-)$ so to an actual morphism $ A \longrightarrow B$ . This is just another way of saying that $ \Delta [1_{\textbf {Set}^{\textbf {A}}}] (\Phi ) = {\textrm {Id}}_{\textbf {A}}$ , the identity profunctor on $ \textbf {A}$ , i.e., the hom functor.

More generally, if $ F$ preserves binary coproducts, a $ t$ as above corresponds to a natural transformation

\begin{equation*} \textbf {B}(B,-) \longrightarrow F(\textbf {A} (A,-))\rlap {\,,} \end{equation*}

another way of viewing the identity

\begin{equation*} \Delta {[}F{]} (\Phi ) = C\!or (F) \end{equation*}

of Proposition4.4.

With the natural transformation version of $ \Delta$ , it is easy to see how $ \Delta {[}F{]} (\Phi ) (A, B)$ is functorial in $ A$ and $ B$ . Given a $ t$ as in (2) and morphisms $ f \colon A' \longrightarrow A$ and $ g \colon B \longrightarrow B'$ , we get pullbacks

the third one because $ F$ is tense.

Similarly, functoriality in $ \Phi$ is clear. For $ \phi \colon \Phi \longrightarrow \Psi$ , we get pullbacks

again using tenseness of $ F$ .

The same goes for the functoriality in $ F$ . If $ \alpha \colon F \longrightarrow G$ is a tense transformation, we get pullbacks

Showing that $ \Delta {[}F{]} \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}}$ is tense in this context is probably no easier than the element-wise proof given for Proposition4.2, but it may be more conceptual. It is a result we need if we want to iterate $ \Delta$ , as we do. So we reprove it.

The proof that $ \Delta {[}F{]}$ preserves the pullbacks of complemented subobjects is basically the same as in Proposition 4.2, but we reproduce it here without reference to particular differences or evaluation functors.

Let

be a pullback of complemented subobjects in $ \textbf {Set}^{\textbf {A}}$ and $ A$ an object of $ \textbf {A}$ . Consider the four squares in $ \textbf {Set}^{\textbf {B}}$

(1) and (4) are pullbacks by definition of $ \Delta$ and (2) because $ F$ is tense. As the pasted rectangle (1) + (2) is equal to (3) + (4), we get that (3) is also a pullback.

As $ \Delta {[}F{]}$ preserves pullbacks of complemented subobjects, it will take a complemented subobject to a mono, but we still have to prove that it’s complemented. We have to prove that for any $ f \colon A' \longrightarrow A$ and $ g \colon B \longrightarrow B'$ ,

is a pullback.

An element of $ \Delta [\Phi ] (A,B)$ is a natural transformation $ t \colon \textbf {B}(B,-) \longrightarrow F (\Phi + \textbf {A}(A,-))$ . To be in $ \Delta [\Phi _0] (A,B)$ means that it factors through . Referring to the following diagram

$ \Delta [\Phi ] (f,g) (t)$ is the composite of the left arrow with the two top arrows, and to say that it is in $ F(\Phi _0 + \textbf {A}(A',-))$ means that there is a $ u$ making the outside boundary commute. The square in a pullback because $ F$ is tense so there exists a unique $ u'$ as shown and as $ F (\Phi _0 + \textbf {A}(A,-))$ is complemented there exists a $ u''$ by Proposition2.2. So $ t$ factors through $ F(\Phi _0 + \textbf {A}(A,-))$ , which is what we wanted.

4.4 Lax chain rule

We saw in Paré (Reference Paré2024) that the chain rule for the single variable functorial difference was expressed as a laxity morphism rather than an isomorphism, and the same applies in the multivariable case. For tense functors $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ and $ G \colon \textbf {Set}^{\textbf {B}} \longrightarrow \textbf {Set}^{\textbf {C}}$ , we will construct a comparison transformation

\begin{equation*} \gamma (\Phi ) \colon \Delta [G] (F(\Phi )) \otimes _{\textbf {B}} \Delta {[}F{]} (\Phi ) \longrightarrow \Delta [GF](\Phi ) \end{equation*}

and establish associativity and unit laws for it. In fact, considering $ \Delta {[}F{]} (\Phi )$ as a profunctor may not mean much unless it composes like a profunctor. Otherwise, it is just an object of $ \textbf {Set}^{\textbf {A}^{op} \times \textbf {B}}$ .

The construction of $ \gamma$ in Paré (Reference Paré2024) is perhaps a bit opaque and the profunctor interpretation clarifies this. We’ll see that it is, in a sense, just composition as it should be.

In the previous section, we described the functoriality of $ \Delta$ in terms of the characterization (2) of Proposition4.6, but for the chain rule the characterization (3) is better, so we reformulate the functorialities in this context. As we will refer to it a lot, let us call a natural transformation $ t$ such that

is a pullback, a PPI transformation (for pullback preserves injections).

