2.1 Finite multisets

A finite multiset on a set A is a function $m:A\to{\mathbb{N}}$ such that the set $\operatorname{\mathsf{supp}}({m})=\{a\in A\mid m(a)\not=0\}$ is finite, and we use $\mathcal M_{{\mathrm{fin}}}({A})$ for the set of all finite multisets of elements of A. The cardinality of m is $\#{m}=\sum_{a\in A}m(a)$ . We use $[\,]$ for the empty multiset (so that $\operatorname{\mathsf{supp}}({[\,]})=\emptyset$ where $\operatorname{\mathsf{supp}}({m})=\{a\in A\mid m(a)\not=0\}$ is the support of m) and if $m_0,m_1\in\mathcal M_{{\mathrm{fin}}}({A})$ then $m_0+m_1\in\mathcal M_{{\mathrm{fin}}}({A})$ is defined by $(m_0+m_1)(a)=m_0(a)+m_1(a)$ . If $a_{1},\dots,a_{n}\in A$ , we use $[{a_{1},\dots,a_{n}}]$ for the $m\in\mathcal M_{{\mathrm{fin}}}({A})$ such that m(a) is the number of $i\in\{1,\dots,n\}$ such that $a_i=a$ . If $m=[{a_{1},\dots,a_{n}}]\in\mathcal M_{{\mathrm{fin}}}({A})$ and $p=[{b_{1},\dots,b_{p}}]\in\mathcal M_{{\mathrm{fin}}}({B})$ , then $m\times p=[{(a_i,b_j)\mid i\in\{1,\dots ,n\}\text{ and }j\in\{1,\dots,p\}}]\in\mathcal M_{{\mathrm{fin}}}({A\times B})$ . If $M=[{m_{0},\dots,m_{n}}]\in\mathcal M_{{\mathrm{fin}}}({\mathcal M_{{\mathrm{fin}}}({A})})$ we set $\Sigma{M}=\sum_{i=0}^nm_i\in\mathcal M_{{\mathrm{fin}}}({A})$ .

2.2 The SMCC of pointed sets

Let $\mathbf{Set}_0$ be the category of pointed sets. We use $0_X$ or simply 0 for the distinguished point of the object X. A morphism $f\in\mathbf{Set}_0(X,Y)$ is a function $f:X\to Y$ such that $f(0_X)=0_Y$ . The terminal object is the singleton $\{0\}$ . The cartesian product ${X}\mathrel{\&}{Y}$ is the ordinary cartesian product, with $0_{{X}\mathrel{\&}{Y}}=(0_X,0_Y)$ . The tensor product ${X}\otimes{Y}$ is defined as:

\begin{align*} {X}\otimes{Y}=\{(x,y)\in X\times Y\mid x=0\Leftrightarrow y=0\}\end{align*}

with $0_{{X}\otimes{Y}}=(0_X,0_Y)$ . The unit of the tensor product is the object $1=\{0,\ast\}$ of $\mathbf{Set}_0$ . This category is enriched over itself, the distinguished point of $\mathbf{Set}_0(X,Y)$ being the constantly $0_Y$ function. Actually, it is monoidal closed with ${X}\multimap{Y}=\mathbf{Set}_0(X,Y)$ and $0_{{X}\multimap{Y}}$ defined by $0_{{X}\multimap{Y}}(x)=0_Y$ for all $x\in X$ . A mono in $\mathbf{Set}_0$ is a morphism of $\mathbf{Set}_0$ which is injective as a function.

Unless explicitly stipulated, all the categories $\mathcal{L}$ we consider in this paper are enriched over pointed sets, so this assumption will not be mentioned any more. In the case of symmetric monoidal categories, this also means that the tensor product of morphisms is “bilinear” with respect to the pointed structure, that is, if $f\in\mathcal{L}(X_0,Y_0)$ then ${f}\otimes{0}=0\in\mathcal{L}({X_0}\otimes{X_1},{Y_0}\otimes{Y_1})$ and by symmetry we have ${0}\otimes{f}=0$ .

2.3 Monoidal and resource categories

Following a well-established tradition, if X is an object of a category $\mathcal{L}$ we use X to denote the identity morphism at X in $\mathcal{L}$ .

A symmetric monoidal category (SMC) is a category $\mathcal{L}$ equipped with a bifunctor $\mathcal{L}^2\to\mathcal{L}$ denoted as $\otimes$ , a monoidal unit 1 which is an object of $\mathcal{L}$ and $\lambda_X\in\mathcal{L}({1}\otimes{ }X,X)$ , $\rho_X\in\mathcal{L}({X}\otimes{1},X)$ , $\alpha_{X_0,X_1,X_2}\in\mathcal{L}({({X_0}\otimes{X_1})}\otimes{X_2},{X_0}\otimes{({X_1}\otimes{X_2})})$ and $\gamma_{X_0,X_1}\in\mathcal{L}({X_0}\otimes{X_1},{X_1}\otimes{X_0})$ as associated isomorphisms satisfying the usual McLane coherence commutations. Given objects $X_{0},\dots,X_{n-1}$ and $i<j$ in $\{0,\dots,n-1\}$ , we use $\gamma_{i,j}$ for the canonical swapping iso in $\mathcal{L}(X_0\otimes\cdots\otimes X_{n-1},X_0\otimes\cdots\otimes X_{i-1}\otimes X_j\otimes X_{i+1}\cdots \otimes X_{j-1}\otimes X_i\otimes X_{j+1}\otimes\cdots\otimes X_{n-1})$ .

2.3.1 Commutative comonoids

Definition 1. In a SMC $\mathcal{L}$ (with the usual notations), a commutative comonoid is a tuple $C=(\underline{C},\mathsf w_{C},\mathsf c_{C})$ where $\underline{C}\in\mathcal{L}$ , $\mathsf w_{C}\in\mathcal{L}(\underline{C},1)$ and $\mathsf c_{C}\in\mathcal{L}(\underline{C},{\underline{C}}\otimes{\underline{C}})$ are such that the following diagrams commute.

The category ${\mathcal {L}}^{\otimes}$ of commutative comonoids has these tuples as objects, and an element of ${\mathcal {L}}^{\otimes}(C,D)$ is an $f\in\mathcal{L}(\underline{C},\underline{D})$ such that the two following diagrams commute

Theorem 1. For any SMC $\mathcal{L}$ the category ${\mathcal {L}}^{\otimes}$ is cartesian. The terminal object is $(1,\operatorname{\mathsf{Id}}_1,({\lambda_1})^{-1})$ (remember that $\lambda_1=\rho_1$ ) simply denoted as 1 and for any object C the unique morphism $C\to1$ is $\mathsf w_{C}$ . The cartesian product of $C_0,C_1\in{\mathcal {L}}^{\otimes}$ is the object ${C_0}\otimes{C_1}$ of ${\mathcal {L}}^{\otimes}$ such that $\underline{{C_0}\otimes{C_1}}={\underline{C_0}}\otimes{\underline{C_1}}$ and the structure maps are defined as:

The projections $\mathsf{pr}_{i}^\otimes\in{\mathcal {L}}^{\otimes}({C_0}\otimes{C_1},C_i)$ are given by:

The proof is straightforward. In a commutative monoid M, multiplication is a monoid morphism $M\times M\to M$ . The following is in the vein of this simple observation.

Lemma 2. If $C\in{\mathcal {L}}^{\otimes}$ , then $\mathsf w_{C}\in{\mathcal {L}}^{\otimes}(C,1)$ and $\mathsf c_{C}\in{\mathcal {L}}^{\otimes}(C,{C}\otimes{C})$ .

Proof. The second statement amounts to the following commutation

which results from the commutativity of C. The first statement is similarly trivial.

2.3.2 Resource categories

The notion of resource category is more general than that of a Seely category in the sense of Melliès (Reference Melliès2009). We keep only the part of the structure and axioms that we need to define our notion of differential structure and keep our setting as general as possible.

An object X of an SMC $\mathcal{L}$ is exponentiable if the functor ${\_}\otimes{ }X$ has a right adjoint, denoted as ${X}\multimap{\_}$ . In that case, we use $\operatorname{\mathsf{ev}}\in\mathcal{L}({\left({X}\multimap{Y}\right)}\otimes{X},Y)$ for the counit of the adjunction and, given $f\in\mathcal{L}({Z}\otimes{X},Y)$ we use $\operatorname{\mathsf{cur}} f$ for the associated morphism $\operatorname{\mathsf{cur}} f\in\mathcal{L}(Z,{X}\multimap{Y})$ .

We say that the SMC $\mathcal{L}$ is closed (is an SMCC) if any object of $\mathcal{L}$ is exponentiable.

A category $\mathcal{L}$ is a resource category if

• $\mathcal{L}$ is an SMC;

• $\mathcal{L}$ is cartesian with terminal object $\top$ (so that 0 is the unique element of $\mathcal{L}(X,\top)$ ) and cartesian product of $X_0$ , $X_1$ denoted $({X_0}\mathrel{\&}{X_1},\mathsf{pr}_{0},\mathsf{pr}_{1})$ and pairing of morphisms $(f_i\in\mathcal{L}(Y,X_i))_{i=0,1}$ denoted $\langle{f_0,f_1}\rangle\in\mathcal{L}(Y,{X_0}\mathrel{\&}{X_1})$ ;

• and $\mathcal{L}$ is equipped with a resource comonad, that is a tuple $(\ !{\_},\operatorname{\mathsf{der}}_{},\operatorname{\mathsf{dig}}_{},\mathsf m^0,\mathsf m^2)$ where $\ !{\_}$ is a functor $\mathcal{L}\to\mathcal{L}$ which is a comonad with counit $\operatorname{\mathsf{der}}_{}$ (dereliction) and comultiplication $\operatorname{\mathsf{dig}}_{}$ (digging), and $\mathsf m^0\in\mathcal{L}(1,\ !{\top})$ and $\mathsf m^2\in\mathcal{L}({\ !{X}}\otimes{\ !{Y}},\ !{({X}\mathrel{\&}{Y})})$ are the Seely isomorphisms subject to conditions that we do not recall here; see for instance Melliès (Reference Melliès2009) apart for the following which explains how $\operatorname{\mathsf{dig}}_{}$ interacts with $\mathsf m^2$ .

