The idea I have in these notes is to expand the notion of cartesian bicategory to the tricategorical context, using spans of groupoids as an illustrative example. One motivation is that the symmetric monoidal structure follows straight away from the cartesian structure, if we assume that 2-products or 3-products are symmetric monoidal (which is true almost by fiat!). Another motivation is that a lot is known about cartesian bicategories, and the methodology seems ripe for extending this notion to tricategories.
The contents of this article are as follows. First we warm up by recalling some of the basic theory of cartesian bicategories, including their property-like nature and their symmetric monoidal structure, and the fact that $Span(C)$, for $C$ a finitely complete category, is a cartesian bicategory. Then, we reprise this development but one dimension higher, first defining cartesian tricategories, sketching their basic structure, and then applying it to the tricategory of spans of groupoids.
A cartesian bicategory is a “bicategory of generalized relations” like $Rel$ or $Span$ or $Prof$, which carries a “product-like” tensor product, that is, a monoidal product which becomes an honest 2-product when restricted to a subbicategory like $Set$ or $Cat$, namely the subbicategory whose 1-cells are the left adjoint 1-cells in the original bicategory.
The very brief description above can be turned into a definition as follows: following Carboni and Walters, define a map in a bicategory $B$ to be a 1-cell in $B$ that has a right adjoint. $Map(B)$ denotes the subbicategory whose 1-cells are precisely the maps in $B$, with all 2-cells between them.
A cartesian structure on $B$ consists of
2-functors $\otimes: B \times B \to B$ and $I: \mathbf{1} \to B$, where $\mathbf{1}$ is the terminal 2-category;
Map-valued lax transformations
where $\Delta: B \to B \times B$ is the diagonal 2-functor and $!: B \to \mathbf{1}$ is the unique 2-functor;
Invertible modifications
satisfying the triangulator coherence conditions.
A cartesian bicategory is a bicategory with a cartesian structure.
Here, we assume throughout that $n$-categorical concepts are weak $n$-categorical, e.g., by a “2-functor”, we mean a weak 2-functor, etc. Under our conventions, a lax transformation $\theta: F \to G$ between 2-functors has (not necessarily invertible) structure 2-cells pointing in the direction
(This convention is opposite to Bénabou’s convention.) There is a
The structure cell $\theta \cdot f$ is invertible if $f$ is a map in $B$.
We say a (lax) transformation $\theta$ is map-valued if each of the components $\theta a$ is a map.
Any 2-functor $F: B \to C$ takes maps to maps, hence induces a 2-functor $Map(F): Map(B) \to Map(C)$. It follows immediately from this and the lemma that on restriction of a cartesian structure to $Map(B)$, the 2-functor
is right biadjoint to the 2-functor
because the famous lemma ensures that the unit and counit
are strong (i.e., pseudo) transformations, which taken together with invertible triangulator 2-cells $s$, $t$ makes $\otimes_|$ precisely a right 2-adjoint to the diagonal $\Delta$, i.e., a 2-product on $Map(B)$. By similar reasoning, the 2-functor $I: \mathbf{1} \to Map(B)$ becomes a right 2-adjoint to $!: Map(B) \to \mathbf{1}$, i.e., a 2-terminal object of $Map(B)$.
Suppose $B$ has a cartesian structure. The symmetric monoidal 2-category structure on $Map(B)$, whose monoidal product is the 2-product on $Map(B)$, extends to a symmetric monoidal 2-category structure on all of $B$.
What does it take to exhibit a symmetric monoidal 2-category structure on $B$? One needs
2-functors $\otimes: B \times B \to B$, $I: \mathbf{1} \to B$;
Strong equivalences $\alpha$, $\lambda$, $\rho$, $\sigma$ (associativity, unit, and symmetry constraints);
A slew of invertible modifications such as $R_{a|b, c}$ (which promote the classical coherence equations for symmetric monoidal categories to isomorphisms);
A bunch of new coherence conditions involving various pasting diagrams of the modification data and equalities between the pastings.
It’s a complicated set of data and axioms, but the following proof shows the whole thing can be finessed if we simply accept that 2-products are symmetric monoidal, according to any reasonable definition of that notion. (Alternatively, Max Kelly has already proven this fact about 2-products, so we could just quote him.)
First, given objects $a, b, c$ of $B$, we may regard them as objects of $Map(B)$, where there is an associativity constraint
on $Map(B)$ which is definable by exploiting 2-universal properties of the 2-product. The associativity thus has an expression in terms of $\otimes$, $\delta$, and $\pi$, which are globally defined on $B$, hence $\alpha$ is globally defined as a lax transformation on $B$. By similar considerations, the symmetry and unit constraints on $Map(B)$ also extend to globally defined lax transformations on $B$. We argue in a moment that these constraints are strong (adjoint) equivalences.