Functoriality of $ \Delta {[}F{]} (\Phi ) (A,B)$ , considered as a set of PPI transformations, is easy. It’s just composition with $ F (\Phi + \textbf {A} (f, -))$ and $ F (\Phi ) + \textbf {B}(g,-)$ respectively.

Functoriality in $ \Phi$ and $ F$ are a bit more complicated as the $ \Phi$ and $ F$ appear in both the domain and codomain of $ t$ . The following characterization will be useful, although it is nothing but a reformulation.

Proposition 4.7. Let $ t \colon F (\Phi ) + \textbf {B} (B,-) \longrightarrow F (\Phi + \textbf {A}(A,-))$ be a PPI transformation.

(1) If $ \phi \colon \Phi \longrightarrow \Psi$ is a natural transformation, then $ \Delta {[}F{]} (\phi ) (A,B) (t)$ is the unique PPI transformation $ t'$ such that

(2) If $ \alpha \colon F \longrightarrow G$ is a tense transformation, then $ \Delta [\alpha ] (\Phi ) (A,B) (t)$ is the unique PPI transformation $ t''$ such that

Theorem 4.2. For tense functors $ F \colon \textbf {Set}^{\textbf {A}} \longrightarrow \textbf {Set}^{\textbf {B}}$ and $ G \colon \textbf {Set}^{\textbf {B}} \longrightarrow \textbf {Set}^{\textbf {C}}$ , there is a natural transformation

\begin{equation*} \gamma \colon (\Delta [G] \circ F) \otimes \Delta {[}F{]} \longrightarrow \Delta [GF] \end{equation*}

which is

1. (1) natural in $ F$ and $ G$

2. (2) associative

3. (3) normal (invertible unitors)

Proof. $ \gamma$ is to be understood pointwise, i.e., as a profunctor morphism

\begin{equation*} \gamma (\Phi): \Delta[G](F(\Phi)) \otimes _{\textbf {B}} \Delta[F](\Phi)\longrightarrow \Delta[GF](\Phi)\end{equation*}

for each $ \Phi \in \textbf {Set}^{\textbf {A}}$ , and furthermore natural in that $ \Phi$ .

Let $ A \in \textbf {A}$ and $ C \in \textbf {C}$ . An element of

\begin{equation*} \big (\Delta [G] (F(\Phi )) \otimes _{\textbf {B}} \Delta {[}F{]} (\Phi )\big ) (A,C) \end{equation*}

is an equivalence class

where $ u$ and $ t$ are PPI transformations. Let $ \gamma (\Phi )(A,C) (u \otimes _B t) = G t \cdot u$ , which is indeed PPI:

We must show that $ \gamma (\Phi ) (A,C)$ is well defined. Suppose we have another pair of transformation related by a single morphism

This means that we have commutative squares

If we apply $ G$ to the second and paste it to the first, we get a commutative diagram which shows that $ G t \cdot u = G t' \cdot u'$ . It follows that $ \gamma (\Phi ) (A,C)$ is well defined.

Naturality in $ A$ and $ C$ is clear as it is just composition with $ F (\Phi + \textbf {A} (f, -))$ and $ F (\Phi ) + \textbf {C}(h,-)$ , respectively, and has nothing to do with the equivalence relation, which is localized at $ B$ . So, we get a profunctor morphism $ \gamma (\Phi )$ .

To show that $ \gamma$ is natural in $ \Phi$ , let $ \phi \colon \Phi \longrightarrow \Psi$ be a natural transformation and consider

where the vertical arrows are induced by $ \phi$ . If we chase an element $ u \otimes _B t$ in the domain, first around the left-bottom we get $ u' \otimes _B t'$ and then $ G t' \cdot u'$ where $ u'$ and $ t'$ are the unique PPI’s such that

On the other hand, going around the top-right we get $ G t \cdot u$ and then $ v'$ the unique PPI such that

If we apply $ G$ to the diagram for $ t'$ above and paste it to the one for $ u'$ , we see that $ G t' \cdot u'$ is such a $ v'$ , and so $ v' = G t' \cdot u'$ . This gives naturality in $ \Phi$ .

We can check naturality in $ F$ and $ G$ separately. First, let $ \alpha \colon F \longrightarrow F'$ be a tense natural transformation. We wish to show that

commutes. $ \alpha$ acting on an element $ u \otimes _B t$ of the domain gives $ u' \otimes _B t'$ which gets sent to $ G t' \cdot u'$ , where

On the other hand we first get $ G t \cdot u$ and then $ v'$ such that

Again, applying $ G$ to the square for $ t'$ and pasting to the one for $ u'$ , we see that $ v' = G u' \cdot t'$ , i.e., naturality in $ F$ .