(1)

Then $\ !{\_}$ inherits a lax symmetric monoidality $\mu^0,\mu^2$ on $\mathcal{L}$ (considered as an SMC). This means that one can define $\mu^0\in\mathcal{L}(1,\ !{1})$ and $\mu^2_{X_0,X_1}\in\mathcal{L}({\ !{X_0}}\otimes{\ !{X_1}},\ !{({X_0}\otimes{X_1})})$ satisfying suitable coherence commutations. Explicitly these morphisms are given by:

Lemma 3. The following diagram commutes:

Proof. This results from the definition of $\mu^2$ and from the following commutation

which results from the observation that $\ !{0}\in\mathcal{L}(\ !\ !{X},\ !{0})$ can be written $\ !{0}=\ !({0\,\operatorname{\mathsf{der}}_{X}})$ .

For any $X\in\mathcal{L}$ , it is possible to define a contraction morphism $\operatorname{\mathsf{contr}}_{X}\in\mathcal{L}(\ !{X},{\ !{X}}\otimes{\ !{X}})$ and a weakening morphism $\operatorname{\mathsf{weak}}_{X}\in\mathcal{L}(\ !{X},1)$ turning $\ !{X}$ into a commutative comonoid. These morphisms are defined as follows:

Lemma 4. The two following diagrams commute in any resource category $\mathcal{L}$ .

Proof. For the first diagram, we have

and

For the second diagram, we compute

\begin{align*} f_1 &=\ !{\langle{{\mathsf{pr}_{0}}\otimes{Y},{\mathsf{pr}_{1}}\otimes{Y}}\rangle} \,\mu^2_{{X_0}\mathrel{\&}{X_1},Y}\\ &=\ !{\langle{{\mathsf{pr}_{0}}\otimes{Y},{\mathsf{pr}_{1}}\otimes{Y}}\rangle} \,\ !{({\operatorname{\mathsf{der}}_{{X_0}\mathrel{\&}{X_1}}}\otimes{\operatorname{\mathsf{der}}_{Y}})} \,\ !{({\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y}})^{-1}} \,\operatorname{\mathsf{dig}}_{X_0\mathrel{\&} X_1\mathrel{\&} Y} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y}\\ &\hspace{10em}\text{by definition of }\mu^2\\ &=\ !({{({\operatorname{\mathsf{der}}_{X_0}}\otimes{\operatorname{\mathsf{der}}_{Y}})}\mathrel{\&}{({\operatorname{\mathsf{der}}_{X_1}}\otimes{\operatorname{\mathsf{der}}_{Y}})}}) \,\ !{\langle{{\ !{\mathsf{pr}_{0}}}\otimes{\ !{Y}}, {\ !{\mathsf{pr}_{1}}}\otimes{\ !{Y}}}\rangle}\\ &\hspace{10em}\,\ !{({\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y}})^{-1}} \,\operatorname{\mathsf{dig}}_{X_0\mathrel{\&} X_1\mathrel{\&} Y} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y} \text{ by naturality of }\operatorname{\mathsf{der}}_{}\\ &=\ !({{({\operatorname{\mathsf{der}}_{X_0}}\otimes{\operatorname{\mathsf{der}}_{Y}})}\mathrel{\&}{({\operatorname{\mathsf{der}}_{X_1}}\otimes{\operatorname{\mathsf{der}}_{Y}})}}) \, f_2 \end{align*}

where

\begin{align*} f_2 &=\ !{\langle{{\ !{\mathsf{pr}_{0}}}\otimes{\ !{Y}}, {\ !{\mathsf{pr}_{1}}}\otimes{\ !{Y}}}\rangle} \,\ !{({\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y}})^{-1}} \,\operatorname{\mathsf{dig}}_{X_0\mathrel{\&} X_1\mathrel{\&} Y} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y}\\ &=\ !({{({\mathsf m^2_{X_0,Y}})^{-1}}\mathrel{\&}{({\mathsf m^2_{X_1,Y}})^{-1}}}) \,\ !{\langle{\ !{\mathsf{pr}_{0}},\ !{\mathsf{pr}_{1}}}\rangle} \,\ !\ !{q} \,\operatorname{\mathsf{dig}}_{X_0\mathrel{\&} X_1\mathrel{\&} Y} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y} \end{align*}

by naturality of $\mathsf m^2$ . In that expression, $\mathsf{pr}_{i}\in\mathcal{L}(X_0\mathrel{\&} Y\mathrel{\&} X_1\mathrel{\&} Y,{X_i}\mathrel{\&}{Y})$ and $q=\langle{\mathsf{pr}_{0},\mathsf{pr}_{2},\mathsf{pr}_{1},\mathsf{pr}_{2}}\rangle \in\mathcal{L}(X_0\mathrel{\&} X_1\mathrel{\&} Y,X_0\mathrel{\&} Y\mathrel{\&} X_1\mathrel{\&} Y)$ . We have used the commutation of the following diagram

which is easily proved by post-composing the two equated morphisms with $\mathsf{pr}_{i}\in\mathcal{L}({({\ !{X_0}}\otimes{\ !{Y}})}\mathrel{\&}{({\ !{X_1}}\otimes{\ !{Y}})},({\ !{X_i}}\otimes{\ !{Y}}))$ for $i=0,1$ .

Observe that

\begin{align*} &\ !({{({\operatorname{\mathsf{der}}_{X_0}}\otimes{\operatorname{\mathsf{der}}_{Y}})}\mathrel{\&}{({\operatorname{\mathsf{der}}_{X_1}}\otimes{\operatorname{\mathsf{der}}_{Y}})}})\\ &\hspace{8em}=\mathsf m^2_{{X_0}\otimes{Y},{X_1}\otimes{Y}} \,{({\ !({{\operatorname{\mathsf{der}}_{X_0}}\otimes{\operatorname{\mathsf{der}}_{Y}}})}\otimes{}) {\ !({{\operatorname{\mathsf{der}}_{X_1}}\otimes{\operatorname{\mathsf{der}}_{Y}}})}} \, ({\mathsf m^2_{{\ !{X_0}}\otimes{\ !{Y}},{\ !{X_1}}\otimes{\ !{Y}}}})^{-1} \end{align*}

by naturality of $\mathsf m^2$ . For the same reason, the following diagram commutes:

and hence

\begin{align*} f_1 &=\mathsf m^2_{{X_0}\otimes{Y},{X_1}\otimes{Y}} \,{({\ !({{\operatorname{\mathsf{der}}_{X_0}}\otimes{\operatorname{\mathsf{der}}_{Y}}})}\otimes{}) {\ !({{\operatorname{\mathsf{der}}_{X_1}}\otimes{\operatorname{\mathsf{der}}_{Y}}})}} \, ({\mathsf m^2_{{\ !{X_0}}\otimes{\ !{Y}},{\ !{X_1}}\otimes{\ !{Y}}}})^{-1}\\ &\hspace{8em} \,\ !({{({\mathsf m^2_{X_0,Y}})^{-1}}\mathrel{\&}{({\mathsf m^2_{X_1,Y}})^{-1}}}) \,\ !{\langle{\ !{\mathsf{pr}_{0}},\ !{\mathsf{pr}_{1}}}\rangle} \,\ !\ !{q} \,\operatorname{\mathsf{dig}}_{X_0\mathrel{\&} X_1\mathrel{\&} Y} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y} \\ &=\mathsf m^2_{{X_0}\otimes{Y},{X_1}\otimes{Y}} \,{({\ !({{\operatorname{\mathsf{der}}_{X_0}}\otimes{\operatorname{\mathsf{der}}_{Y}}})}\otimes{}) {\ !({{\operatorname{\mathsf{der}}_{X_1}}\otimes{\operatorname{\mathsf{der}}_{Y}}})}} \,({\ !{({\mathsf m^2_{X_0,Y}})^{-1}}}\otimes{\ !{({\mathsf m^2_{X_1,Y}})^{-1}}})\\ &\hspace{8em} \,({\mathsf m^2_{\ !{\left({X_0}\mathrel{\&}{Y}\right)},\ !{\left({X_1}\mathrel{\&}{Y}\right)}}})^{-1} \,\ !{\langle{\ !{\mathsf{pr}_{0}},\ !{\mathsf{pr}_{1}}}\rangle} \, f_3 \end{align*}

where, by naturality of $\operatorname{\mathsf{dig}}_{}$ ,

\begin{align*} f_3 &=\ !\ !{q} \,\operatorname{\mathsf{dig}}_{X_0\mathrel{\&} X_1\mathrel{\&} Y} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y}\\ &=\operatorname{\mathsf{dig}}_{X_0\mathrel{\&} Y\mathrel{\&} X_1\mathrel{\&} Y} \,\ !{q} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y} \in\mathcal{L}({\ !{\left({X_0}\mathrel{\&}{X_1}\right)}}\otimes{\ !{Y}}, \ !\ !{(X_0\mathrel{\&} Y\mathrel{\&} X_1\mathrel{\&} Y)}) \end{align*}

and hence, by the diagram (1)

\begin{align*} \ !{\langle{\ !{\mathsf{pr}_{0}},\ !{\mathsf{pr}_{1}}}\rangle}\, f_3 &=\ !{\langle{\ !{\mathsf{pr}_{0}},\ !{\mathsf{pr}_{1}}}\rangle} \,\operatorname{\mathsf{dig}}_{X_0\mathrel{\&} Y\mathrel{\&} X_1\mathrel{\&} Y} \,\ !{q} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y}\\ &={\mathsf m^2_{\ !({{X_0}\mathrel{\&}{Y}}),\ !({{X_1}\mathrel{\&}{Y}})}} \,({\operatorname{\mathsf{dig}}_{{X_0}\mathrel{\&}{Y}}}\otimes{\operatorname{\mathsf{dig}}_{{X_1}\mathrel{\&}{Y}}}) \,({\mathsf m^2_{{X_0}\mathrel{\&}{Y},{X_1}\mathrel{\&}{Y}}})^{-1} \,\ !{q} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y}\,. \end{align*}

