1. Parametrized Categories and Protocalibrations

Categories varying over (Street Reference Street1981) (“fibered over” Bénabou Reference Bénabou1975, “indexed by” Johnstone et al. Reference Johnstone, Paré, Rosebrugh, Schumacher, Wood and Wraith1978, or “parametrized by” Schumacher and Street Reference Schumacher and Street1988) a category or bicategory $\mathscr{C}$ can be regarded as pseudofunctors $\mathbb{X} : \mathscr{C}^{\textrm{op}}\to \textrm{Cat}$ . We think of an object of $\mathbb{X}U$ as a generalized object of the $\mathscr{C}$ -variable category $\mathbb{X}$ . We write

\begin{equation*}\textrm{CAT}\mathscr{C} = \textrm{Ps}( \mathscr{C}^{\textrm{op}}, \textrm{Cat})\end{equation*}

for the (strict) bicategory of pseudofunctors (called “homomorphisms” in Bénabou Reference Blumberg and Hill1967), pseudonatural transformations, and modifications.

A $\mathscr{C}$ -variable category $\mathbb{X}\in \textrm{CAT}\mathscr{C}$ is cocomplete with respect to a morphism $j : \mathbb{A} \to \mathbb{B}$ in $\textrm{CAT}\mathscr{C}$ (see Section 7 of Street Reference Street1981) when every morphism $f : \mathbb{A} \to \mathbb{X}$ has a pointwise (see Section 4 of Street Reference Street1974) left extension along $j$ . Generally, pointwiseness translates to a lax form of the Chevalley-Beck (CB) condition. However, the pseudo form is relevant when dealing with groupoid fibrations as we will now remind the reader.

A morphism $\kappa : z\to x$ in a category $E$ is cartesian for a functor $p : E \to B$ when the following square (1.1) is a pullback in $\textrm{Set}$ for all objects $k$ of $E$ .

(1.1)

A functor $p : E\to B$ is a groupoid fibration (over $B$ ) when every morphism of $E$ is cartesian for $p$ and, for all $x\in E$ and $\phi : b\to px$ in $B$ , there exists a morphism $\kappa : z\to x$ in $E$ and an isomorphism $b\cong pz$ whose composite with $p\kappa$ is $\phi$ . A functor $p : E\to B$ is a groupoid opfibration (over $B$ ) when $p : E^{\textrm{op}}\to B^{\textrm{op}}$ is a groupoid fibration.

A morphism $p : E\to B$ in a bicategory $\mathscr{C}$ is a groupoid fibration when, for all objects $U\in \mathscr{C}$ , the functor $\mathscr{C}(U,p) : \mathscr{C}(U,E)\to \mathscr{C}(U,B)$ is a groupoid fibration. The proof of Proposition 1.1 is essentially as for Proposition 23 of Street (Reference Street1974); alternatively, check it in $\textrm{Cat}$ and use Yoneda since all concepts are representable. The term bipullback (or “bicategorical pullback”) is used here in the sense of Item (1.23) of Street (Reference Street1980) and Section 2 of Street (Reference Street2020).

Proposition 1.1. In any finitely complete bicategory, suppose $p : E\to B$ is a groupoid opfibration. In diagram (1.2), the 2-morphism $\sigma$ exhibits $k$ as a pointwise left extension of $f$ along $p$ if and only if, for all morphisms $b$ with the square a bipullback, the pasted composite at $p$ exhibits $kb$ as a left extension of $f\tilde {b}$ along $\tilde {p}$ .

(1.2)

If $\mathscr{R}$ is a set of morphisms of a bicategory $\mathscr{C}$ , we say $\mathbb{X}$ is $\mathscr{R}$ -cocomplete when it is cocomplete with respect to all morphisms $\mathscr{C}(\!-,j) : \mathscr{C}(\!-,U)\to \mathscr{C}(\!-,V)$ for $j : U\to V$ in $\mathscr{R}$ . A pseudonatural transformation $\theta : \mathbb{X}\to \mathbb{Y}$ is $\mathscr{R}$ -cocontinuous when it preserves left extensions along the $\mathscr{C}(\!-,j)$ for $j\in \mathscr{R}$ .

Let $\textrm{CAT}_{\mathscr{R}}\mathscr{C}$ denote the sub-bicategory of $\textrm{CAT}\mathscr{C}$ consisting of the $\mathscr{R}$ -cocomplete objects and $\mathscr{R}$ -cocontinuous morphisms.

We are interested in those $\mathscr{R}$ in possession of most of the properties in common with the “calibrations” of Street (Reference Street2020), and the calibrations of Bénabou (Reference Blumberg and Hill1975), when $\mathscr{C}$ is a category. In the case where $\mathscr{C}$ is a category of $G$ -sets, these conditions plus closure under coproducts (which we will also require in Section 6) are those called “indexing categories” in equivariant stable homotopy theory; see Chan (Reference Chan2024).

Proposition 1.4 (Compare Street Reference Street1980, Proposition 9.12, Street Reference Street1981, Proposition 7.12). Let $\mathscr{R}$ be a protocalibration of the bicategory $\mathscr{C}$ . A $\mathscr{C}$ -variable category $\mathbb{X}$ is $\mathscr{R}$ -cocomplete if and only if, for all $f \in \mathscr{R}$ , the functor $\mathbb{X}f$ has a left adjoint $\mathbb{X}_*f$ and, for all bicategorical pullback squares

(1.3)

in $\mathscr{E}$ , the mate

\begin{equation*}\mathbb{X}_*f_g \ \circ \ \mathbb{X}g_f\ \Rightarrow \ \mathbb{X}g \ \circ \ \mathbb{X}_*f \end{equation*}

of the isomorphism

\begin{equation*}\mathbb{X}g_f \ \circ \ \mathbb{X}f \ \cong \ \mathbb{X}f_g \ \circ \ \mathbb{X}g\end{equation*}

is invertible. A pseudonatural transformation $\theta : \mathbb{X}\to \mathbb{Y}$ is $\mathscr{R}$ -cocontinuous if and only if, for all $U\xrightarrow {f}W$ in $\mathscr{R}$ , the mate $\mathring {\theta }_f : \mathbb{Y}_*f\circ \theta _U\to \theta _W\circ \mathbb{X}_*f$ of $\theta _f : \theta _U\circ \mathbb{X}f\to \mathbb{Y}f\circ \theta _W$ is invertible.

Proposition 1.8. Suppose $\mathscr{C}$ has binary biproducts and $\mathscr{R}$ is a protocalibration of $\mathscr{C}$ . For $\mathbb{Y}$ and $\mathbb{Z}$ in $\textrm{CAT}_{\mathscr{R}}\mathscr{C}$ , the object $[\mathbb{Y},\mathbb{Z}]_{\mathscr{R}} \in \textrm{CAT}\mathscr{C}$ , defined on objects by

\begin{eqnarray*} [\mathbb{Y},\mathbb{Z}]_{\mathscr{R}}W = \textrm{CAT}_{\mathscr{R}}\mathscr{C}(\mathbb{Y},\mathbb{Z}(\!-\times W)) \end{eqnarray*}

and on morphisms $W'\xrightarrow {h}W$ by composition with $\mathbb{Z}(\!-\times h)$ , is in $\textrm{CAT}_{\mathscr{R}}\mathscr{C}$ .

Proof. By the last sentence of Proposition 1.4, we need to see that, if $\theta : \mathbb{Y}\to \mathbb{Z}(\!-\times W)$ is $\mathscr{R}$ -cocontinuous, so too is $\mathbb{Z}(\!-\times h)\circ \theta$ . Take $V\xrightarrow {r}U$ in $\mathscr{R}$ . The right square in the diagram

is the value of $\mathbb{Z}$ on a bipullback in $\mathscr{C}$ . The invertibility of the mate of the pasted 2-morphism follows from $\mathscr{R}$ -cocontinuity of $\theta$ and $\mathscr{R}$ -cocompleteness of $\mathbb{Z}$ .