Symmetric monoidal structure on $B$ also demands various structural modifications (such as a Yang-Baxter modification $R_{\bullet |\bullet, \bullet}$) satisfying various coherence conditions, but the components of such modifications ($R_{a|b, c}$, say) are defined by regarding their arguments $a, b, c$ as objects of $Map(B)$ and using the corresponding modifications there. Again, each such modification on $Map(B)$ is defined in terms of 2-adjunction data $\otimes$, $I$, $\delta$, $\pi$, $\varepsilon$, $s$, $t$, $u$ which are globally defined on $B$, so each such structure is a modification on $B$. Various coherence conditions on the modifications must be checked, but the conditions hold at every choice of objects of $B$ by regarding them as objects of $Map(B)$ and using the symmetric monoidal structure there, so the conditions hold on $B$.
Now we check that the structural transformations $\alpha$ are strong adjoint equivalences. Indeed, when regarded as being defined on $Map(B)$, the constraint $\alpha$ is an equivalence and so has a left adjoint (with invertible unit and counit) $\alpha^-$ with components
and just like $\alpha$, $\alpha^-$ is definable in terms of global structure on $B$, making $\alpha^-$ a transformation on $B$. Then $\alpha$, and by similar reasoning the symmetry and unit constraints, are strong transformations on $B$ by the lemma which follows.
For 2-categories $B$, $C$, let $Hom_l(C, B)$ denote the 2-category of 2-functors, lax transformations, and modifications. Let $Hom_s(C, B)$ denote the 2-category of 2-functors, strong transformations, and modifications.
If $\alpha$ is a right adjoint in $Hom_l(C, B)$, then the transformation $\alpha$ is strong. Consequently, if $\alpha^- \dashv \alpha$ is an adjoint equivalence, so that both $\alpha^- \dashv \alpha$ and $\alpha \dashv \alpha^-$, then $\alpha^- \dashv \alpha$ is a strong adjoint equivalence in $Hom_s(C, B)$.
Only the first statement requires proof. Given $r: c \to d$ in $C$, let $ev_c: Hom_l(C, B) \to B$ denote the 2-functor which evaluates at $c$ (keep in mind that the 1-cells in $Hom_l(C, B)$ are lax transformations), and let $ev_r: ev_c \to ev_d$ denote the evident transformation; this is oplax under our convention. Then, by dualizing the famous lemma, $ev_r(\alpha) = \alpha \cdot r$ is an isomorphism if $\alpha$ is a right adjoint. Since $\alpha \cdot r$ is an isomorphism for all 1-cells $r$ in $C$, it follows that $\alpha$ is strong.
This completes the argument that the symmetric monoidal structure on $Map(B)$ extends one on $B$.
Another thing worth pointing out about cartesian bicategories is that cartesian structure is really a property, i.e., up to equivalence, there is at most one cartesian structure that can obtain on a bicategory. This is well-known to the cognoscenti; indeed, one can use cartesian structure on $B$ to see that the local hom-categories $B(b, c)$ have cartesian products, and that a cartesian structure on $B$ can be entirely retrieved from
The 2-products on $Map(B)$,
The products on local hom-categories $B(b, c)$
Therefore, since 2-products and products are unique up to appropriate canonical equivalence, so must be cartesian structure. For now, we content ourselves with a sketch of these facts.
The binary product on $B(b, c)$ is given by
where $f^*$ denotes the right adjoint of a map $f$. The terminal object of $B(b, c)$ is given by
To see that this prescription for binary products works, one notes first that the “laxified 2-adjunction” $\Delta \dashv_{lax} \otimes$ on a cartesian bicategory induces an adjoint pair
for each $b \in Ob(B)$, $c = (c_1, c_2) \in Ob(B \times B)$. Thus, we have that for any $b' \in Ob(B)$, the composite
which is the same as the diagonal $B(b, b') \to (B \times B)(\Delta b, \Delta b') \simeq B(b, b') \times B(b, b')$, is left adjoint to the composite
which, applied to a pair $(r: b \to b', s: b \to b')$, gives the product $r \times s$ as we have described it.
The argument that the precription for the terminal object works is entirely similar.