It follows that

\begin{align*} f_1 &=\mathsf m^2_{{X_0}\otimes{Y},{X_1}\otimes{Y}} \,{({\ !({{\operatorname{\mathsf{der}}_{X_0}}\otimes{\operatorname{\mathsf{der}}_{Y}}})}\otimes{}) {\ !({{\operatorname{\mathsf{der}}_{X_1}}\otimes{\operatorname{\mathsf{der}}_{Y}}})}} \,({\ !{({\mathsf m^2_{X_0,Y}})^{-1}}}\otimes{}) {\ !{({\mathsf m^2_{X_1,Y}})^{-1}}}\\ &\hspace{4em} ({\operatorname{\mathsf{dig}}_{{X_0}\mathrel{\&}{Y}}}\otimes{\operatorname{\mathsf{dig}}_{{X_1}\mathrel{\&}{Y}}}) \,({\mathsf m^2_{{X_0}\mathrel{\&}{Y},{X_1}\mathrel{\&}{Y}}})^{-1} \,\ !{q} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y}\\ &=\mathsf m^2_{{X_0}\otimes{Y},{X_1}\otimes{Y}} \,{({\ !({{\operatorname{\mathsf{der}}_{X_0}}\otimes{\operatorname{\mathsf{der}}_{Y}}})}\otimes{}) {\ !({{\operatorname{\mathsf{der}}_{X_1}}\otimes{\operatorname{\mathsf{der}}_{Y}}})}} \,({\ !{({\mathsf m^2_{X_0,Y}})^{-1}}}\otimes{}) {\ !{({\mathsf m^2_{X_1,Y}})^{-1}}}\\ &\hspace{4em} ({\operatorname{\mathsf{dig}}_{{X_0}\mathrel{\&}{Y}}}\otimes{\operatorname{\mathsf{dig}}_{{X_1}\mathrel{\&}{Y}}}) \,({\mathsf m^2_{X_0,Y}}\otimes{\mathsf m^2_{X_1,Y}})\\ &\hspace{4em} ({({\mathsf m^2_{X_0,Y}})^{-1}}\otimes{({\mathsf m^2_{X_1,Y}})^{-1}}) \,({\mathsf m^2_{{X_0}\mathrel{\&}{Y},{X_1}\mathrel{\&}{Y}}})^{-1} \,\ !{q} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y}\\ &=\mathsf m^2_{{X_0}\otimes{Y},{X_1}\otimes{Y}} \,({\mu^2_{X_0,Y}}\otimes{\mu^2_{X_1,Y}}) \,({({\mathsf m^2_{X_0,Y}})^{-1}}\otimes{({\mathsf m^2_{X_1,Y}})^{-1}}) \,({\mathsf m^2_{{X_0}\mathrel{\&}{Y},{X_1}\mathrel{\&}{Y}}})^{-1} \,\ !{q} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y} \end{align*}

hence,

\begin{align*} f_1\,({\mathsf m^2_{X_0,X_1}}\otimes{\ !{Y}}) &=\mathsf m^2_{{X_0}\otimes{Y},{X_1}\otimes{Y}} \,({\mu^2_{X_0,Y}}\otimes{\mu^2_{X_1,Y}}) \,({({\mathsf m^2_{X_0,Y}})^{-1}}\otimes{({\mathsf m^2_{X_1,Y}})^{-1}}) \,({\mathsf m^2_{{X_0}\mathrel{\&}{Y},{X_1}\mathrel{\&}{Y}}})^{-1}\\ &\hspace{4em} \,\ !{q} \,\mathsf m^2_{{X_0}\mathrel{\&}{X_1},Y} \,({\mathsf m^2_{X_0,X_1}}\otimes{\ !{Y}})\\ &=\mathsf m^2_{{X_0}\otimes{Y},{X_1}\otimes{Y}} \,({\mu^2_{X_0,Y}}\otimes{\mu^2_{X_1,Y}}) \,({\mathsf m^4_{X_0,Y,X_1,Y}})^{-1} \,\ !{q} \,\mathsf m^3_{X_0,X_1,Y}\\ &=\mathsf m^2_{{X_0}\otimes{Y},{X_1}\otimes{Y}} \,({\mu^2_{X_0,Y}}\otimes{\mu^2_{X_1,Y}}) \,\gamma_{2,3} \,(\ !{X_0}\otimes\ !{X_1}\otimes\operatorname{\mathsf{contr}}_{Y}\ !) \end{align*}

by the monoidality properties of the Seely isomorphisms, where we have used $\mathsf m^k$ for their k-ary version.

2.3.3 Coalgebras of the resource comonad

A $\ !$ -coalgebra is a pair $P=(\underline{P},h_{P})$ where $\underline{P}$ is an object of $\mathcal{L}$ and $h_{P}\in\mathcal{L}(\underline{P},\ !{\underline{P}})$ satisfies

Given coalgebras P and Q, a coalgebra morphism from P to Q is an $f\in\mathcal{L}(\underline{P},\underline{Q})$ such that the following square commutes

The category so defined is the Eilenberg–Moore category ${\mathcal {L}}^!$ associated with the comonad $\ !{\_}$ . We will use the following standard result for which we refer to Melliès (Reference Melliès2009).

Theorem 2. The Eilenberg–Moore category ${\mathcal{L}}^!$ of a resource category $\mathcal{L}$ is cartesian with final object $(1,\mu^0)$ simply denoted as 1 and cartesian product of $P_0,P_1$ the coalgebra $({\underline{P_0}}\otimes{\underline{P_1}},\mu^2_{\underline{P_0},\underline{P_1}} \,({h_{P_0}}\otimes{h_{P_1}}))$ denoted as ${P_0}\otimes{P_1}$ with projection $\mathsf{pr}_{0}^\otimes\in{\mathcal{L}}^!({P_0}\otimes{P_1},P_0)$ defined as the following composition of morphisms

and similarly for $\mathsf{pr}_{1}^\otimes\in{\mathcal{L}}^!({P_0}\otimes{P_1},P_1)$ . And given $f_i\in{\mathcal{L}}^!(Q,P_i)$ for $i=0,1$ , the unique morphism $\langle{f_0,f_1}\rangle^\otimes\in{\mathcal{L}}^!(Q,{P_0}\otimes{P_1})$ such that $\mathsf{pr}_{i}^\otimes\,\langle{f_0,f_1}\rangle^\otimes=f_i$ is defined as the following composition of morphisms

Last, the unique morphism $P\to1$ in ${\mathcal{L}}^!$ is

An immediate consequence of this theorem is the following observation.

Proposition 5. Let P be an object of ${\mathcal{L}}^!$ , $u\in{\mathcal{L}}^!(P,1)$ and $d\in{\mathcal{L}}^!(P,{P}\otimes{P})$ be such that

commute. Then, $u=\operatorname{\mathsf{weak}}_{P}\,h_{P}$ and $d=\langle{\underline{P},\underline{P}}\rangle^\otimes =({\operatorname{\mathsf{der}}_{\underline{P}}}\otimes{\operatorname{\mathsf{der}}_{\underline{P}}}) \,\operatorname{\mathsf{contr}}_{\underline{P}} \,h_{P}$ and the following diagram commutes in $\mathcal{L}$ .

Proof. The first equation results from the universal property of the terminal object. The second one results from the universal property of the cartesian product and from the commutation of

since $ \mathsf{pr}_{0}^\otimes\, d =\lambda_{\underline{P}} \,({\operatorname{\mathsf{weak}}_{\underline{P}}}\otimes{\underline{P}}) \,({h_{P}}\otimes{\underline{P}}) \, d =\lambda_{\underline{P}} \,({u}\otimes{\underline{P}}) \, d=\operatorname{\mathsf{Id}}_{\underline{P}} $ and similarly for $\mathsf{pr}_{1}^\otimes$ .

For the last commutation, we have

2.3.4 Lafont categories and the free exponential

In many interesting models of ${\textsf{LL}}$ , the exponential resource modality is completely determined by the tensor product; in that case, one says that the exponential is free. We provide the precise definition of such categories and give some of their properties that we will use in the paper.

Let $\mathcal{L}$ be an SMC. Remember from Melliès (Reference Melliès2009) that $\mathcal{L}$ is a Lafont category if the forgetful functor $U:{\mathcal {L}}^{\otimes}\to\mathcal{L}$ has a right adjoint $E:\mathcal{L}\to{\mathcal {L}}^{\otimes}$ . We use $(\ !{X},\operatorname{\mathsf{weak}}_{X},\operatorname{\mathsf{contr}}_{X})$ for the commutative comonoid EX. In that case, we use $(\ !{\_},\operatorname{\mathsf{der}}_{},\operatorname{\mathsf{dig}}_{})$ for the associated comonad UE called the free exponential of the SMC $\mathcal{L}$ .

More explicitly, this means that for any object X of $\mathcal{L}$ , for any commutative comonoid $C=(\underline{C},\mathsf w_{C}:\underline{C}\to1, \mathsf c_{C}:\underline{C}\to{\underline{C}}\otimes{\underline{C}})$ and any $f\in\mathcal{L}(\underline{C},X)$ , there is exactly one morphism $f^\otimes\in\mathcal{L}/X((\underline{C},f),(\ !{X},\operatorname{\mathsf{der}}_{X}\ !))$ which is a comonoid morphism. In other words, there is exactly one morphism $f^\otimes\in\mathcal{L}(\underline{C},\ !{X})$ such that the three following diagrams commute.

Lemma 6. Let $\mathcal{L}$ be a Lafont category. For any commutative comonoid C, there is exactly one morphism $\delta_C\in\mathcal{L}(\underline{C},\ !{\underline{C}})$ such that the following diagrams commute.

Moreover $(\underline{C},\delta_C)$ is a $\ !$ -coalgebra.

Proof. The first part of the statement is just a special case of the universal property with $X=\underline{C}$ and $f=\operatorname{\mathsf{Id}}_X$ . For the second part, we only have to prove

Setting $f_1=\ !{\delta_C}\,\delta_C$ and $f_2=\operatorname{\mathsf{dig}}_{\underline{C}}\,\delta_C$ , observe first that $f_1,f_2\in{\mathcal {L}}^{\otimes}(C,(\ !\ !{\underline{C}},\mathsf c_{\ !{\underline{C}}},\mathsf w_{\ !{\underline{C}}}))$ because both are defined by composing morphisms in that category. The equation $f_1=f_2$ follows by universality, observing that

for $i=1,2$ , which readily results from the naturality of $\operatorname{\mathsf{der}}_{}$ and from the definition of a comonad.

Here are two important special cases of the above. First, there is exactly one morphism $\mu^0\in\mathcal{L}(1,\ !{1})$ such that

Next, there is exactly one morphism $\mu^2_{X,Y}\in\mathcal{L}({\ !{X}}\otimes{\ !{Y}},\ !({{X}\otimes{Y}}))$ such that

These two morphisms turn $\ !{\_}$ into a lax monoidal comonad on the SMC $\mathcal{L}$ .