2. $\mathscr{R}$ -Cocompletion

Suppose $\mathscr{R}$ is a protocalibration of the bicategory $\mathscr{C}$ . The $\mathscr{R}$ -cocompletion $\overrightarrow {\mathbb{X}}$ of $\mathbb{X}\in \textrm{CAT}\mathscr{C}$ is described as follows (see Bénabou Reference Bénabou1975; Street Reference Street1972, Reference Street1981). An object of the category $\overrightarrow {\mathbb{X}}U$ is a pair $(U\xleftarrow {u}S, x\in \mathbb{X}S)$ with $u\in \mathscr{R}$ ; we might think of such an object as a $U$ -indexed family of generalized objects of $\mathbb{X}$ . A morphism

\begin{equation*}(U\xleftarrow {u}S, x\in \mathbb{X}S)\to (U\xleftarrow {u'}S', x'\in \mathbb{X}S')\end{equation*}

of $\overrightarrow {\mathbb{X}}U$ is an isomorphism class of triples $(S\xrightarrow {w}S', u'w\cong u, x\xrightarrow {\xi } (\mathbb{X}w)x')$ where $\xi$ is a morphism of $\mathbb{X}S$ ; if we identify representables with their representing objects, the picture of such a morphism is a lax morphism (2.4) of spans in $\textrm{CAT}\mathscr{C}$ .

(2.4)

For $r : V\to U$ in $\mathscr{C}$ , define the functor $\overrightarrow {\mathbb{X}}r :\overrightarrow {\mathbb{X}}U\to \overrightarrow {\mathbb{X}}V$ to take $(u,x)$ to $(u_r, (\mathbb{X}r_u)x)$ where

(2.5)

are bipullbacks, and $(w,\xi )$ to $(\bar {w},\bar {\xi })$ where $\bar {w}$ is defined by compatible $u'_r\bar {w} \cong u_r$ , $r_{u'}\bar {w} \cong wr_u$ , and $\bar {\xi }$ by

\begin{equation*}\bar {\xi } : = \left ( (\mathbb{X}r_u)x\xrightarrow {(\mathbb{X}r_u)\xi }(\mathbb{X}r_u)(\mathbb{X}w)x' \cong (\mathbb{X}\bar {w})(\mathbb{X}r_{u'})x'\right ).\end{equation*}

For a 2-morphism $\alpha : r\Rightarrow s : V\to U$ , let $S\xleftarrow {s_u}Q\xrightarrow {u_s}V$ be the bipullback of the cospan $S\xrightarrow {u}U\xleftarrow {s}V$ . Since $u$ is a groupoid opfibration, there exists a cocartesian 2-morphism $\kappa : r_u\Rightarrow k$ and isomorphism $su_r\cong uk$ such that $u\kappa$ is the composite $ur_u\cong ru_r\xrightarrow {\alpha u_r}su_r\cong uk$ . If $r\in \mathscr{R}$ then $\overrightarrow {\mathbb{X}}r$ has a left adjoint $\overrightarrow {\mathbb{X}}_*r : \overrightarrow {\mathbb{X}}V\to \overrightarrow {\mathbb{X}}U$ taking $(V\xleftarrow {v}T, y\in \mathbb{X}T)$ to $(U\xleftarrow {rv}T, y\in \mathbb{X}T)$ (noting that $rv\in \mathscr{R}$ by condition (C)). To verify the pseudo-Chevalley-Beck condition, we refer to the bipullbacks (2.6) with $f, u\in \mathscr{R}$ .

(2.6)

Then we see that

\begin{eqnarray*} (\overrightarrow {\mathbb{X}}g\circ \overrightarrow {\mathbb{X}}_*f) (u,x) & = & (\overrightarrow {\mathbb{X}}g) (fu,x) \ \cong \ (f_gu_{g_f}, (\mathbb{X}g_{fu})x) \\ & = & (\overrightarrow {\mathbb{X}}_*f_g)(u_{g_f},(\mathbb{X}g_{fu})x) \\ & \cong & (\overrightarrow {\mathbb{X}}_*f_g\circ \mathbb{X}g_f)(u,x) \end{eqnarray*}

naturally in $(u,x)\in \overrightarrow {\mathbb{X}}U$ . So $\overrightarrow {\mathbb{X}}\in \textrm{CAT}\mathscr{E}$ is $\mathscr{R}$ -cocomplete by Proposition 1.4.

Remark 2.3. The coproduct completion $\textrm{Fam}X$ of a category $X$ can be constructed as the lax comma category

Analogously, the $\mathscr{R}$ -cocompletion $\overrightarrow {\mathbb{X}}$ of $\mathbb{X}\in \textrm{CAT}\mathscr{C}$ can be constructed as the lax comma category

in $\textrm{Ps}(\mathscr{C}^{\textrm{op}},\textrm{CAT})$ , where the bottom arrow is the restricted Yoneda embedding and the right arrow corresponds to $\mathbb{X}$ under the bicategorical Yoneda Lemma.

Directly from the explicit descriptions, we can verify the following.

There is a pseudonatural transformation $\eta _{\mathbb{X}} : \mathbb{X}\to \overrightarrow {\mathbb{X}}$ whose component $\eta _{\mathbb{X} U} : \mathbb{X}U\to \overrightarrow {\mathbb{X}}U$ at $U\in \mathscr{E}$ is defined by $\eta _{\mathbb{X} U}x = (U\xleftarrow {1_U}U, x\in \mathbb{X} U)$ .

We think of the left adjoint to $\eta _{\mathbb{X}}$ as assigning a coproduct to the indexed families of generalized objects.

Much as in Section 2 of Street (Reference Street1974) and Section 3 of Street (Reference Street1980), for each protocalibration $\mathscr{R}$ of $\mathscr{C}$ , we see we have a Kock-Zöberlein (Kock Reference Kock1972-73, Reference Kock1995; Zöberlein Reference Zöberlein1973) pseudomonad

\begin{equation*}\overrightarrow {(\!-\!)}_{\mathscr{R}} = (\overrightarrow {(\!-\!)}, \eta , \mu )\end{equation*}

on $\textrm{CAT}\mathscr{C}$ .

The opposite $\mathbb{X}^{\textrm{op}}$ of $\mathbb{X}\in \textrm{CAT}\mathscr{C}$ is obtained by composition with the 2-isomorphism $(\!-\!)^{\textrm{op}} : \textrm{Cat}\to \textrm{Cat}^{\textrm{co}}$ . This is the value on objects for a 2-isomorphism

(2.7) \begin{eqnarray} (\!-\!)^{\textrm{op}} : \textrm{CAT}\mathscr{C}\to (\textrm{CAT}\mathscr{C}^{\textrm{co}})^{\textrm{co}}. \end{eqnarray}

Suppose $\mathscr{L}$ is a protocalibration of $\mathscr{C}^{\textrm{co}}$ . We define $\mathbb{X}\in \textrm{CAT}\mathscr{C}$ to be $\mathscr{L}$ -complete when $\mathbb{X}^{\textrm{op}}$ is $\mathscr{L}$ -cocomplete. It follows that the $\mathscr{L}$ -completion $\overleftarrow {\mathbb{X}}$ of $\mathbb{X}\in \textrm{CAT}\mathscr{C}$ is $\overrightarrow {(\mathbb{X}^{\textrm{op}})}^{\textrm{op}}$ . If $\mathbb{X}$ is $\mathscr{L}$ -complete and $V\xrightarrow {f}U$ is in $\mathscr{L}$ , we write $\mathbb{X}_!f$ for the right adjoint of $\mathbb{X}f : \mathbb{X}U\to \mathbb{X}V$ .

We write $_{\mathscr{L}\,}{\textrm{CAT}\mathscr{C}}{ }$ for the locally full sub-bicategory of $\textrm{CAT}\mathscr{C}$ consisting of the $\mathscr{L}$ -complete objects and $\mathscr{L}$ -continuous morphisms. The 2-isomorphism (2.7) induces an isomorphism

(2.8) \begin{eqnarray} _{\mathscr{L}}{\textrm{CAT}}{ }\mathscr{C} \cong ({\textrm{CAT}}{_{\mathscr{L}} }\mathscr{C}^{\textrm{co}})^{\textrm{co}}. \end{eqnarray}

We have a dual Kock-Zöberlein pseudomonad on $\textrm{CAT}\mathscr{C}$ for which the Eilenberg-Moore pseudo-algebras are the $\mathscr{L}$ -complete objects where the algebra action on the object is right adjoint to the unit.

