Todd Trimble Spans of groupoids

Contents

Idea

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.

What is a cartesian bicategory?

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.

• Mike Shulman has often remarked that for the $Prof$-like examples, where Morita equivalence is weaker than equivalence, it makes sense to consider such examples instead as framed bicategories. I agree, but in our main application below to spans of groupoids, where Morita equivalence of groupoids implies equivalence of groupoids, we won’t have to worry about this point.

Definition

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.

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

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 $Map(B)$.

Symmetric monoidal structure of a cartesian bicategory

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.)

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.

This completes the argument that the symmetric monoidal structure on $Map(B)$ extends one on $B$.

Cartesian structure is a property

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.

Local products

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.

Cartesian structure in terms of 2-products and local products

• $\otimes$ is definable in terms of 2-products on $Map(B)$ and local products in $B$.

(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.

• The structures of lax unit $\delta: 1 \to \otimes \Delta$ and lax counit $\pi: \Delta \otimes \to 1$ are definable from 2-products and local products.

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.

Left adjoints in $Span$

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.

The cartesian bicategory $Span$

As before, $C$ is a finitely complete category.

Frobenius condition

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?

Cartesian tricategories

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.)

Left 2-adjoints in $Span(Gpd)$

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.

The proof is wholly analogous to the proof we gave for theorem 2 above, the one that characterizes maps or left adjoints in $Span(C)$.