The correspondence $C\mapsto(\underline{C},\delta_C)$ can be turned into a functor $\mathsf A:{\mathcal {L}}^{\otimes}\to{\mathcal{L}}^!$ acting as the identity on morphisms. Let indeed $f\in{\mathcal {L}}^{\otimes}(C,D)$ , it suffices to prove that $\delta_D\, f=\ !{f}\,\delta_C\in\mathcal{L}(\underline{C},\ !{\underline{D}})$ . Let $f_0=\delta_D\, f$ and $f_1=\ !{f}\,\delta_C$ . By the universal property, it suffices to prove that the three following diagrams commute for $i=0,1$ :

These commutations follow from the commutations satisfied by $\delta_C$ and $\delta_D$ and from the fact that $f\in{\mathcal {L}}^{\otimes}(C,D)$ . As an example of these computations, we have

\begin{align*} \operatorname{\mathsf{contr}}_{\underline{D}}\, f_0 &=\operatorname{\mathsf{contr}}_{\underline{D}}\,\delta_D\, f\\[2pt] &=({\delta_D}\otimes{\delta_D})\,\mathsf c_{D}\, f\\[2pt] &=({\delta_D}\otimes{\delta_D})\,({f}\otimes{f})\,\mathsf c_{C}\\[2pt] &=({f_0}\otimes{f_0})\,\mathsf c_{C}\end{align*}

and

\begin{align*} \operatorname{\mathsf{contr}}_{\underline{D}}\, f_1 &=\operatorname{\mathsf{contr}}_{\underline{D}}\,\ !{f}\,\delta_C\\[2pt] &=({\ !{f}}\otimes{\ !{f}})\,\operatorname{\mathsf{contr}}_{\underline{C}}\, \delta_C\\[2pt] &=({\ !{f}}\otimes{\ !{f}}) \,({\delta_C}\otimes{\delta_C})\,\mathsf c_{C}\\[2pt] &=({f_1}\otimes{f_1})\,\mathsf c_{C} \,.\end{align*}

Conversely given a $\ !$ -coalgebra $P=(\underline{P},h_{P})$ , one can define a commutative comonoid structure on $\underline{P}$ by the following two morphisms:

that we respectively denote as $\mathsf w_{P}$ and $\mathsf c_{P}$ . This correspondence $P\mapsto\mathsf M(P)=(\underline{P},\mathsf w_{P},\mathsf c_{P})$ can be turned into a functor $\mathsf M:{\mathcal{L}}^!\to{\mathcal {L}}^{\otimes}$ acting as the identity on morphisms.

Theorem 3. For any Lafont SMC $\mathcal{L}$ , the functors $\mathsf A$ and $\mathsf M$ define an isomorphism of categories between ${\mathcal {L}}^{\otimes}$ and ${\mathcal{L}}^!$ .

Proof. Let $C\in{\mathcal {L}}^{\otimes}$ and let $P=\mathsf A(C)$ so that $\underline{P}=\underline{C}$ and $h_{P}=\delta_C$ . Let $D=\mathsf M(P)$ so that $\underline{D}=\underline{C}$ ,

\begin{align*} \mathsf w_{D}&=\operatorname{\mathsf{weak}}_{\underline{P}}\,h_{P} =\operatorname{\mathsf{weak}}_{\underline{C}}\,\delta_C=\mathsf w_{C}\\[2pt] \mathsf c_{D}&=({\operatorname{\mathsf{der}}_{\underline{P}}}\otimes{\operatorname{\mathsf{der}}_{\underline{P}}}) \,\operatorname{\mathsf{contr}}_{\underline{P}} \,h_{P}\\[2pt] &=({\operatorname{\mathsf{der}}_{\underline{C}}}\otimes{\operatorname{\mathsf{der}}_{\underline{C}}}) \,\operatorname{\mathsf{contr}}_{\underline{C}} \,\delta_C\\[2pt] &=({\operatorname{\mathsf{der}}_{\underline{C}}}\otimes{\operatorname{\mathsf{der}}_{\underline{C}}}) ({\delta_C}\otimes{\delta_C}) \,\operatorname{\mathsf{contr}}_{\underline{C}}\\[2pt] &=\operatorname{\mathsf{contr}}_{\underline{C}}. \end{align*}

Conversely let $P\in{\mathcal{L}}^!$ . Let $C=\mathsf M(P)$ so that $\underline{C}=\underline{P}$ , $\mathsf w_{C}=\operatorname{\mathsf{weak}}_{\underline{P}}\,h_{P}$ and $\mathsf c_{C}=({\operatorname{\mathsf{der}}_{\underline{P}}}\otimes{\operatorname{\mathsf{der}}_{\underline{P}}})$ $ \,\operatorname{\mathsf{contr}}_{\underline{P}} \,h_{P}$ . Let $Q=\mathsf A(C)=(\underline{P},\delta_C)$ . To prove that $\delta_C=h_{P}$ , it suffices to show that the following diagrams commute

which results from the definition of C and from the fact that P is a coalgebra. Let us check for instance the last one:

\begin{align*} ({h_{P}}\otimes{h_{P}}) \,\mathsf c_{C} &=({h_{P}}\otimes{h_{P}}) \,({\operatorname{\mathsf{der}}_{\underline{P}}}\otimes{\operatorname{\mathsf{der}}_{\underline{P}}}) \,\operatorname{\mathsf{contr}}_{\underline{P}} \,h_{P}\\ &=({\operatorname{\mathsf{der}}_{\ !{\underline{P}}}}\otimes{\operatorname{\mathsf{der}}_{\ !{\underline{P}}}}) \,({\ !{h_{P}}}\otimes{\ !{h_{P}}}) \,\operatorname{\mathsf{contr}}_{\underline{P}} \,h_{P}\\ &=({\operatorname{\mathsf{der}}_{\ !{\underline{P}}}}\otimes{\operatorname{\mathsf{der}}_{\ !{\underline{P}}}}) \,\operatorname{\mathsf{contr}}_{\ !{\underline{P}}} \,\ !{h_{P}} \,h_{P}\\ &=({\operatorname{\mathsf{der}}_{\ !{\underline{P}}}}\otimes{\operatorname{\mathsf{der}}_{\ !{\underline{P}}}}) \,\operatorname{\mathsf{contr}}_{\ !{\underline{P}}} \,\operatorname{\mathsf{dig}}_{\underline{P}} \,h_{P}\\ &=({\operatorname{\mathsf{der}}_{\ !{\underline{P}}}}\otimes{\operatorname{\mathsf{der}}_{\ !{\underline{P}}}}) ({\operatorname{\mathsf{dig}}_{\underline{P}}}\otimes{\operatorname{\mathsf{dig}}_{\underline{P}}}) \,\operatorname{\mathsf{contr}}_{\underline{P}} \,h_{P}\\ &=\,\operatorname{\mathsf{contr}}_{\underline{C}} \,h_{P} \end{align*}

where we have used in particular the fact that for any $X\in\mathcal{L}$ , one has $\operatorname{\mathsf{dig}}_{X}\in{\mathcal {L}}^{\otimes}(E(X),E(\ !{X}))$ by the fact that the comonad $\ !{\_}$ is induced by the adjunction $U\mathrel{\dashv} E$ .

This shows that $\mathsf M$ and $\mathsf A$ define a bijective correspondence on objects and since both functors act as the identity on morphisms, our contention is proven.

In that way, we retrieve the fact that ${\mathcal{L}}^!$ is cartesian since ${\mathcal {L}}^{\otimes}$ is always cartesian by Theorem 1 (even if $\mathcal{L}$ is not Lafont). Remember that in the general (not necessarily Lafont) case the fact that ${\mathcal{L}}^!$ is cartesian could be proven under the additional assumption that $\mathcal{L}$ is a resource category. Remember also that a cartesian Lafont SMC is automatically a resource category; see Melliès (Reference Melliès2009).

Lemma 7. Let $C_0,C_1\in{\mathcal {L}}^{\otimes}$ . Remember that we use ${C_0}\otimes{C_1}$ for the cartesian product of $C_0$ and $C_1$ in ${\mathcal {L}}^{\otimes}$ (see Theorem 1). Then, we have

\begin{align*} \delta_1=\mu^0 \delta_{{C_0}\otimes{C_1}} =\mu^2_{{\underline{C_0}},{\underline{C_1}}} \,({\delta_{C_0}}\otimes{\delta_{C_1}}) \in\mathcal{L}({\underline{C_0}}\otimes{\underline{C_1}}, \ !({{\underline{C_0}}\otimes{\underline{C_1}}}))\,. \end{align*}

Proof. One just checks that the right-hand morphisms satisfy the three diagrams of Lemma 6.

Theorem 4. Let $\mathcal{L}$ be a Lafont category and let $C\in{\mathcal {L}}^{\otimes}$ . Then the following diagrams commute

Proof. We deal with the second diagram, the argument for the first one being completely similar. By Lemma 2 we have $\mathsf c_{C}\in{\mathcal {L}}^{\otimes}(C,{C}\otimes{C})$ and hence (since $\mathsf A$ is the identity on morphisms) we have $\mathsf c_{C}\in{\mathcal{L}}^!(\mathsf A(C),\mathsf A({C}\otimes{C}))$ which is exactly the diagram under consideration by Lemma 7.

2.4 The category of sets and relations

This category is a well-known categorical model of classical ${\textsf{LL}}$ that we briefly recall here. It is perhaps the simplest example of a Lafont resource category.

The category $\operatorname{\mathbf{Rel}}$ has sets as objects and $\operatorname{\mathbf{Rel}}(X,Y)={\mathcal P}\left({X\times Y}\right)$ , that is, a morphism from the set X to the set Y is a relation from X to Y. The identity morphism $\operatorname{\mathsf{Id}}_X$ is the diagonal relation on X and composition is the usual composition of relations. An iso in $\operatorname{\mathbf{Rel}}$ is a relation which is (the graph of) a bijection.