An object of $\overleftarrow {\mathbb{X}}U$ is a pair $(U\xleftarrow {u}S, x\in \mathbb{X}S)$ with $u\in \mathscr{L}$ , as for $\overrightarrow {\mathbb{X}}U$ . A morphism

\begin{equation*}(U\xleftarrow {u}S, x\in \mathbb{X}S)\to (U\xleftarrow {u'}S', x'\in \mathbb{X}S')\end{equation*}

of $\overleftarrow {\mathbb{X}}U$ is an isomorphism class of triples $(S'\xrightarrow {w}S, u'\cong uw, (\mathbb{X}w)x\xrightarrow {\zeta }x')$ where $\zeta$ is a morphism of $\mathbb{X}S'$ .

(2.9)

For $r : V\to U$ in $\mathscr{E}$ , the functor $\overleftarrow {\mathbb{X}}r :\overleftarrow {\mathbb{X}}U\to \overleftarrow {\mathbb{X}}V$ takes the object $(u,x)$ to $(u_r, (\mathbb{X}r_u)x)$ with notation as in (2.5), which is used also in defining $\overleftarrow {\mathbb{X}}r$ on morphisms. The unit $\rho _{\mathbb{X}} : \mathbb{X}\to \overleftarrow {\mathbb{X}}$ for the pseudomonad has components at $U$ the functor $\rho _U$ taking $x\in \mathbb{X}U$ to $(1_U, x)\in \overleftarrow {\mathbb{X}}U$ and $\zeta : x \to x'$ to $(1_U,\zeta )$ . For emphasis, we state the dual form of Proposition 2.5:

We think of the right adjoint to $\rho _{\mathbb{X}}$ as assigning a product to the indexed families of generalized objects of $\mathbb{X}$ .

3. $\mathscr{R}$ -Spans and $\mathscr{R}$ -Cocompleteness

We look at slight variants of Bénabou’s bicategory of spans (Bénabou Reference Blumberg and Hill1967; also see Day Reference Day1974 and Carboni et al. Reference Carboni, Kelly, Walters and Wood2007). Let $\mathscr{R}$ be a protocalibration of the bicategory $\mathscr{C}$ ; see Definition 1.2. A (left) $\mathscr{R}$ -span from $U$ to $V$ in $\mathscr{C}$ is a diagram $U\xleftarrow {u}S\xrightarrow {v}V$ in $\mathscr{C}$ with $u\in \mathscr{R}$ . There is a bicategory $\textrm{Spn}_{\mathscr{R}}\mathscr{C}$ whose objects are those of $\mathscr{C}$ , whose morphisms are the $\mathscr{R}$ -spans, and whose 2-morphisms (denoted $[f,\sigma ]$ by abuse of language) are the isomorphism classes of diagrams (3.10).Footnote ²

(3.10)

The composition of $\mathscr{R}$ -spans is defined on morphisms using bipullback. To define the composition on 2-morphisms, we use the constructions (in dual form for groupoid opfibrations rather than fibrations) given in Proposition 5.2 (iii) and Proposition 8.4 of Street (Reference Street2020). Explicitly, consider diagram (3.11)

(3.11)

where we have bipullbacks (3.12).

(3.12)

Using the groupoid opfibration property of $r'$ , we obtain a 2-morphism $\kappa : gv_r\Rightarrow k$ and $r'k\cong v'fr_v$ such that $r'\kappa$ is the composite

\begin{equation*}r'gv_r\cong rv_r\cong vr_v\xrightarrow {\sigma r_v} v'fr_v\cong r'k.\end{equation*}

Now define $h : P\to P'$ , up to isomorphism using the bipullback property of $P'$ , by $r'_{v'}h\cong f r_v$ and $v'_{r'}h\cong k$ . Then, we have the diagram

where $\rho$ is the composite $gv_r\xrightarrow {\kappa }k\cong v'_{r'}h$ . Now, we define the desired composite of the two span morphisms $[f, \sigma ]$ and $[g, \tau ]$ appearing in diagram (3.11) to be $[h, s'\rho ]$ .

We have a pseudofunctor

(3.13) \begin{eqnarray} (\!-\!)_* : \mathscr{C}\to \textrm{Spn}_{\mathscr{R}}\mathscr{C} \end{eqnarray}

taking the morphism $U\xrightarrow {r} V$ to the $\mathscr{R}$ -span $r_* = (U\xleftarrow {1_U}U\xrightarrow {r}V)$ and the 2-morphism $\alpha : r\Rightarrow s$ to the 2-morphism

If $r\in \mathscr{R}$ then $r_*$ has a right adjoint $r^* = (V\xleftarrow {r}U\xrightarrow {1_U}U)$ in $\textrm{Spn}_{\mathscr{R}}\mathscr{E}$ . Every morphism $U\xleftarrow {u}S\xrightarrow {v}V$ of $\textrm{Spn}_{\mathscr{R}}\mathscr{E}$ decomposes as $v_*\circ u^*$ .

Restriction along the opposite of (3.13) defines a pseudofunctor

(3.14) \begin{eqnarray} \textrm{CATSpn}_{\mathscr{R}}\mathscr{C} \to \textrm{CAT}\mathscr{C} \end{eqnarray}

which is pseudomonadic since $(\!-\!)_*$ is the identity on objects and the restriction has both left and right biadjoints. Re-identification of the pseudo-algebras as follows is an objective variant of Lindner’s result (Lindner Reference Lindner1976).

Proposition 3.1. Restriction along the opposite of (3.13) defines a bicategorical equivalence

\begin{eqnarray*} \textrm{CAT}(\textrm{Spn}_{\mathscr{R}}\mathscr{C}) \sim \textrm{CAT}_{\mathscr{R}}\mathscr{C}. \end{eqnarray*}

Proof. We shall describe the inverse biequivalence. Each $\mathbb{X}\in \textrm{CAT}_{\mathscr{R}}\mathscr{C}$ extends along the opposite of $(\!-\!)_*$ to a pseudofunctor $\mathbb{X}_{\textrm{sp}} : (\textrm{Spn}_{\mathscr{R}}\mathscr{C})^{\textrm{op}} \to \textrm{Cat}$ , which agrees with $\mathbb{X}$ on objects and is defined on morphisms by $\mathbb{X}_{\textrm{sp}}(v_*\circ u^*) = \mathbb{X}_*u\circ \mathbb{X}v$ . For 2-morphisms, we define $\mathbb{X}_{\textrm{sp}}[f,\sigma ] : \mathbb{X}_*u\circ \mathbb{X}v\to \mathbb{X}_*u'\circ \mathbb{X}v'$ to be the mate under the adjunction $\mathbb{X}_*u\dashv \mathbb{X}u$ of the composite

\begin{eqnarray*} \mathbb{X}v\xrightarrow {\mathbb{X}\sigma } \mathbb{X}f\circ \mathbb{X}v'\xrightarrow {\mathbb{X}f\circ \eta _{u'}\circ \mathbb{X}v'}\mathbb{X}f\circ \mathbb{X}u'\circ \mathbb{X}_*u'\circ \mathbb{X}v' \cong \mathbb{X}u\circ \mathbb{X}_*u'\circ \mathbb{X}v' \end{eqnarray*}

where $\eta _{u'}$ is the unit for $\mathbb{X}_*{u'}\dashv \mathbb{X}u'$ . The composition-preservation constraints come from the pointwiseness in $\mathscr{R}$ -cocompleteness: referring to the diagram (3.11), we have

\begin{eqnarray*} \mathbb{X}_{\textrm{sp}}(v_*\circ u^*)\circ \mathbb{X}_{\textrm{sp}}(s_*\circ r^*) & \cong & \mathbb{X}_*u\circ \mathbb{X}v\circ \mathbb{X}_*r\circ \mathbb{X}s \\[5pt] & \cong & \mathbb{X}_*u\circ \mathbb{X}_*r_v\circ \mathbb{X}v_r\circ \mathbb{X}s \\[5pt] & \cong & \mathbb{X}_*(u\circ r_v)\circ \mathbb{X}(s\circ v_r) \\[5pt] & \cong & \mathbb{X}_{\textrm{sp}}((s\circ v_r)_* \circ (u\circ r_v)^*) \\[5pt] & \cong & \mathbb{X}_{\textrm{sp}}(s_*\circ {v_r}_* \circ {r_v}^*\circ u^*) \\[5pt] & \cong & \mathbb{X}_{\textrm{sp}}(s_*\circ r^*\circ v_*\circ u^*). \end{eqnarray*}

Clearly $\mathbb{X}_{\textrm{sp}}\circ (\!-\!)_*\simeq \mathbb{X}$ .