For naturality in $ G$ , let $ \beta \colon G \longrightarrow G'$ be a tense natural transformation. We’ll show that

commutes. An element $ u \otimes t$ of the domain, goes down to $ u' \otimes t$ and then $ G' u' \cdot t$ for $ u'$ such that

$ u \otimes t$ goes across to $ G t \cdot u$ and then down to $ v'$ such that

If we paste the diagram for $ u'$ with the naturality square

and compare with the diagram for $ v'$ we see that $ v' = G' t \cdot u'$ , which gives naturality in $ G$ .

Let

\begin{equation*} \textbf {Set}^{\textbf {A}} {\xrightarrow{F\hspace{8pt}}} \textbf {Set}^{\textbf {B}} {\xrightarrow{\hspace{8pt}G\hspace{8pt}}} \textbf {Set}^{\textbf {C}} {\xrightarrow{\hspace{8pt}H\hspace{8pt}}} \textbf {Set}^{\textbf {D}} \end{equation*}

be tense functors. Associativity involves taking an element $ v \otimes u \otimes t$ of

\begin{equation*} \Delta [H] (GF(\Phi )) \otimes _{\textbf {B}} \Delta [G] (F\Phi ) \otimes _{\textbf {C}} \Delta {[}F{]} (\Phi ) \end{equation*}

at $ (A,D)$ and applying $ \gamma$ in two different ways to reduce it to elements of $ \Delta [HGF](\Phi )$ , and seeing that they are equal. This is for any PPI transformations

\begin{align*} t \colon F (\Phi ) + \textbf {B} (B,-) & \longrightarrow F (\Phi + \textbf {A} (A,-))\\[8pt] u \colon GF (\Phi ) + \textbf {C} (C,-) & \longrightarrow G(F(\Phi ) + \textbf {B} (B,-))\\[8pt] v \colon HGF (\Phi ) + \textbf {D} (D,-) &\longrightarrow H (GF(\Phi ) + \textbf {C} (C,-))\rlap. \end{align*}

And indeed, we get

For the unit laws, first assume that $ \textbf {B} = \textbf {A}$ and that $ F = {\textrm {id}}_{\textbf {Set}^{\textbf {A}}}$ . Then, $ \gamma (\Phi )$ takes the form

\begin{equation*} \gamma (\Phi ) \colon \Delta [G] \otimes _{\textbf {A}} \Delta [{\textrm {id}}_{\textbf {Set}^{\textbf {A}}}] (\Phi ) \longrightarrow \Delta [G] (\Phi ) \end{equation*}

and an element of the domain is an equivalence class $ u \otimes t$ for PPI’s

\begin{equation*} \Phi + \textbf {A} (A',-) \xrightarrow{\hspace{6pt}t\hspace{6pt}} \Phi + \textbf {A} (A,-) \quad G(\Phi ) + \textbf {C} (C,-) \xrightarrow{\hspace{6pt}u\hspace{6pt}} G(\Phi + \textbf {A}(A',-))\rlap . \end{equation*}

For $ t$ to be a PPI, it must be of the form

\begin{equation*} \Phi + \textbf {A}(A',-) \xrightarrow{\Phi + \textbf {A} (f,-)} \Phi + \textbf {A}(A,-) \end{equation*}

and every equivalence class has a unique representative where $ f$ is $ 1_A$ . Then, $ \gamma (\Phi )(u \otimes 1) = u$ gives our bijective right unitor.

For the left unitor, let $ \textbf {B} = \textbf {C}$ and $ G = {\textrm {id}}_{\textbf {Set}^{\textbf {C}}}$ . Then, $ \gamma$ takes the form

\begin{equation*} \gamma (\Phi ) \colon \Delta [{\textrm {id}}_{\textbf {Set}^{\textbf {C}}}] (F (\Phi ) \otimes \Delta {[}F{]} (\Phi )) \longrightarrow \Delta {[}F{]} (\Phi ) \end{equation*}

and an element of its domain is an equivalence class $ u \otimes t$ with PPI’s

\begin{equation*} F(\Phi ) + \textbf {C}(C',-) \xrightarrow{\hspace{5pt}t\hspace{5pt}} F(\Phi + \textbf {A} (A,-))\quad F(\Phi ) + \textbf {C} (C,-) \xrightarrow{\hspace{5pt}u\hspace{5pt}} F(\Phi ) + \textbf {C} (C',-) \end{equation*}

For $ u$ to be a PPI it must be of the form $ F(\Phi ) + \textbf {C}(g,-)$ . Again every equivalence class contains a unique representative with $ g = 1_C$ . Then,

\begin{equation*} \gamma (\Phi ) (1 \otimes t) = t \end{equation*}

gives the bijective unitor.