(It may help for the reader to just figure this out himself, based on how it works in $Rel$. The cartesian product of a relation $R$ from $A$ to $B$ and a relation $S$ from $C$ to $D$ is given by the formula $(R \times S)(\langle a, c \rangle, \langle b, d \rangle) = R(a, b) \wedge S(c, d)$.)
For objects, it is obvious: regard objects $b$ and $c$ in $B$ as belonging to $Map(B)$, and define $b \otimes c$ to be their 2-product there.
For morphisms $r: a \to b$ and $s: c \to d$, the morphism $r \otimes s: a \times c \to b \times d$ is the local product in the category $B(a \times c, b \times d)$ of
and
This formula extends to 2-cells as well by whiskering. The proof that this works is easily discovered by writing down the definitions in terms of string diagrams.
Indeed, by regarding an object $c$ as belonging to $Map(B)$, the component $\delta c$ is the 2-diagonal $c \to c \times c$. If $r: c \to d$ is a morphism of $B$, then the structure 2-cell $\delta \cdot r: (\delta d)r \to (r \otimes r)(\delta c)$ is uniquely determined as the mate of the local diagonal
Similarly, by regarding objects $c$ and $d$ as belonging to $Map(B)$, the component $\pi(c, d)$ is the pair or 2-projections $(\pi_1: c \times d \to c, \pi_2: c \times d \to d)$. If $r: a \to c$ and $s: b \to d$ are morphisms, then the structure 2-cell $\pi \cdot (r, s)$ is a pair of 2-cells
uniquely determined as mates of the local projections
(according to how we retrieve $r \otimes s$ as a local product).
Similarly, the structure of the lax unit $\varepsilon: 1_B \to I!$ is definable in terms of the 2-terminal object in $Map(B)$ and the characterization of local terminal objects.
The invertible modifications $s$, $t$, $u$ on $B$ are uniquely determined by regarding them as the triangulators for the 2-product structure on $Map(B)$.
This completes the argument that cartesian structure is a property.
This section is a routine graduate-course exercise, as a warm-up to the slightly more elaborate calculations on spans of groupoids below.
Let $C$ be a finitely complete category, and let $Span(C)$ be the usual (weak) 2-category whose 1-cells are spans in $C$. There is an inclusion
which sends each object to itself, and a morphism $f: c \to d$ to the span
(This is the usual way to turn functions into relations.) Given $f, g: c \to d$ in $C$, a 2-cell $i(f) \to i(g)$ can only be an identity 2-cell. So, regarding $C$ as a locally discrete 2-category, $i$ becomes a locally full inclusion.
Furthermore, each $i(f)$ is a left adjoint; its right adjoint is the same span read in reverse:
This is a very easy and routine exercise which can be left to the reader.
If $X \stackrel{r}{\leftarrow} S \stackrel{s}{\to} Y$ has a right adjoint in $Span(C)$, then $r$ is an isomorphism. As a result, every left adjoint in $Span(C)$ is isomorphic to a “function” $X \stackrel{1_X}{\leftarrow} X \stackrel{f}{\to} Y$, where $f = s r^{-1}$, and the sub-2-category
is 2-equivalent to the sub-2-category $i: C \to Span(C)$.
All diagrams of 2-cells commute in $Map(Span(C))$ (because $C$ is a locally discrete 2-category).
Let $Y \stackrel{t}{\leftarrow} T \stackrel{u}{\to} X$ be the right adjoint. Then a structure of unit $\eta$ for the adjunction amounts to a diagram
in which $r r' = 1_X$ and $u u' = 1_X$. Now we need to see $r' r = 1_S$. The evident morphism of spans $S \eta: S \to S T S$ amounts to a diagram of shape
and whatever the counit $\varepsilon: S T \to 1_Y$ is, the fact that the composite 2-cell $(\varepsilon S)(S \eta)$ is given by the identity on $S$ forces $r' r = 1_S$, as desired.
As before, $C$ is a finitely complete category.
$Span(C)$ is a cartesian bicategory.
Routine; we sketch the main points. The functor $\otimes$ takes a pair of spans $A \stackrel{r}{\leftarrow} S \stackrel{s}{\to} B$, $C \stackrel{t}{\leftarrow} T \stackrel{u}{\to} D$ to the evident cartesian product
The lax transformation $\delta: 1 \to \otimes \Delta$ has components
(more pedantically, we mean the span $i(\delta c)$), which is obviously map-valued. The structure 2-cell $\delta \cdot (r, s)$ is provided by $\delta S$ in the commutative diagram
which expresses naturality of $\delta$ in $C$.