For any $\mathscr{R}$ -cocontinuous pseudonatural transformation $\theta : \mathbb{X}\to \mathbb{Y}$ , we have the diagram

for all $S\xrightarrow {u}U$ in $\mathscr{R}$ and $S\xrightarrow {v}V$ in $\mathscr{C}$ (using terminology from Proposition 1.4 for the mate of $\theta _u$ ). The pasted composite diagram exhibits the extension of $\theta$ to a pseudonatural transformation $\theta _{\textrm{sp}} : \mathbb{X}_{\textrm{sp}}\to \mathbb{Y}_{\textrm{sp}}$ . For a modification $\theta \Rightarrow \phi : \mathbb{X}\to \mathbb{Y}$ , we can keep the same components on objects to define the extended modification $\theta _{\textrm{sp}} \Rightarrow \phi _{\textrm{sp}} : \mathbb{X}_{\textrm{sp}}\to \mathbb{Y}_{\textrm{sp}}$ .

Let $\mathscr{L}$ be a protocalibration of the bicategory $\mathscr{C}^{\textrm{co}}$ . So we can define

\begin{equation*}_{\mathscr{L}}{\textrm{Spn}}{}\mathscr{C} = (\textrm{Spn}_{\mathscr{L}}\mathscr{C}^{\textrm{co}})^{\textrm{co}}.\end{equation*}

The 2-morphisms are represented by diagrams of the form (3.15).

(3.15)

For completeness, we state the dual form of Proposition 3.1.

Proposition 3.2. Restriction along the opposite of $(\!-\!)_* : \mathscr{C} \to _{\mathscr{L}}{\textrm{Spn}}{}\mathscr{C}$ defines a bicategorical equivalence

\begin{eqnarray*} \textrm{CAT}(_{\mathscr{L}}{\textrm{Spn}}{}\mathscr{C}) \sim _{\mathscr{L}}{\textrm{CAT}}{}\mathscr{C}. \end{eqnarray*}

4. A Distributive Law From Compatibility

Let $\mathscr{E}$ be a category with pullbacks. The functor defined by pulling back along $u : S\to U$ is denoted by $\Delta _u : \mathscr{E}_{/ U}\to \mathscr{E}_{/ S}$ . There is always a left adjoint to $\Delta _u$ denoted by $\Sigma _u$ and obtained by composition with $u$ . Recall that a morphism $u : S\to U$ is called powerful (or “exponentiable”) when $\Delta _u$ has a right adjoint $\Pi _u$ . Then, for each $a : A\to U$ , the counit of the adjunction $\Delta _u\dashv \Pi _u$ gives a morphism $e$ as in diagram (4.16).

(4.16)

Let $\mathscr{P}$ denote the set of powerful morphisms in $\mathscr{E}\,$ ; by Corollary 2.6 of Street & Verity (Reference Street and Verity2010), $\mathscr{P}$ is a protocalibration in the sense of Definition 1.2.

For this section, fix two protocalibrations $\mathscr{L}$ and $\mathscr{R}$ of the category $\mathscr{E}$ . We are particularly interested in the $\mathscr{R}$ -cocompletion and the $\mathscr{L}$ -completion. So, in this section, $\overrightarrow {\mathbb{X}}$ will mean the $\mathscr{R}$ -cocompletion of $\mathbb{X}\in \textrm{CAT}\mathscr{E}$ and $\overleftarrow {\mathbb{X}}$ will mean the $\mathscr{L}$ -completion of $\mathbb{X}$ .

Our Definition 4.1 is motivated by concepts in Blumberg-Hill (Reference Bénabou2018, Reference Bénabou2022) and Chan (Reference Chan2024).

(4.17)

Suppose $(\mathscr{L},\mathscr{R})$ is compatible. There is a bicategorical distributive law (in the sense of Marmolejo Reference Marmolejo1999) of the form

\begin{equation*}\lambda : \overleftarrow {(\!-\!)}\circ \overrightarrow {(\!-\!)} \longrightarrow \overrightarrow {(\!-\!)}\circ \overleftarrow {(\!-\!)} \end{equation*}

where the component functor

\begin{equation*}\lambda _U :\overleftarrow {\overrightarrow {\mathbb{X}}}U\longrightarrow \overrightarrow {\overleftarrow {\mathbb{X}}}U\end{equation*}

takes $(U\xleftarrow {u}S, S\xleftarrow {a}A, x \in \mathbb{X}A)$ , where $u\in \mathscr{L}$ and $a\in \mathscr{R}$ , to

\begin{equation*}(U\xleftarrow {\Pi _ua}B, B\xleftarrow {\bar {u}}P, (\mathbb{X}e)x\in \mathbb{X}P)\end{equation*}

in the notation of diagram (4.16).

Proof. Suppose $V\xrightarrow {r}U$ is in $\mathscr{L}$ . We will use the right adjoints $\Pi _r$ and $\mathbb{X}_!r$ available because of powerfulness and of $\mathscr{L}$ -completeness of $\mathbb{X}$ .

We must show that $\overrightarrow {\mathbb{X}}r$ has a right adjoint. For each $(V\xleftarrow {v}T,y\in \mathbb{X}T)\in \overrightarrow {\mathbb{X}}V$ , we must produce an object $(U\xleftarrow {v'}R,y'\in \mathbb{X}R)\in \overrightarrow {\mathbb{X}}U$ and a bijection

(4.18) \begin{eqnarray} \overrightarrow {\mathbb{X}}V((\Delta _ru,(\mathbb{X}\bar {r})x),(v,y)) \cong \overrightarrow {\mathbb{X}}U((u,x),(v',y')) \end{eqnarray}

natural in $(u,x)\in \overrightarrow {\mathbb{X}}U$ , where the following square is a pullback.

Take an element $(w,\xi )$ as shown in (4.19) of the left-hand side of (4.18).

(4.19)

Since $r$ is powerful, morphisms $w$ making the left triangle of (4.19) commute are in natural bijection with morphisms $w' : S \to R$ such that $\Pi _rv\circ w' = u$ where $R$ is the domain of $\Pi _rv$ . Since $\bar {r}$ is in $\mathscr{L}$ and $\mathbb{X}$ is $\mathscr{L}$ -complete, morphisms $\xi : (\mathbb{X}\bar {r})x\to (\mathbb{X}w)y$ are in natural bijection with morphisms $\bar {\xi } : x\to (\mathbb{X}_!\bar {r})(\mathbb{X}w)y$ . The diagram below shows how $w$ and $w'$ are related.

Also, we have the Chevalley-Beck isomorphism $(\mathbb{X}w') (\mathbb{X}_!\tilde {r}) \cong (\mathbb{X}_!\bar {r})(\mathbb{X}\Delta _{\tilde {r}}w')$ . Consequently, we have $(\mathbb{X}_!\bar {r})(\mathbb{X}w) \cong (\mathbb{X}_!\bar {r})(\mathbb{X}\Delta _{\tilde {r}}w')(\mathbb{X}e)\cong (\mathbb{X}w') (\mathbb{X}_!\tilde {r})(\mathbb{X}e)$ . So $\bar {\xi } : x\to (\mathbb{X}_!\bar {r})(\mathbb{X}w)y$ are in bijection with $\bar {\xi } : x\to (\mathbb{X}w') (\mathbb{X}_!\tilde {r})(\mathbb{X}e)y$ . Therefore, we have the desired bijection (4.18) with $v'=\Pi _rv$ and $y'= (\mathbb{X}_!\tilde {r})(\mathbb{X}e)y$ .

Since $\overrightarrow {\mathbb{X}}$ is $\mathscr{R}$ -cocomplete, it already satisfies the CB-condition for left adjoints $\overrightarrow {\mathbb{X}}_*r$ . By replacing all functors in that invertibility condition by their right adjoints when $r\in \mathscr{L}$ , we induce an invertible mate yielding the CB-condition for the right adjoints $\overrightarrow {\mathbb{X}}_!r$ . So $\overrightarrow {\mathbb{X}}$ is $\mathscr{L}$ -complete.