The category $\operatorname{\mathbf{Rel}}$ is monoidal with monoidal product ${X_0}\otimes{X_1}=X_0\times X_1$ and monoidal unit $1=\{\ast\}$ . Given $s_i\in\operatorname{\mathbf{Rel}}(X_i,Y_i)$ for $i=0,1$ , the relation ${s_0}\otimes{s_1}\in\operatorname{\mathbf{Rel}}({X_0}\otimes{X_1},{Y_0}\otimes{Y_1})$ is defined as:

\begin{align*} {s_0}\otimes{s_1} =\{((a_0,a_1),(b_0,b_1))\mid (a_i,b_i)\in s_i\text{ for }i=0,1\}\end{align*}

which turns $\otimes$ into a functor and $\operatorname{\mathbf{Rel}}$ into a SMC (with obvious symmetric monoidality isos). It is also closed with ${X}\multimap{Y}=X\times Y$ as internal hom object and evaluation morphism:

\begin{align*} \operatorname{\mathsf{ev}}=\{(((a,b),a),b)\mid a\in X\text{ and }b\in Y\} \in\operatorname{\mathbf{Rel}}({\left({X}\multimap{Y}\right)}\otimes{X},Y)\,.\end{align*}

With dualizing object ${\mathord{\perp}}=1$ , this category is $\ast$ -autonomous.

The category $\operatorname{\mathbf{Rel}}$ is not complete, but it is cartesian. Given a family $(X_i)_{i\in I}$ of sets, their product is

\begin{align*} (\mathop{\&}_{i\in I}X_i,(\mathsf{pr}_{i})_{i\in I})\end{align*}

where $\mathop{\&}_{i\in I}X_i=\bigcup_{i\in I}\{i\}\times X_i$ and the projections are $\mathsf{pr}_{i}=\{((i,a),a)\mid i\in I\text{ and }a\in X_i\}\in\operatorname{\mathbf{Rel}}(\ !\mathop{\&}_{j\in I}X_j,X_i)$ . Given a family of morphisms $(s_i\in\operatorname{\mathbf{Rel}}(Y,X_i))_{i\in I}$ , the unique morphism $\langle{s_i}\rangle_{i\in I}\in\operatorname{\mathbf{Rel}}(Y,\mathop{\&}_{i\in I}X_i)$ such that $\mathsf{pr}_{i}\,\langle{s_j}\rangle_{j\in I}=s_i$ is

\begin{align*} \langle{s_i}\rangle_{i\in I}= \{(b,(i,a))\mid i\in I\text{ and }(b,a)\in s_i\}\,.\end{align*}

The terminal object is $\top=\emptyset$ .

As an SMC, $\operatorname{\mathbf{Rel}}$ is a Lafont category. The associated resource comonad $(\ !,\operatorname{\mathsf{der}}_{},\operatorname{\mathsf{dig}}_{}\ !)$ on $\operatorname{\mathbf{Rel}}$ is given by $\ !{X}=\mathcal M_{{\mathrm{fin}}}({X})$ (see Section 2.1) with functorial action given by:

\begin{align*} \ !{s}=\{([{a_{1},\dots,a_{n}}],[{b_{1},\dots,b_{n}}]) \mid n\in{\mathbb{N}}\text{ and }\forall i\ (a_i,b_i)\in s\} \in\operatorname{\mathbf{Rel}}(\ !{X},\ !{Y})\end{align*}

for $s\in\operatorname{\mathbf{Rel}}(X,Y)$ . The counit is $\operatorname{\mathsf{der}}_{X}=\{([{a}],a)\mid a\in X\}\in\operatorname{\mathbf{Rel}}(\ !{X},X)$ and the comultiplication is $\operatorname{\mathsf{dig}}_{X} =\{(m_1+\cdots+m_k,[{m_{1},\dots,m_{k}}])\mid k\in{\mathbb{N}} \text{ and}m_{1},\dots,m_{k}\in\ !{X}\}$ . Its strong symmetric monoidality from the SMC $(\operatorname{\mathbf{Rel}},\mathrel{\&},\top)$ to the SMC $(\operatorname{\mathbf{Rel}},\otimes,1)$ is given by the isos $\mathsf m^0\in\operatorname{\mathbf{Rel}}(1,\ !{\top})$ and $\mathsf m^2_{X_0,X_1}\in\operatorname{\mathbf{Rel}}({\ !{X_0}}\otimes{\ !{X_1}},\ !{\left({X_0}\mathrel{\&}{X_1}\right)})$ given by $\mathsf m^0=\{(\ast,[]\,)\}$ and

3.1 A monad structure on $\mathsf S$

We already noticed that there is a natural transformation $\iota_{0}\in\mathcal{L}(X,\mathsf S X)$ . As also mentioned the morphisms $\pi_{i}\,\pi_{j}\in\mathcal{L}(\mathsf S^2 X,X)$ (for all $i,j\in\{0,1\}$ ) are summable so that the morphisms $\pi_{0}\,\pi_{0},\pi_{1}\,\pi_{0}+\pi_{0}\,\pi_{1}\in\mathcal{L}(\mathsf S^2X,\mathsf S X)$ are summable by Theorem 5, let $\tau=\langle{\pi_{0}\,\pi_{0},\pi_{1} \,\pi_{0}+\pi_{0} \,\pi_{1}}\rangle_{\mathsf S}\in\mathcal{L}(\mathsf S^2X,\mathsf S X)$ be the witness of this summability.

Similarly, using the naturality of $\pi_{1}$ , we have $\pi_{1}\,(\mathsf S f)\,\tau_{X} =f\,\pi_{1}\tau_X =f\,(\pi_{0}\,\pi_{1}+\pi_{1}\,\pi_{0}) =f\,\pi_{0}\,\pi_{1}+f\,\pi_{1}\,\pi_{0}$ and $\pi_{1}\,\tau_Y\,(\mathsf S^2f) =(\pi_{0}\,\pi_{1}+\pi_{1}\,\pi_{0})\,(\mathsf S^2f) =\pi_{0}\,\pi_{1}\,(\mathsf S^2f)+\pi_{1}\,\pi_{0}\,(\mathsf S^2f) =f\,\pi_{0}\,\pi_{1}+$ $f\,\pi_{1}\,\pi_{0}$ . The other naturalities are proved in the same way.

One proves $\tau_X\,\tau_{\mathsf S X}=\tau_X\,\mathsf S\tau_X$ by showing in the same manner that $\pi_{0}\,\tau_X\,\tau_{\mathsf S X}=\pi_{0}\,\pi_{0}\,\pi_{0} =\pi_{0}\,\tau_X\,\mathsf S\tau_X $ and that $\pi_{1}\,\tau_X\,\tau_{\mathsf S X}=\pi_{0}\,\pi_{0}\,\pi_{1} +\pi_{0}\,\pi_{1}\,\pi_{0} +\pi_{1}\,\pi_{0}\,\pi_{0} =\pi_{1}\,\tau_X\,\mathsf S\tau_X$ . The commutations involving $\tau$ and $\iota_{0}$ are proved in the same way. The last equation results from $\pi_{i}\,\pi_{j}\,\mathsf c=\pi_{j}\,\pi_{i}$

Just as in tangent categories, this monad structure will be crucial for expressing that the differential is a linear morphism.

3.2 Summable symmetric monoidal category

We assume now that $\mathcal{L}$ is a SMC, with monoidal product $\otimes$ , unit 1 and isomorphisms $\rho_X\in\mathcal{L}({X}\otimes{1},X)$ , $\lambda_X\in\mathcal{L}({1}\otimes{ }X,X)$ , $\alpha_{X_0,X_1,X_2}\in\mathcal{L}({({X_0}\otimes{X_1})}\otimes{X_2},{X_0}\otimes{({X_1}\otimes{X_2})})$ and $\gamma_{X_0,X_1}\in\mathcal{L}({X_0}\otimes{X_1},{X_1}\otimes{X_0})$ . Most often these isos will be kept implicit to simplify the presentation.

Assume that $\mathcal{L}$ is also equipped with a summability structure. We assume now that the following property holds, which expresses that the tensor distributes over the (partially defined) sum.

(S $\otimes$ -dist) If $(f_{00},f_{01})$ is a summable pair of morphisms in $\mathcal{L}(X_0,Y_0)$ and $f_1\in\mathcal{L}(X_1,Y_1)$ , then $({f_{00}}\otimes{f_1},{f_{01}}\otimes{f_1})$ is a summable pair of morphisms in $\mathcal{L}({X_0}\otimes{X_1},{Y_0}\otimes{Y_1})$ , and moreover

\begin{align*} {f_{00}}\otimes{f_1}+{f_{01}}\otimes{f_1}={(f_{00}+f_{01})}\otimes{f_1}\end{align*}

As a consequence, using the symmetry of $\otimes$ , if $(f_{00},f_{01})$ is summable in $\mathcal{L}(X_0,Y_0)$ and $(f_{10},f_{11})$ is summable in $\mathcal{L}(X_1,Y_1)$ , the family $({f_{00}}\otimes{f_{10}},{f_{00}}\otimes{f_{11}},{f_{01}}\otimes{f_{10}},{f_{01}}\otimes{f_{11}})$ is summable in $\mathcal{L}({X_0}\otimes{X_1},{Y_0}\otimes{Y_1})$ and we have

\begin{align*} {(f_{00}+f_{01})}\otimes{(f_{10}+f_{11})} ={f_{00}}\otimes{f_{10}}+{f_{00}}\otimes{f_{11}}+{f_{01}}\otimes{f_{10}}+{f_{01}}\otimes{f_{11}}\,.\end{align*}

We can define a natural transformation $\phi^1_{X_0,X_1}\in\mathcal{L}({X_0}\otimes{\mathsf S X_1},\mathsf S({X_0}\otimes{X_1}))$ by setting $\phi^1_{X_0,X_1}=\langle{{X_0}\otimes{\pi_{0}},{X_0}\otimes{\pi_{1}}}\rangle_{\mathsf S}$ which is well defined by (S $\otimes$ -dist). We use $\phi^0_{X_0,X_1}\in\mathcal{L}({\mathsf S X_0}\otimes{X_1},\mathsf S({X_0}\otimes{X_1}))$ for the natural transformation defined from $\phi^1$ using the symmetry isomorphism of the SMC, that is, $\phi^0_{X_0,X_1}=\phi^1_{X_1,X_0}\,\gamma=\langle{{\pi_{0}}\otimes{X_1},{\pi_{1}}\otimes{X_1}}\rangle_{\mathsf S}\in\mathcal{L}({\mathsf S X_0}\otimes{X_1},\mathsf S({X_0}\otimes{X_1}))$ .