The lax transformation $\pi: \Delta \otimes \to 1$ has components
(more pedantically, we mean the pair of spans $(i(\pi_1), i(\pi_2))$), which are obviously map-valued. Structure 2-cells are provided by morphisms $(\pi_1: S \times T \to S, \pi_2: S \times T \to T)$ which occur in a commutative diagram which expresses naturality of the projections $\pi_1$, $\pi_2$ in $C$.
The 2-functor $I: \mathbf{1} \to Span(C)$ names the terminal object of $C$ regarded as an object of $Span(C)$. The lax transformation $\varepsilon: 1 \to I !$ is has components $\varepsilon c$ given by the unique morphism $c \to 1$ in $C$.
The required triangulator modifications, since they are essentially valued in the locally discrete 2-category $C$ (via the equivalence $C \simeq Map(Span(C))$, are identities; they are the equations expressed by commutative diagrams of the form
and the triangulator coherence conditions of course hold by corollary 1. In summary, $C$ as a 2-category has 2-products since $C$ has products, and therefore $Span(C)$ has a cartesian structure.
One other thing worth pointing out about $Span(C)$ is the Frobenius condition: letting $\delta^*$ be the right adjoint of the map $\delta$, the 2-cell
namely the mate of the coassociativity 2-cell $id: (\delta \otimes 1)\delta \to (1 \otimes \delta)\delta$, is invertible, and similarly the mate
of the inverse coassociativity $id: (1 \otimes \delta)\delta \to (\delta \otimes 1)\delta$ is also invertible. This fact can be exploited to show that $Span(C)$ is a compact monoidal 2-category, where each object $c$ is 0-dual to itself. (The unit is
and the counit is the opposite span
the construction of the triangulators is based on the Frobenius isomorphisms and is left to the reader.)
This same fact indicates that the Frobenius condition does not hold in the cartesian bicategory $Prof$, since the 0-dual of a category $C$ is not $C$ but rather $C^{op}$; however, the cartesian bicategory consisting of groupoids and profunctors between them is again Frobenius (or discrete, in the language of Carboni and Walters in Cartesian Bicategories I).
Steve Lack and others (this is to be looked up) have a result which abstractly characterizes those cartesian bicategories of the form $Span(C)$. We may see whether this generalizes to the case of cartesian tricategories to follow: is there a similar characterization of cartesian tricategories which arise as spans of groupoids?
In principle, the definition is a straightforward generalization of the definition we wrote down for cartesian bicategories. The procedure is basically this:
Write down, in elementary (nuts-and-bolts) terms, the explicit definition of 3-adjunction.
For a (as always, weak) 3-category $C$, write down the complete set of data which would stipulate that $\Delta: C \to C \times C$ and $!: C \to 1$ are left (3-)adjoints, i.e., so that $C$ has finite 3-products.
Then laxify this data as much as possible so that the restriction of the laxified data to the sub-3-category $Map(C)$, whose 1-cells are the left (2-)adjoints in $C$, results in $Map(C)$ having 3-products.
(Some work to be done here.)
Let $Span(Gpd)$ be the usual (weak) 3-category whose 1-cells are spans of groupoids. There is an inclusion
which sends each groupoid to itself, and a functor $F: G \to H$ to the span
It sends a natural transformation $\alpha: F \to F'$ to the 2-cell
It is clear that given natural transformations $\alpha, \beta: F \to F'$ in $Gpd$, a 3-cell $i(\alpha) \to i(\beta)$ can only be an identity 2-cell. So, regarding $Gpd$ as a 2-locally discrete 3-category, $i$ becomes a 2-locally full inclusion.
Furthermore, each 1-cell $i(F)$ is a left 2-adjoint in $Span(Gpd)$; its right 2-adjoint is the same span read in reverse:
This is a very easy and routine exercise which can be left to the reader.
If a span of groupoids $G \stackrel{r}{\leftarrow} X \stackrel{s}{\to} H$ is a left 2-adjoint, then $r$ is an equivalence. As a result, every left 2-adjoint in $Span(Gpd)$ is equivalent to a functorial span $G \stackrel{1_G}{\leftarrow} G \stackrel{F}{\to} H$, where $F = s r^{-1}$ ($r^{-1}$ here denoting a quasi-inverse of $r$), and the sub-2-category
is 2-equivalent to the sub-3-category $i: Gpd \to Span(Gpd)$.
The proof is wholly analogous to the proof we gave for theorem 2 above, the one that characterizes maps or left adjoints in $Span(C)$.
All pasting diagrams of 3-cells in $Map(Span(Gpd))$ commute (because it is equivalent to the 2-locally discrete 3-category $Gpd$).