The second sentence of the proposition follows on checking that the canonical natural transformation

is invertible.

The pseudomonad $\overrightarrow {(\!-\!)}$ on $\textrm{CAT}\mathscr{E}$ therefore lifts to $_{\mathscr{L}}{\textrm{CAT}}{}\mathscr{E}$ .

(4.20)

The pseudo-algebras for the lifted pseudomonad are those of the composite pseudomonad $\mathbb{X}\mapsto \overrightarrow {\overleftarrow {\mathbb{X}}}$ on $\textrm{CAT}\mathscr{E}$ form the 2-category we denote by $_{\mathscr{L}}{\textrm{CAT}}{_{\mathscr{R}}}\mathscr{E}$ at the top of the diamond (4.21) of inclusion 2-functors. Using Lemma 4.3 and Proposition 3.1, we have biequivalences

\begin{eqnarray*} _{\mathscr{L}}{\textrm{CAT}}{_{\mathscr{R}}}\mathscr{E} \sim \textrm{CAT}_{\mathscr{R}_*}(_{\mathscr{L}}{\textrm{Spn}}{}\mathscr{E}\,)\sim \textrm{CAT}(\textrm{Spn}_{\mathscr{R}_*}(_{\mathscr{L}}{\textrm{Spn}}{}\mathscr{E}\,)). \end{eqnarray*}

While the composite pseudomonad is not Kock-Zöberlein or dual Kock-Zöberlein, the pseudo-algebra structures are unique up to isomorphism: it is a property of an $\mathscr{E}$ -variable category to be a pseudo-algebra for both $\overleftarrow {(\!-\!)}$ and $\overrightarrow {(\!-\!)}$ , and to satisfy the distributive law.

(4.21)

5. Polynomials as Spans of Spans

Let $\mathscr{E}$ be a category with pullbacks.

There is a bicategory $\textrm{Ply}\mathscr{E}$ of polynomials in $\mathscr{E}\,$ ; in the notation of Walker (Reference Walker2019) it is $\textrm{Poly}(\mathscr{E}\,)$ and in the notation of Proposition 8.4 of Street (Reference Street2020) it is the bicategory $\textrm{PolySpn}{\mathscr{E}}$ . The objects are those of $\mathscr{E}$ . The morphisms from $X$ to $Y$ are polynomials $p = (X\xleftarrow {r} A\xrightarrow {n}B\xrightarrow {t} Y)$ in $\mathscr{E}$ with $n$ powerful. The morphisms between polynomials are as in Subsection 2.11 of Gambino and Kock (Reference Gambino and Kock2013) and Definition 8.2 of Street (Reference Street2020): they are isomorphism classes of commutative diagrams

(5.22)

wherein the square as indicated is a pullback. Composition uses bipullback of spans of spans (see Gambino and Kock Reference Gambino and Kock2013; Street Reference Street2020; Weber Reference Weber2015). As a consequence of Theorem 5.1 and Proposition 6.6 below, finite coproducts exist in each $\textrm{Ply}{\mathscr{E}}(X,Y)$ and are preserved by precomposition with morphisms, yet generally not by postcomposition making the bicategory $\textrm{Ply}{\mathscr{E}}$ enriched in a certain skew-monoidal bicategory $\textrm{Cat}_{+ \textrm{sk}}$ (extending a bit the terminology of Garner and Lemay Reference Garner and Lemay2021).

Recall the philosophy of Street (Reference Street2020) that it is useful to regard $p$ as a span

(5.23) \begin{eqnarray} p = (X\xleftarrow {(n,A,r)} B\xrightarrow {(1_B,B,t)} Y) \end{eqnarray}

in $\textrm{Spn}\mathscr{E}$ . Then the morphisms interpret as diagrams

(5.24)

We modify this a bit by taking a compatible pair $(\mathscr{L},\mathscr{R})$ of protocalibrations on $\mathscr{E}$ and restricting the polynomials (5.23) to those with $n\in \mathscr{L}$ and $t\in \mathscr{R}$ . Restricting the morphisms of $\textrm{Ply}{\mathscr{E}}$ in this way, we obtain a sub-bicategory which we denote by $_{\mathscr{L}}{\textrm{Ply}}{_{\mathscr{R}}}\mathscr{E}$ .

Theorem 5.1. If $(\mathscr{L},\mathscr{R})$ is a compatible pair of protocalibrations on the category $\mathscr{E}$ and ${{\mathscr{R}}_*}$ is as in Lemma 4.3 then

\begin{equation*}(_{\mathscr{L}}{\textrm{Ply}}{_{\mathscr{R}}}\mathscr{E}\,)^{\textrm{op}} = \textrm{Spn}_{{{\mathscr{R}}_*}}(_{\mathscr{L}}{\textrm{Spn}}{}\mathscr{E}\,).\end{equation*}

Proof. The 2-morphisms of $\textrm{Spn}_{\mathscr{R}_*}(_{\mathscr{L}}{\textrm{Spn}}{}\mathscr{E}\,)$ are isomorphism classes of morphisms as pictured in (5.25).

(5.25)

Unpacking this shows that the data involved is exactly as in the diagram (5.22) in $\mathscr{E}$ . While unpacking, just note that the 2-morphism $\lambda$ in $_{\mathscr{L}}{\textrm{Spn}}{}\mathscr{E}$ is in the reverse direction as a morphism of spans.

Proof. Using Theorem 5.1 and Propositions 3.1 and 3.2, we have

\begin{eqnarray*} \phantom {aaaaaaaaaa} \textrm{Ps}(_{\mathscr{L}}{\textrm{Ply}}{_{\mathscr{R}}}\mathscr{E}, \textrm{Cat}) &\sim & \textrm{CAT}(\textrm{Spn}_{{{\mathscr{R}}_*}}(_{\mathscr{L}}{\textrm{Spn}}{}\mathscr{E}\,)) \\ &\sim & \textrm{CAT}_{\mathscr{R}_*}(_{\mathscr{L}}{\textrm{Spn}}{}\mathscr{E}\,) \\ & \sim & _{\mathscr{L}}{\textrm{CAT}}{_{\mathscr{R}}}\mathscr{E} \ . \ \phantom {aaaaaaaaaaaaaaaa} \end{eqnarray*}

(5.26)

6. Finite Coproducts for Spans and Polynomials

Examples of the concepts of this section are provided by lextensive categories. Recall that a category $\mathscr{E}$ is lextensive (in the sense used by Schanuel Reference Schanuel2000) when it has finite limits, finite coproducts, and, equivalences

\begin{equation*}\mathbf{1} \simeq \mathscr{E}_{/0} \ \text{ and } \ \mathscr{E}_{/U}\times \mathscr{E}_{/V}\simeq \mathscr{E}_{/U+V}.\end{equation*}

The second equivalence takes an object $(A\xrightarrow {h}U, B\xrightarrow {k}V)\in \mathscr{E}_{/U}\times \mathscr{E}_{/V}$ to the object $(A+B\xrightarrow {h+k}U+V)\in \mathscr{E}_{/U+V}$ ; the fact that this inverse equivalence is fully faithful means that a commutative triangle

(6.27)

implies there are unique $r : A\to A'$ and $s : B\to B'$ such that $f = r+s$ , $h = h'r$ and $h'=k's$ .

It follows that the initial object $0$ is strictly initial and, for every coproduct diagram $U\xrightarrow {i}U+V\xleftarrow {j}V$ in $\mathscr{E}$ and every $R\xrightarrow {f}U+V$ , there is a diagram

(6.28)

in which the squares are pullbacks and the top row is also a coproduct diagram.

This should motivate the following bicategorical concepts.