Theorem 8. The natural transformation $\phi^1$ is a strength for the monad $(\mathsf S,\iota_{0},\tau)$ and the following diagram commutes:

Therefore, equipped with the strength $\phi^1$ , the monad $(\mathsf S,\iota_{0},\tau)$ is commutative.

Proof. The fact that $\phi^1$ is a strength means that the following two diagrams commute:

Let us prove for instance the second one. We have

The fact that $(\mathsf S,\iota_{0},\tau,\phi^1)$ is a commutative monad means that, moreover, the following diagram commutes:

which results from a stronger property, namely that, as announced, the following diagram commutes:

and from Theorem 7. This commutation is proved as follows:

We set

\begin{align*} \mathsf L_{X_0,X_1} &=\tau\,(\mathsf S\phi^0_{X_0,X_1})\,\phi^1_{\mathsf S X_0,X_1}\\ &=\tau\,(\mathsf S\phi^1_{X_0,X_1})\,\phi^0_{X_0,\mathsf S X_1}\\ &=\langle{{\pi_{0}}\otimes{\pi_{0}}, {\pi_{1}}\otimes{\pi_{0}}+{\pi_{0}}\otimes{\pi_{1}}}\rangle_{\mathsf S}\\ &\in\mathcal{L}({\mathsf S X_0}\otimes{\mathsf S X_1},\mathsf S({X_0}\otimes{X_1}))\,.\end{align*}

It is well known that in such a commutative monad situation, the associated tuple $(\mathsf S,\iota_{0},\tau,\mathsf L)$ is a symmetric monoidal monad on the SMC $\mathcal{L}$ . In particular, we will use the following equation:

(3) \begin{equation} \mathsf L\,({\mathsf S X_0}\otimes{\iota_{0}}) =\phi^0_{X_0,X_1}=\langle{{\pi_{0}}\otimes{X_1},{\pi_{1}}\otimes{X_1}}\rangle_{\mathsf S} \in\mathcal{L}({\mathsf S X_0}\otimes{X_1},\mathsf S({X_0}\otimes{X_1}))\end{equation}

and symmetrically for $\mathsf L\,({\iota_{0}}\otimes{\mathsf S X_1})$ .

4.1 Differential structure

(S $\mathrel{\&}$ -pres) The functor $\mathsf S$ preserves all finite cartesian products. In other words, the morphisms $0\in\mathcal{L}(\mathsf S\top,\top)$ and $\langle{\mathsf S{\mathsf{pr}_{0}},\mathsf S{\mathsf{pr}_{1}}}\rangle\in\mathcal{L}(\mathsf S({X_0}\mathrel{\&}{X_1}), {\mathsf S{X_0}}\mathrel{\&}{\mathsf S{X_1}})$ are isos.

A differential structure on a summable resource category $\mathcal{L}$ consists of a natural transformation $\partial_X\in\mathcal{L}(\ !{\mathsf S X},\mathsf S\ !{X})$ which satisfies the following conditions:

It is standard that this condition allows one to extend (in the sense of Power and Watanabe Reference Power and Watanabe2002, Definition 4.5) the functor $\ !{\_}$ to the Kleisli category $\mathcal{L}_{\mathsf S}$ of the monad $\mathsf S$ . In this Kleisli category, a morphism $X\to Y$ can be seen as a pair $(f_0,f_1)$ of two summable morphisms in $\mathcal{L}(X,Y)$ , and composition is defined by $g\mathrel\circ f=(g_0\, f_0,g_1\, f_0+g_0\, f_1)$ , a definition which is very reminiscent of the multiplication of dual numbers.

This condition allows us to extend the functor $\mathsf S$ to the Kleisli category ${\mathcal L}_!$ . We obtain in that way the functor $\widetilde{\mathsf D}:{\mathcal L}_!\to{\mathcal L}_!$ defined as follows: on objects, we set $\widetilde{\mathsf D} X=\mathsf S X$ . Next, given $f\in{\mathcal L}_!(X,Y)=\mathcal{L}(\ !{X},Y)$ , the morphism $\widetilde{\mathsf D} f\in{\mathcal L}_!(\mathsf S X,\mathsf S Y)=\mathcal{L}(\ !{\mathsf S X},\mathsf S Y)$ is defined by $\widetilde{\mathsf D} f=(\mathsf S f)\,\partial_X$ . The purpose of the two commutations is precisely to make this operation functorial, and this functoriality is a categorical version of the chain rule of calculus, exactly as in tangent categories since, as we will see, this functor $\widetilde{\mathsf D}$ essentially computes the derivative of f.

In other words, we have explicit expressions for $\partial_\top$ and $\partial_{{X_0}\mathrel{\&}{X_1}}$ :

(4)

(5)

This diagram, involves the canonical flip $\mathsf c$ and expresses a kind of commutativity of the second derivative.

We assume that $\mathcal{L}$ is a differential summable resource category.

4.2 Derivatives and partial derivatives in the Kleisli category

The Kleisli category ${\mathcal L}_!$ of the comonad $(\ !{},\operatorname{\mathsf{der}}_{},\operatorname{\mathsf{dig}}_{}\ !)$ is well known to be cartesian, where we use “ $\mathrel\circ$ ” for the composition of morphisms. In general, it is not a differential cartesian category in the sense of Alvarez-Picallo and Lemay (Reference Alvarez-Picallo, Lemay, Goubault-Larrecq and König2020) because it is not required to be left additive. We postpone the precise axiomatization of this kind of partially additive ^{Footnote 6} Our running example of coherence spaces is an example of such a category which is not a differential category.

There is an inclusion functor $\operatorname{\mathsf{Der}}:\mathcal{L}\to{\mathcal L}_!$ which maps X to X and $f\in\mathcal{L}(X,Y)$ to $f\,\operatorname{\mathsf{der}}_{X}\in{\mathcal L}_!(X,Y)$ , and it is faithful but not full in general and allows us to see any morphism of $\mathcal{L}$ as a “linear morphism” of ${\mathcal L}_!$ .

We have already mentioned the functor $\widetilde{\mathsf D}:{\mathcal L}_!\to{\mathcal L}_!$ , remember that $\widetilde{\mathsf D} X=\mathsf S X$ and $\widetilde{\mathsf D} f=(\mathsf S f)\,\partial_X$ when $f\in{\mathcal L}_!(X,Y)$ . Then we have $\widetilde{\mathsf D}\mathrel\circ\operatorname{\mathsf{Der}}=\operatorname{\mathsf{Der}}\mathrel\circ\mathsf S$ which allows us to extend simply the monad structure of $\mathsf S$ to $\widetilde{\mathsf D}$ by setting $\zeta_X=\operatorname{\mathsf{Der}}{\iota_{0}}\in{\mathcal L}_!(X,\widetilde{\mathsf D} X)$ and $\theta_X=\operatorname{\mathsf{Der}}{\tau}\in{\mathcal L}_!(\widetilde{\mathsf D}^2 X,\widetilde{\mathsf D} X)$ .

Similarly,

Since $\mathsf S$ preserves cartesian products, we can equip easily this monad $(\widetilde{\mathsf D},\zeta,\theta)$ on ${\mathcal L}_!$ with a commutative strength $\psi^1_{X_0,X_1}\in{\mathcal L}_!({X_0}\mathrel{\&}{\widetilde{\mathsf D}{X_1}},\widetilde{\mathsf D}{\left({X_0}\mathrel{\&}{X_1}\right)})$ which is the following composition in $\mathcal{L}$ :

where $\eta={\langle{\mathsf S{\mathsf{pr}_{0}},\mathsf S{\mathsf{pr}_{1}}}\rangle}^{-1}$ is the canonical iso of (S $\mathrel{\&}$ -pres).

Given $f\in{\mathcal L}_!({X_0}\mathrel{\&}{X_1},Y)$ , we can define the partial derivatives $\widetilde{\mathsf D}_0f\in{\mathcal L}_!({\widetilde{\mathsf D} X_0}\mathrel{\&}{X_1},\widetilde{\mathsf D} Y)$ and $\widetilde{\mathsf D}_1f\in{\mathcal L}_!({X_0}\mathrel{\&}{\widetilde{\mathsf D} X_1},\widetilde{\mathsf D} Y)$ as $\widetilde{\mathsf D} f\mathrel\circ\psi^0$ and $\widetilde{\mathsf D} f\mathrel\circ\psi^1$ where we use $\psi^0$ for the strength ${\widetilde{\mathsf D} X_0}\mathrel{\&}{X_1}\to\widetilde{\mathsf D}\left({X_0}\mathrel{\&}{X_1}\right)$ defined from $\psi^1$ using the symmetry of $\mathrel{\&}$ . We have

In other words, $\widetilde{\mathsf D}_0f$ is fully characterized by the two following equations:

(6)

(7)

and of course there are symmetric equations characterizing $\widetilde{\mathsf D}_1f$ .

\begin{align*} \partial_X=\langle{d_0,d_1}\rangle:\ !{\left({X}\mathrel{\&}{X}\right)}\to{\ !{X}}\mathrel{\&}{\ !{X}}\end{align*}

where $d_0=\ !{\mathsf{pr}_{0}}$ and $d_1$ is the following composition of morphisms:

where $\operatorname{\overline{\mathsf{der}}}_{X}:X\to\ !{X}$ is the codereliction morphism and $\operatorname{\overline{\mathsf{contr}}}_{X}$ is the cocontraction morphism of the differential ${\textsf{LL}}$ model structure.

This fact has been proven in Spring 2021 by Aymeric Walch during his Master Internship and the proof will be made available soon.

4.3 Deciphering the diagrams

After this rather terse list of categorical axioms, it is fair to provide the reader with intuitions about their mathematical meaning; this is the purpose of this section.