Suppose the bicategory $\mathscr{C}$ admits the binary bicoproduct $U\xrightarrow {i}U+V\xleftarrow {j}V$ and $\mathscr{R}$ is a protocalibration of $\mathscr{C}$ . For the pseudofunctor $\mathbb{C}_{\mathscr{R}} : \mathscr{C}^{\textrm{op}}\to \textrm{Cat}$ of Definition 2.2, we have a canonical comparison functor

\begin{equation*}\mathbb{C}_{\mathscr{R}}(U+V)\to \mathbb{C}_{\mathscr{R}}U\times \mathbb{C}_{\mathscr{R}}V\end{equation*}

taking $Z\xrightarrow {w}U+V$ to the pair $(X\xrightarrow {w_i}U, Y\xrightarrow {w_j}V)$ defined by bipullbacks

Lemma 6.1. Suppose the bicategory $\mathscr{C}$ admits finite bicoproducts and the protocalibration $\mathscr{R}$ is closed under them. Then the canonical comparison functors

\begin{eqnarray*} \mathbb{C}_{\mathscr{R}}0 \to \mathbf{1} \ \text{ and } \ \mathbb{C}_{\mathscr{R}}(U+V)\to \mathbb{C}_{\mathscr{R}}U\times \mathbb{C}_{\mathscr{R}}V \end{eqnarray*}

have left adjoints, respectively, defined on objects by

\begin{equation*} (0\in \mathbf{1}) \mapsto (0\xrightarrow {1_0}0) \ \text{ and } \ (X\xrightarrow {u\in \mathscr{R}}U, Y\xrightarrow {v\in \mathscr{R}}V)\mapsto (X+Y\xrightarrow {u+v\in \mathscr{R}}U+V).\end{equation*}

Definition 6.2. Suppose $\mathscr{R}$ is a protocalibration of the bicategory $\mathscr{C}$ . We say a bicategory $\mathscr{C}$ is $\mathscr{R}$ -extensive when it admits finite bicoproducts and these are taken into finite biproducts by the pseudofunctor $\mathbb{C}_{\mathscr{R}} : \mathscr{C}^{\textrm{op}}\to \textrm{Cat}$ of Definition 2.2.

\begin{eqnarray*} \mathbb{C}_{\mathscr{R}}0 \simeq \mathbf{1} \ \text{ and } \ \mathbb{C}_{\mathscr{R}}(U+V)\simeq \mathbb{C}_{\mathscr{R}}U\times \mathbb{C}_{\mathscr{R}}V \end{eqnarray*}

When $\mathscr{R}$ is closed under finite bicoproducts, the inverse equivalences are the left adjoints in Definition 6.1.

Proof. The bi-initial object of $\textrm{Spn}_{\mathscr{R}}\mathscr{C}$ is that of $\mathscr{C}$ denoted $0$ . For the 1-morphism part of the universal property: if $0\xleftarrow {u}S\xrightarrow {v}Y$ is a left $\mathscr{R}$ -span then $u$ is an equivalence (using $\mathbb{C}_{\mathscr{R}}0 \sim \mathbf{1}$ ); so $S$ is bi-initial and we conclude that all such spans are isomorphic in $\textrm{Spn}_{\mathscr{R}}\mathscr{C}$ while the 2-morphism property of $0$ follows from that of $0\in \mathscr{C}$ .

We will show that a bicoproduct $U\xrightarrow {i}U+V\xleftarrow {j}V$ in $\mathscr{C}$ gives one $U\xrightarrow {i_*}U+V\xleftarrow {j_*}V$ in $\textrm{Spn}_{\mathscr{R}}\mathscr{C}$ . Note that coprojections $i$ , $j$ and codiagonals are in $\mathscr{R}$ . Take spans $U\xleftarrow {u}S\xrightarrow {v}W$ and $V\xleftarrow {r}T\xrightarrow {s}W$ with $u, r\in \mathscr{R}$ (and hence $u+r \in \mathscr{R}$ ). We obtain $U+V\xleftarrow {u+r\in \mathscr{R}}S+T\xrightarrow {[v,s]}W$ whose composites with $i_*$ and $j_*$ are isomorphic to the given two spans because our assumptions give the bipullbacks

The 2-morphism part of the universal property uses $\mathscr{R}$ -extensivity and the properties of $\mathscr{R}$ :

The coproduct of $X\xleftarrow {u\in \mathscr{R}}S\xrightarrow {v}Y$ and $X\xleftarrow {u'\in \mathscr{R}}S'\xrightarrow {v'}Y$ in $\textrm{Spn}_{\mathscr{R}}\mathscr{C}(X,Y)$ is $X\xleftarrow {[u,u']\in \mathscr{R}}S+S'\xrightarrow {[v,v']}Y$ . Contemplation of the diagram

(where the diamond is a bipullback) reveals the claimed right preservation property.

For any bicategory $\mathscr{D}$ admitting finite bicoproducts, we write $\textrm{fpCAT}\mathscr{D}$ for the full sub-bicategory of $\textrm{CAT}\mathscr{D}$ consisting of the finite biproduct preserving pseudofunctors $\mathscr{D}^{\textrm{op}}\to \textrm{Cat}$ . If $\mathscr{T}$ is a protocalibration of $\mathscr{D}$ , we write $\textrm{fpCAT}_{\mathscr{T}}\mathscr{D}$ for the intersection of $\textrm{fpCAT}\mathscr{D}$ and $\textrm{CAT}_{\mathscr{T}}\mathscr{D}$ .

Proposition 6.7. Under the hypotheses of Proposition 6.6, the biequivalence of Proposition 3.1 restricts to a biequivalence

\begin{eqnarray*} \textrm{fpCAT}(\textrm{Spn}_{\mathscr{R}}\mathscr{C}) \sim \textrm{fpCAT}_{\mathscr{R}}\mathscr{C}. \end{eqnarray*}

Proof. For this proof, put $\mathscr{C} = _{\mathscr{L}}{\textrm{Spn}}{}\mathscr{E}$ . The injections into a bicoproduct and the codiagonals are in $\mathscr{R}_*$ . The bicoproduct of $f_*$ and $f'_*$ is isomorphic to $(f+f')_*$ and so in $\mathscr{R}_*$ . It remains to prove $\mathbb{C}_{\mathscr{R}_*}$ preserves biproducts. Up to equivalence, we can take the objects of $\mathbb{C}_{\mathscr{R}_*}U$ to be morphisms $X\xrightarrow {a}U$ in $\mathscr{R}$ since every morphism $X\to U$ in $\mathscr{R}_*$ is isomorphic to some $X\xrightarrow {a_*}U$ with $a\in \mathscr{R}$ . Moreover, every morphism

in $\mathbb{C}_{\mathscr{R}_*}U$ has a morphism of the form $f_*$ in the isomorphism class $[u,S,v]$ . Thus we have an equivalence of categories $\mathbb{C}_{\mathscr{R}_*}U\simeq \mathbb{E}_{\mathscr{R}}U$ .Footnote ³ Also, the canonical functor $\mathbb{C}_{\mathscr{R}_*}(U+V)\to \mathbb{C}_{\mathscr{R}_*}U \times \mathbb{C}_{\mathscr{R}_*}V$ transports across the equivalences to the lextensivity equivalence $\mathbb{E}_{\mathscr{R}}(U+V) \simeq \mathbb{E}_{\mathscr{R}}U\times \mathbb{E}_{\mathscr{R}}V$ .

As a corollary of Theorem 5.1, Proposition 6.6 and Lemma 6.9, we obtain:

7. Objective Mackey Functors

Suppose the bicategory $\mathscr{C}$ is finitely bicocomplete and $\mathscr{R}$ -extensive for a protocalibration $\mathscr{R}$ which, when regarded as a wide sub-bicategory of $\mathscr{C}$ , is closed under finite bicoproducts.

Definition 7.1. An objective $\mathscr{R}$ -Mackey functor on $\mathscr{C}$ is an $\mathscr{R}$ -cocomplete pseudofunctor $\mathbb{M} : \mathscr{C}^{\textrm{op}} \to \textrm{Cat}$ which preserves finite biproducts (in the bicategorical sense). We define the bicategory of objective $\mathscr{R}$ -Mackey functors by $\textrm{OMky}_{\mathscr{R}}\mathscr{C} = \textrm{fpCAT}_{\mathscr{R}}\mathscr{C}$ . By Proposition 6.7,

(7.29) \begin{eqnarray} \textrm{fpCAT}(\textrm{Spn}_{\mathscr{R}}\mathscr{C}) \sim \textrm{OMky}_{\mathscr{R}}\mathscr{C}. \end{eqnarray}

When $\mathscr{R} = \mathscr{C}$ , we drop the subscripts $\mathscr{C}$ and refer to objective Mackey functors.

Example 7.3. In the terminology of Proposition 1.8, if $\mathbb{Z}$ is an objective $\mathscr{R}$ -Mackey functor, so is $[\mathbb{Y},\mathbb{Z}]_{\mathscr{R}}$ . For, we have

\begin{eqnarray*} [\mathbb{Y},\mathbb{Z}]_{\mathscr{R}}(W+W') & \simeq & \textrm{CAT}_{\mathscr{R}}\mathscr{E}(\mathbb{Y},\mathbb{Z}(\!-\times (W+W')) \\ & \simeq & \textrm{CAT}_{\mathscr{R}}\mathscr{E}(\mathbb{Y},\mathbb{Z}(\!-\times W)\times \mathbb{Z}(\!-\times W')) \\ & \simeq & \textrm{CAT}_{\mathscr{R}}\mathscr{E}(\mathbb{Y},\mathbb{Z}(\!-\times W)) \times \textrm{CAT}_{\mathscr{R}}\mathscr{E}(\mathbb{Y}, \mathbb{Z}(\!-\times W')) \\ & \simeq & [\mathbb{Y},\mathbb{Z}]_{\mathscr{R}}W\times [\mathbb{Y},\mathbb{Z}]_{\mathscr{R}}W'. \end{eqnarray*}

(We are using the last sentence of Corollary 2.7 in the third step.)

As in Example 6.8, we consider the symmetric closed monoidal bicategory $\textrm{Cat}_{+}$ of categories with finite coproducts and finite-coproduct-preserving functors; the internal hom $[\mathscr{A},\mathscr{B}]_{+}$ of $\textrm{Cat}_{+}$ is $\textrm{Cat}_{+}(\mathscr{A},\mathscr{B})$ with pointwise coproduct. Cartesian product $A\times B$ of categories is a direct sum in $\textrm{Cat}_{+}$ and the terminal category $\mathbf{1}$ is also initial.

Proof. By assumption, the codiagonal $\nabla : U+U\to U$ and unique $! : 0\to U$ are in $\mathscr{R}$ . The diagonal of $\mathbb{M}U$ is isomorphic to the composite

\begin{equation*}\mathbb{M}U \xrightarrow {\mathbb{M}\nabla } \mathbb{M}(U+U)\simeq \mathbb{M}U\times \mathbb{M}U\end{equation*}

and so has a left adjoint by Proposition 1.4. Thus, $\mathbb{M}U$ has binary coproducts. Similarly, $\mathbb{M}U \xrightarrow {\mathbb{M}!} \mathbb{M}0\simeq \mathbf{1}$ has a left adjoint so $\mathbb{M}U$ has an initial object. For $f : U\to V$ , the functor $\mathbb{M}f$ preserves these finite coproducts as can be seen by applying the CB property to the pullbacks

in our $\mathscr{R}$ -extensive bicategory $\mathscr{C}$ .

Corollary 7.6. If $\mathbb{M}$ is an objective $\mathscr{R}$ -Mackey functor and $U\xrightarrow {i} U+V\xleftarrow {j}V$ is a coproduct in $\mathscr{C}$ then

exhibits a (bicategorical) direct sum in $\textrm{Cat}_{+}$ .

Corollary 7.7. For all $f : X'\to X$ in $\mathscr{C}$ , the functor $\mathbb{C}_{\mathscr{R}}f : \mathbb{C}_{\mathscr{R}}X \to \mathbb{C}_{\mathscr{R}}X'$ preserves finite coproducts and the pseudofunctor

\begin{eqnarray*} \mathscr{C}^{\textrm{op}}\times \mathscr{C}\xrightarrow {(\!-\!)_*\times (\!-\!)_*} (\textrm{Spn}_{\mathscr{R}})^{\textrm{op}}\times \textrm{Spn}_{\mathscr{R}}\xrightarrow {\textrm{Spn}_{\mathscr{R}}(\!-,-)}\textrm{Cat} \end{eqnarray*}

factors through the inclusion $\textrm{Cat}_{+}\hookrightarrow \textrm{Cat}$ of the sub-2-category of categories with finite coproducts and functors which preserve them.

We end this section with a consequence of Proposition 6.5.

8. A Monoidal Structure on $\textrm{OMky}_{\mathscr{R}}\mathscr{E}$

The construction $[\mathbb{Y},\mathbb{Z}]_{\mathscr{R}}$ defined and discussed in Proposition 1.8 and Example 7.3 provides a closed structure (in the sense of Day and Street Reference Day and Street1997) on the bicategory $\textrm{OMky}_{\mathscr{R}}\mathscr{C}$ . The closed structure is symmetric monoidal: there is a symmetric tensor product $\mathbb{M}\star \mathbb{N}$ defined by a pseudonatural equivalence

\begin{eqnarray*} \textrm{OMky}_{\mathscr{R}}\mathscr{C} (\mathbb{M}\star \mathbb{N}, \mathbb{L}) \simeq \textrm{OMky}_{\mathscr{R}}\mathscr{C} (\mathbb{M}, [\mathbb{N},\mathbb{L}]_{\mathscr{R}}). \end{eqnarray*}

There is a bicategorically universal pseudonatural transformation

\begin{equation*}\omega _{\mathbb{M},\mathbb{N}} : \mathbb{M}\times \mathbb{N} \to \mathbb{M}\star \mathbb{N}\end{equation*}

which is $\mathscr{R}$ -cocontinuous in each variable separately. The unit for this tensor product is the objective $\mathscr{R}$ -Burnside functor $\mathbb{C}_{\mathscr{R}}$ (see Example (7.2)); for

\begin{eqnarray*} [\mathbb{C}_{\mathscr{R}},\mathbb{L}]_{\mathscr{R}} & = & \textrm{CAT}_{\mathscr{R}}\mathscr{C}(\overrightarrow {\mathbf{1}},\mathbb{L}(\!-\times W)) \\ & \simeq & \textrm{CAT}\mathscr{C}(\mathbf{1},\mathbb{L}(\!-\times W)) \\ & \simeq & \textrm{lim} \ \mathbb{L}(\!-\times W) \\ & \simeq & \mathbb{L}(\mathbf{1}\times W) \\ & \simeq & \mathbb{L}W \end{eqnarray*}

where the (bicategorical) limit is over the category $\mathscr{C}^{\textrm{op}}$ which has the (bicategorical) initial object $\mathbf{1}$ .

The monoidales (pseudomonoids) for the convolution monoidal structure $\mathbb{A}\star \mathbb{B}$ we call objective $\mathscr{R}$ -Green functors on $\mathscr{C}$ and write $\textrm{OGrn}_{\mathscr{R}}\mathscr{C}$ for the bicategory they, with the strong monoidal morphisms, form. An objective $\mathscr{R}$ -Green functor is symmetric when it is symmetric as a monoidale.

The tensor unit for a symmetric monoidal structure is always a symmetric monoidale. So the objective $\mathscr{R}$ -Burnside functor $\mathbb{C}_{\mathscr{R}}$ is an example of an objective $\mathscr{R}$ -Green functor.

Let us now describe the symmetric monoidal structure obtained by transporting the one on $\textrm{OMky}_{\mathscr{R}}\mathscr{C}$ across the biequivalence (7.29).