One should think of the objects of $\mathcal{L}$ as partial commutative monoids (with additional structures depending on the considered category), and $\mathsf S X$ as the object of pairs (x,u) of elements $x,u\in X$ such that $x+u\in X$ is defined. The morphisms in $\mathcal{L}$ are linear in the sense that they preserve 0 and these partially defined sums, whereas the morphisms of ${\mathcal L}_!$ should be thought of as functions which are not linear but admit a “derivative.” More precisely, $f\in{\mathcal L}_!(X,Y)$ can be seen as a function $X\to Y$ and, given $(x,u)\in\mathsf S X$ we have

\begin{align*} \widetilde{\mathsf D} f(x,u)=(f(x),\frac{d{f(x)}}{d{x}}\cdot{u})\in\mathsf S Y\,,\end{align*}

where $\frac{d{f(x)}}{d{x}}\cdot{u}$ is just a notation for the second component of the pair $\widetilde{\mathsf D} f(x,u)$ which, by construction, is such that the sum $f(x)+\frac{d{f(x)}}{d{x}}\cdot{u}$ is a well-defined element of Y. Now we assume that this derivative $\frac{d{f(x)}}{d{x}}\cdot{u}$ obeys the standard rules of differential calculus, and we will see that the above axioms about $\partial$ correspond to these rules.

• ( $\partial$ -local) means that the first component of $\widetilde{\mathsf D} f(x,u)$ is f(x), justifying our intuitive notation:

  \begin{align*} \widetilde{\mathsf D} f(x,u)=(f(x),\frac{d{f(x)}}{d{x}}\cdot{u})\in\mathsf S Y. \end{align*}

• The first diagram of ( $\partial$ -chain) means that if $f\in{\mathcal L}_!(X,Y)$ is linear ^{Footnote 7} in the sense that there is $g\in\mathcal{L}(X,Y)$ such that $f=g\,\operatorname{\mathsf{der}}_{X}=\operatorname{\mathsf{Der}} g$ , then $\frac{d{f(x)}}{d{x}}\cdot{u}=f(u)$ . Notice that it prevents differentiation from being trivial by setting $\frac{d{f(x)}}{d{x}}\cdot{u}=0$ for all f and all x,u. Consider now $f\in{\mathcal L}_!(X,Y)$ and $g\in{\mathcal L}_!(Y,Z)$ ; the second diagram means that $\widetilde{\mathsf D}(g\mathrel\circ f)=\widetilde{\mathsf D} g\mathrel\circ\widetilde{\mathsf D} f$ , which amounts to

  \begin{align*} \frac{d{g(f(x))}}{d{x}}\cdot{u}=\frac{d{g(y)}}{d{y}}(f(x))\cdot{(\frac{d{f(x)}}{d{x}}\cdot{u})} \end{align*}

  which is exactly the chain rule.

• The “second derivative” $\widetilde{\mathsf D}^2f\in{\mathcal L}_!(\mathsf S^2X,\mathsf S^2Y)$ of $f\in{\mathcal L}_!(X,Y)$ is $(\mathsf S^2f)\,(\mathsf S\partial_X)\,\partial_{\mathsf S X}$ . Remember that $\widetilde{\mathsf D} f(x,u)=(f(x),\frac{d{f(x)}}{d{x}}\cdot{u})$ , therefore applying the standard rules of differential calculus we have

  \begin{align*} \widetilde{\mathsf D}^2f((x,u),(x',u')) &=(\widetilde{\mathsf D} f(x,u),\frac{d{\widetilde{\mathsf D} f(x,u)}}{d{(x,u)}}\cdot{(x',u')})\\ &=((f(x),\frac{d{f(x)}}{d{x}}\cdot{u}),\frac{\partial{(f(x), \frac{d{f(x)}}{d{x}}\cdot{u})}}{\partial{x}}\cdot{x'}+\frac{\partial{(f(x), \frac{d{f(x)}}{d{x}}\cdot{u})}}{\partial{u}}\cdot{u'})\\ &=((f(x),\frac{d{f(x)}}{d{x}}\cdot{u}),(\frac{d{f(x)}}{d{x}}\cdot{x'}, \frac{d^{2}{f(x)}}{d{x}^{2}}\cdot{(u,x')}+\frac{d{f(x)}}{d{x}}\cdot{u'})) \end{align*}

  where we have used the fact that f(x) does not depend on u and that $\frac{d{f(x)}}{d{x}}\cdot{u}$ is linear in u. We have used ( $\partial$ -lin) to prove Theorem 10 whose main content is the naturality of $\zeta$ and $\theta$ . This second naturality means that $\widetilde{\mathsf D} f\mathrel\circ\theta_X=\theta_Y\mathrel\circ\widetilde{\mathsf D}^2f$ , that is, by the computation above $\frac{d{f(x)}}{d{x}}\cdot{(u+x')}=\frac{d{f(x)}}{d{x}}\cdot{u}+\frac{d{f(x)}}{d{x}}\cdot{x'}$ since, intuitively, $\theta_X((x,u),(x',u'))=(x,u+x')$ . Similarly, the naturality of $\zeta$ means that $\frac{d{f(x)}}{d{x}}\cdot{0}=0$ . So the condition ( $\partial$ -lin) means that the derivative is a function which is linear with respect to its second parameter.

• We have assumed that $\mathcal{L}$ is cartesian and hence ${\mathcal L}_!$ is also cartesian. Intuitively, ${X_0}\mathrel{\&}{X_1}$ is the space of pairs $(x_0,x_1)$ with $x_i\in X_i$ , and our assumption (S $\mathrel{\&}$ -pres) means that $\mathsf S\left({X_0}\mathrel{\&}{X_1}\right)$ is the space of pairs $((x_0,x_1),(u_0,u_1))$ such that $(x_i,u_i)\in\mathsf S X_i$ , and the sum of such a pair is $(x_0+u_0,x_1+u_1)\in {X_0}\mathrel{\&}{X_1}$ . Then, given $f\in{\mathcal L}_!({X_0}\mathrel{\&}{X_1},Y)$ the second diagram of ( $\partial$ - $\mathord{\mathrel{\&}}$ ) means that

  \begin{align*} \frac{d{f(x_0,x_1)}}{d{(x_0,x_1)}}\cdot{(u_0,u_1)} =\frac{\partial{f(x_0,x_1)}}{\partial{x_0}}\cdot{u_0}+\frac{\partial{f(x_0,x_1)}}{\partial{x_1}}\cdot{u_1} \end{align*}

  which can be seen by the following computation of $\pi_{1}\,\widetilde{\mathsf D} f$ :

  

  Then Theorem 9 means that $\frac{d{f(x,x)}}{d{x}}\cdot{u} =\frac{\partial{f(x_0,x_1)}}{\partial{x_0}}(x,x)\cdot{u}+\frac{\partial{f(x_0,x_1)}}{\partial{x_1}}(x,x)\cdot{u}$ which is the essence of the Leibniz rule of calculus.

• The object $\mathsf S^2 X$ consists of pairs ((x,u),(x’,u’)) such that x, u, x’ and u’ are globally summable. Then $\mathsf c\in\mathcal{L}(\mathsf S^2X,\mathsf S^2X)$ maps ((x,u),(x’,u’)) to ((x,x’),(u,u’)). Therefore, using the same computation of $\widetilde{\mathsf D}^2f((x,u),(x',u'))$ as in the case of ( $\partial$ -lin), we see that ( $\partial$ -Schwarz) expresses that $\frac{d^{2}{f(x)}}{d{x}^{2}}\cdot{(u,x')}=\frac{d^{2}{f(x)}}{d{x}^{2}}\cdot{(x',u)}$ (upon taking $u'=0$ ). So this diagram means that the second derivative is a symmetric bilinear function, a property of sufficiently regular differentiable functions often referred to as Schwarz Theorem.

4.4 A differentiation in coherence spaces

Now we exhibit such a differentiation in $\mathbf{Coh}$ . We define $\ !{E}$ as follows: ${}|{\ !{E}}{}|$ is the set of finite multisets ^{Footnote 8} m of elements of ${}|{E}{}|$ such that $\operatorname{\mathsf{supp}}({m})\in\mathsf{Cl}({E})$ (such an m is called a finite multiclique). Given $m_0,m_1\in{}|{\ !{E}}{}|$ , we have if $m_0+ m_1\in{}|{\ !{E}}{}|$ . This operation is a functor $\mathbf{Coh}\to\mathbf{Coh}$ : given $s\in\mathbf{Coh}(E,F)$ one sets

\begin{align*} \ !{s}= \{([{a_{1},\dots,a_{n}}],[{b_{1},\dots,b_{n}}])\mid n\in{\mathbb{N}},\ (a_i,b_i)\in s \text{ for }i=1,\dots,n\text{ and }[{a_{1},\dots,a_{n}}]\in{}|{\ !{E}}{}|\}\end{align*}

which actually belongs to $\mathsf{Cl}({{\ !{E}}\multimap{\ !{F}}})$ because $s\in\mathsf{Cl}({{E}\multimap{F}})$ . The comonad structure of this functor and the associated commutative comonoid structure are given by:

• $\operatorname{\mathsf{der}}_{E}=\{([{a}],a)\mid a\in{}|{E}{}|\}$

• $\operatorname{\mathsf{dig}}_{E}=\{(m,[{m_1,\dots,m_n}])\in{}|{{\ !{E}}\multimap{\ !\ !{E}}}{}|\mid m=m_1+\cdots+ m_n\}$

• $\operatorname{\mathsf{weak}}_{E}=\{([\,],\ast)\}$

• and $\operatorname{\mathsf{contr}}_{E}=\{(m,(m_1,m_2))\in{}|{{\ !{E}}\multimap{({\ !{E}}\otimes{\ !{E}})}}{}|\mid m=m_1+ m_2\}$ .

Composition in ${\mathbf{Coh}}_!$ can be described directly as follows: let $s\in\mathsf{Cl}({{\ !{E}}\multimap{F}})$ and $t\in\mathsf{Cl}({{\ !{F}}\multimap{G}})$ , then $t\mathrel\circ s\in\mathcal{L}{\left({\ !{E}}\multimap{G}\right)}$ is $\{(m,c)\in{}|{{\ !{E}}\multimap{G}}{}|\mid\exists n\in{\mathbb{N}}\,\exists (m_1,b_1),\dots,(m_n,b_n)\in s\ m_1+\cdots+m_n=m\text{ and }([{b_{1},\dots,b_{n}}],c)\in t\}$ . A morphism $s\in{\mathbf{Coh}}_!(E,F)$ induces a function $\widehat{s}:\mathsf{Cl}({E})\to\mathsf{Cl}({F})$ by $\widehat{s}(x)=\{b\mid\exists m\in\mathcal M_{{\mathrm{fin}}}({x})\ (m,b)\in s\}$ . The functions $f:\mathsf{Cl}({E})\to\mathsf{Cl}({F})$ definable in that way are exactly the stable functions: f is stable if for any $x\in\mathsf{Cl}({E})$ and any $b\in f(x)$ there is exactly one minimal subset $x_0$ of x such that $b\in f(x_0)$ , and moreover this $x_0$ is finite. When moreover this $x_0$ is always a singleton f is said to be linear and such linear functions are in bijection with $\mathbf{Coh}(E,F)$ (given $t\in\mathbf{Coh}(E,F)$ , and the associated linear function $\mathsf{Cl}({E})\to\mathsf{Cl}({F})$ is the map $x\mapsto{t}\cdot{x}$ ).

Notice that for a given stable function $f:\mathsf{Cl}({E})\to\mathsf{Cl}({F})$ , there can be infinitely many $s\in{\mathbf{Coh}}_!(E,F)$ such that $f=\widehat{s}$ since the definition of $\widehat{s}$ does not take into account the multiplicities in the multisets m such that $(m,b)\in s$ . For instance, if $a\in{}|{E}{}|$ and $b\in{}|{F}{}|$ then $\{([{a}],b)\}$ and $\{([{a,a}],b)\}$ define exactly the same stable (actually linear) function.

Up to trivial iso we have ${}|{\ !{\mathsf S E}}{}|=\{(m_0,m_1)\in{}|{\ !{E}}{}|\mid \operatorname{\mathsf{supp}}({m_0})\cap\operatorname{\mathsf{supp}}({m_1})=\emptyset\text{ and }m_0+ m_1\in{}|{\ !{E}}{}|\}$ and if $m_{00}+ m_{01}+ m_{10}+ m_{11}\in{}|{\ !{X}}{}|$ and $\operatorname{\mathsf{supp}}({m_{00}+ m_{10}})\cap\operatorname{\mathsf{supp}}({m_{01}+ m_{11}})=\emptyset$ . With this identification, we define $\partial_E\subseteq{}|{{\ !{\mathsf S E}}\multimap{\mathsf S{\ !{E}}}}{}|$ as follows:

(8)

We think useful to check directly that $\partial_E\in\mathbf{Coh}(\ !{\mathsf S E},\mathsf S{\ !{E}})$ although this checking is not necessary since we will see in Section 5.6 that this property results from a much simpler one. Let $((m_{j0},m_{j1}),(i_j,m_j))\in\partial_E$ for $j=0,1$ and assume that

(9)

By symmetry, there are three cases to consider.

• If $i_0=i_1=0$ then, we have $m_{j1}=[\,]$ and $m_{j0}=m_j$ for $j=0,1$ . Then, we have by our assumption (9), and if $(0,m_0)=(0,m_1)$ then $(m_{00},m_{01})=(m_{10},m_{11})$ .

• Assume now that $i_0=i_1=1$ . We have $m_{j1}=[{a_j}]$ for $a_j\in{}|{E}{}|$ , with $a_j\notin \operatorname{\mathsf{supp}}({m_{j0}})$ and $m_{j}=m_{j0}+[{a_j}]$ . Our assumption (9) means that $m_{00}+ m_{10}+[{a_0,a_1}]\in{}|{\ !{E}}{}|$ and $\operatorname{\mathsf{supp}}({m_{00}+ m_{10}})\cap\{a_0,a_1\}=\emptyset$ . Therefore, $m_0+ m_1\in{}|{\ !{E}}{}|$ and hence . Assume moreover that $m_0=m_1$ , that is, $m_{00}+[{a_0}]=m_{10}+[{a_1}]$ . This implies $m_{00}=m_{10}$ and $a_0=a_1$ since we know that $a_1\notin \operatorname{\mathsf{supp}}({m_{00}})$ and $a_0\notin \operatorname{\mathsf{supp}}({m_{10}})$ .

• Last assume that $i_0=1$ and $i_1=0$ . So we have $m_{01}=[{a}]$ with $a\notin \operatorname{\mathsf{supp}}({m_{00}})$ and $m_0=m_{00}+[{a}]$ ; $m_{11}=[\,]$ and $m_1=m_{10}$ . By (9), we know that $\operatorname{\mathsf{supp}}({m_0+ m_1})\in\mathsf{Cl}({\ !{E}})$ . Coming back to the definition of the coherence in $\mathsf S F$ (for a coherence space F), we must also prove that $m_0\not=m_1$ : this results from (9) which entails that $a\notin \operatorname{\mathsf{supp}}({m_1})=m_{10}$ , whereas we know that $a\in \operatorname{\mathsf{supp}}({m_0})$ .

We do not prove the required commutations for the already mentioned reason that they will be reduced in Section 5.6 to a much simpler verification.

Given $x\in\mathsf{Cl}({E})$ , we can define a coherence space $E_x$ (the local sub-coherence space at x) as follows: ${}|{E_x}{}|=\{a\in{}|{E}{}|\setminus x\mid x\cup\{a\}\in\mathsf{Cl}({X})\}$ and if . Then, given $s\in{\mathbf{Coh}}_!(E,F)$ , we can define the differential of s at x as:

\begin{align*} \frac{d s(x)}{d x} =\{(a,b)\in{}|{E_x}{}|\times{}|{F}{}|\mid\exists m\in{}|{\ !{E}}{}|\ (m+[{a}],b)\in s\text{ and } \operatorname{\mathsf{supp}}({m})\subseteq x\} \subseteq{}|{{E_x}\multimap{Y}}{}|\,.\end{align*}

Theorem 11. Let $s\in{\mathbf{Coh}}_!(E,F)$ . Then $\widetilde{\mathsf D} s\in{\mathbf{Coh}}_!(\mathsf S E,\mathsf S F)$ satisfies

\begin{align*} \forall (x,u)\in\mathsf{Cl}({\mathsf S E}) \widehat{\widetilde{\mathsf D} s}(x,u)=(\widehat{s}(x),{\frac{d s(x)}{d x}}\cdot{u}) \end{align*}

Assume first that $(i,b)\in \widehat{\widetilde{\mathsf D} s}(x,u)$ and let $(m_0,m_1)$ be as above. If $i=0$ , we have $(m_0,b)\in s$ and $\operatorname{\mathsf{supp}}({m})_0\subseteq x$ and hence $b\in\widehat{s}(x)$ , that is $(i,b)\in(\widehat{s}(x),{\frac{d s(x)}{d x}}\cdot{u})$ . If $i=1$ let $a\in{}|{E}{}|\setminus\operatorname{\mathsf{supp}}({m_0})$ be such that $m_1=[{a}]$ , $m_0+[{a}]\in{}|{\ !{E}}{}|$ , $(m_0+[{a}],b)\in s$ and $\operatorname{\mathsf{supp}}({m_0,[{a}]})\subseteq(x,u)$ (remember that we consider the elements of $\mathsf{Cl}({\mathsf S E})$ as pairs of cliques), that is $\operatorname{\mathsf{supp}}({m_0})\subseteq x$ and $a\in u$ . Then we know that $a\in{}|{E_x}{}|$ since $x\cup u\in\mathsf{Cl}({E})$ and $x\cap u=\emptyset$ . Therefore, $(i,b)\in(\widehat{s}(x),{\frac{d s(x)}{d x}}\cdot{u})$ .

We have proven $\widehat{\widetilde{\mathsf D} s}(x,u)\subseteq(\widehat{s}(x),{\frac{d s(x)}{d x}}\cdot{u})$ , and we prove the converse inclusion. Let $(i,b)\in (\widehat{s}(x),{\frac{d s(x)}{d x}}\cdot{u})$ . If $i=0$ , we have $b\in \widehat{s}(x)$ , and hence there is a uniquely defined $m_0\in{}|{\ !{E}}{}|$ such that $\operatorname{\mathsf{supp}}({m_0})\subseteq x$ and $(m_0,b)\in s$ . It follows that $((m_0,[\,]),(0,b))\in\partial_E\,\mathsf S s$ and hence $(i,b)\in \widehat{\widetilde{\mathsf D} s}(x,u)$ . Assume now that $i=1$ so that $b\in{\frac{d s(x)}{d x}}\cdot{u}$ , and hence there is $a\in u$ (which implies $a\notin x$ ) such that $(a,b)\in\frac{d s(x)}{d x}$ . So there is $m_0\in{}|{\ !{E}}{}|$ such that $\operatorname{\mathsf{supp}}({m_0})\subseteq x$ and $(m_0+[{a}],b)\in s$ (notice that $a\notin\operatorname{\mathsf{supp}}({m_0})$ since $\operatorname{\mathsf{supp}}({m_0})\subseteq x$ and $a\notin x$ ). It follows that $((m_0,[{a}]),(1,m_0+[{a}]))\in\partial_E$ and hence $((m_0,[{a}]),(1,b))\in (\mathsf S s)\,\partial_E$ so that $(1,b)\in\widehat{\widetilde{\mathsf D} s}(x,u)$ .

This is an effect of the uniformity of the construction $\ !{E}$ , that is, of the fact that for $m\in\mathcal M_{{\mathrm{fin}}}({{}|{E}{}|})$ to be in ${}|{\ !{E}}{}|$ , it is required that $\operatorname{\mathsf{supp}}({m})$ be a clique. Indeed, it is only because of this uniformity requirement in the definition of $\ !{E}$ that a coherence space E can be defined by means of a reflexive coherence relation (or an antireflexive strict coherence relation ) in the sense that one cannot define, in these coherence spaces, a resource modality $\ !{E}$ such that ${}|{\ !{E}}{}|=\mathcal M_{{\mathrm{fin}}}({{}|{E}{}|})$ . But an effect of this simplicity in the axiomatization of coherence is that the summability of two cliques requires their disjointedness, and a consequence of this is the slightly unsatisfactory behavior of differentiation explained in Remark 36. Do well notice however that this peculiarity does not prevent coherence spaces from satisfying all of our new axioms of summability and differentiation.

This can be remedied, without breaking the main feature of our construction, namely that it is compatible with the determinism ^{Footnote 9} of the model, by using nonuniform coherence spaces instead, where it becomes possible to take ${}|{\ !{E}}{}|=\mathcal M_{{\mathrm{fin}}}({E})$ , see Bucciarelli and Ehrhard (Reference Bucciarelli and Ehrhard2001), Boudes (Reference Boudes2011), but where the coherence relation is no more necessarily reflexive (nor antireflexive), see Section 6.1.