By Lemma 1.6, we see that finite product in $\mathscr{C}$ supplies the bicategory $\textrm{Spn}_{\mathscr{R}}\mathscr{C}$ with a symmetric monoidal structure. Using Proposition 7.4, we see that the bicategory $\textrm{fpCATSpn}_{\mathscr{R}}\mathscr{C}$ is the $\textrm{Cat}_{+}$ -enriched bicategory $\textrm{Ps}((\textrm{Spn}_{\mathscr{R}}\mathscr{C})^{\textrm{op}}, \textrm{Cat}_{+})_{+}$ of such $\textrm{Cat}_{+}$ -enriched pseudofunctors. The desired transported symmetric closed monoidal structure $\mathbb{A}\star \mathbb{B}$ on $\textrm{fpCATSpn}_{\mathscr{R}}\mathscr{C}$ is the Day convolution structure with internal hom $[\mathbb{B},\mathbb{D}]$ defined by

\begin{eqnarray*} [\mathbb{B},\mathbb{D}]W \ \simeq \ \textrm{CAT}(\textrm{Spn}_{\mathscr{R}}\mathscr{C})(\mathbb{B},\mathbb{C}(\!-\times W)). \end{eqnarray*}

9. An Objective Mackey Functor of Mackey Functors

Take a lextensive category $\mathscr{E}$ equipped with a protocalibration $\mathscr{R}$ closed under finite coproducts. We will consider the bicategory $\textrm{Spn}_{\mathscr{R}}{\mathscr{E}}$ in the notation of Section 3 and will write $\textrm{clSpn}_{\mathscr{R}}{\mathscr{E}}$ for the classifying category of $\textrm{Spn}_{\mathscr{R}}{\mathscr{E}}$ ; the morphisms are isomorphism classes of spans (see Bénabou Reference Blumberg and Hill1967). We write $\textrm{Spn}{\mathscr{E}}$ and $\textrm{clSpn}{\mathscr{E}}$ for $\textrm{clSpn}_{\mathscr{R}}{\mathscr{E}}$ and $\textrm{clSpn}_{\mathscr{R}}{\mathscr{E}}$ in the case where $\mathscr{R} = \mathscr{E}$ .

As we have pointed out, the bicategory $\textrm{Spn}_{\mathscr{R}}{\mathscr{E}}$ is enriched in the closed symmetric monoidal bicategory $\textrm{Cat}_{+}$ of categories with finite coproducts and finite-coproduct-preserving functors. The symmetric monoidal structure on the bicategory $\textrm{Spn}_{\mathscr{R}}{\mathscr{E}}$ is $\textrm{Cat}_{+}$ -enriched. The more abstract category $\textrm{clSpn}_{\mathscr{R}}{\mathscr{E}}$ , along with its symmetric monoidal structure, is enriched in the usual symmetric closed monoidal category $\textrm{CMon}$ of commutative monoids.

An $\mathscr{R}$ -Mackey functor on $\mathscr{E}$ is a $\textrm{CMon}$ -enriched (or equally, a finite-product-preserving) functor

\begin{equation*}M : (\textrm{clSpn}_{\mathscr{R}}{\mathscr{E}})^{\textrm{op}}\to \textrm{CMon}.\end{equation*}

This is consistent with Lindner (Reference Lindner1976) who looked at the case $\mathscr{R}=\mathscr{E}$ .

Put

\begin{equation*}\textrm{Mky}_{\mathscr{R}}\mathscr{E} = [(\textrm{clSpn}_{\mathscr{R}}{\mathscr{E}})^{\textrm{op}},\textrm{CMon}]_+ \ ,\end{equation*}

the $\textrm{CMon}$ -enriched functor category. It is a monoidal $\textrm{CMon}$ -enriched category by Day convolution (Day Reference Day1970). The monoids for this convolution structure are $\mathscr{R}$ -Green functors on $\mathscr{E}$ ; see Panchadcharam and Street (Reference Panchadcharam and Street2007) for more details in the case $\mathscr{R} = \mathscr{E}$ . We write $\textrm{Grn}_{\mathscr{R}}\mathscr{E}$ for the $\textrm{CMon}$ -enriched category of Green functors. The Burnside functor $E_{\mathscr{R}}$ is an example.

Let $\textrm{RLxt}$ denote the 2-category whose objects are pairs $(\mathscr{A} , \mathscr{O})$ consisting of a lextensive category $\mathscr{A}$ and a protocalibration $\mathscr{O}$ closed under finite coproducts, and, whose morphisms $F : (\mathscr{A} , \mathscr{O})\to (\mathscr{A}' , \mathscr{O}')$ are functors $F : \mathscr{A}\to \mathscr{A}'$ preserving pullback and finite coproducts, and taking $\mathscr{O}$ to $\mathscr{O}'$ . Then we have a pseudofunctor

\begin{equation*}\textrm{Mky}_{R} : \textrm{RLxt} \to \textrm{Cat}_{+}\end{equation*}

taking $(\mathscr{A} , \mathscr{O})$ to $\textrm{Mky}_{\mathscr{O}}\mathscr{A}$ regarded as a category with finite coproducts.

It is useful to notice that the objective Burnside functor $\mathbb{E}_{\mathscr{R}} : \mathscr{E}^{\textrm{op}}\to \textrm{Cat}_+$ lands in the category $\textrm{RLxt}$ over $\textrm{Cat}_{+}$ by equipping each lextensive category $\mathbb{E}_{\mathscr{R}}(X) (\subseteq \mathscr{E}_{/X})$ with the protocalibration $\mathscr{R}_{/X}$ .

Proposition 9.2. The composite pseudofunctor

\begin{equation*}\textrm{Mky}_{\textrm{R}\mathscr{R}} : = \mathscr{E}^{\textrm{op}}\xrightarrow {\mathbb{E}_{\mathscr{R}}}\textrm{RLxt}\xrightarrow {\textrm{Mky}_{R}}\textrm{Cat}_{+}\end{equation*}

becomes a symmetric objective $\mathscr{R}$ -Green functor when equipped with the multiplication induced by the finite-coproduct-preserving-in-each-variable functors

\begin{equation*}\textrm{Mky}_{\textrm{R}}\mathbb{E}_{\mathscr{R}}X \times \textrm{Mky}_{\textrm{R}}\mathbb{E}_{\mathscr{R}}Y \to \textrm{Mky}_{\textrm{R}}\mathbb{E}_{\mathscr{R}}(X\times Y)\end{equation*}

taking $(M,N)$ to $M\boxtimes N$ where

\begin{equation*}(M\boxtimes N)(S\xrightarrow {(a,b)}X\times Y) = M(S\xrightarrow {a}X)\otimes N(S\xrightarrow {b}Y).\end{equation*}

10. Objective Tambara Functors

Suppose $\mathscr{E}$ is finitely complete and the pair $(\mathscr{L},\mathscr{R})$ satisfies the hypotheses of Lemma 6.9.

Definition 10.1. An objective $(\mathscr{L},\mathscr{R})$ -Tambara functor on $\mathscr{E}$ is a finite biproduct preserving pseudofunctor $\mathbb{T} : \mathscr{E}^{\textrm{op}} \to \textrm{Cat}$ which is in $_{\mathscr{L}}{\textrm{CAT}}_{\mathscr{R}}\mathscr{E}$ . We define the bicategory of objective $(\mathscr{L},\mathscr{R})$ -Tambara functors by

\begin{equation*}_{\mathscr{L}}{\textrm{Tmb}}_{\mathscr{R}}\mathscr{E} : = \textrm{fp}_{\mathscr{L}}{\textrm{CAT}}_{\mathscr{R}}\mathscr{E}.\end{equation*}

By Proposition 6.7 and Theorem 5.1,

(10.30) \begin{eqnarray} \textrm{fpCAT}\left ( (_{\mathscr{L}}{\textrm{Ply}}_{\mathscr{R}}\mathscr{E}\,)^{\textrm{op}}\right ) \sim _{\mathscr{L}}{\textrm{Tmb}}_{\mathscr{R}}\mathscr{E}. \end{eqnarray}

We write $\mathbb{T}_{\textrm{pl}} : \textrm{Ply}\mathscr{E} \to \textrm{Cat}$ for the inverse image of $\mathbb{T}$ under this biequivalence.

Therefore, Proposition 6.5 applies to both $\mathscr{R}$ and $\mathscr{L}$ yielding:

An $(\mathscr{L},\mathscr{R})$ -Tambara functor on $\mathscr{E}$ is a finite-product-preserving functor

\begin{equation*}T : \textrm{cl}_{\mathscr{L}}{\textrm{Ply}}_{\mathscr{R}}\mathscr{E}\to \textrm{CMon}.\end{equation*}

This concept, for the case $\mathscr{L}=\mathscr{R}=\mathscr{E}= G\text{-}\textrm{set}$ , arose in the paper by Tambara (Reference Tambara1993); also see Chan (Reference Chan2024), Hoyer (Reference Hoyer2014), and Mazur (Reference Mazur2013) for a version of $\mathscr{L}$ and $\mathscr{R}$ .