### A Tetracategory of Spans (or, What Is a Monoidal Tricategory?)

#### Posted by Alexander Hoffnung

Spans are a wonderfully simple idea, and, as such, they are ubiquitous mathematics. Why? Well, for one, any span, which is a pair of arrows with common domain, from a space (set, groupoid, object, etc.) $A$ to a space $B$:$\begin{matrix}&S&\\&\swarrow \searrow&\\B&&A\\\end{matrix}$can be turned around without any “fuss” about injectivity or surjectivity to obtain a span from the space $B$ to the space $A$:$\begin{matrix}&S&\\&\swarrow \searrow&\\A&&B\\\end{matrix}$See, I just did it!

But before we get carried away, spans have an ugly, dark side as well. Composition of spans is *not* associative. So spans, considered as morphisms between sets, for example, do not even form a category. However, with a sunny disposition and a healthy dose of optimism, unable to have a category, we happily settle for a (weak) $2$-category, or bicategory, of spans.

In fact, Bénabou defined bicategories to handle exactly this type of situation. By defining a suitable notion of ‘maps between spans’, Benabou was able to produce, as an early example of a bicategory, a structure consisting of:

- sets as objects,
- spans of sets as $1$-morphisms, and
- maps of spans of sets as $2$-morphisms.

So how are spans composed? Given composable spans$\begin{matrix}&S&&R&\\&\swarrow \searrow&&\swarrow \searrow&\\C&&B&&A\\\end{matrix}$we can form a composite span$\begin{matrix}&&S R&&\\&&\swarrow\searrow &&\\&S&&R&\\&\swarrow \searrow&&\swarrow \searrow&\\C&&B&&A\\\end{matrix}$

We haven’t yet defined $S R$. Let’s continue to consider the example of spans of sets a bit longer. The category of sets is complete, meaning that it has all limits. In particular, we can define $SR$ to be the pullback, sometimes called the fibered product. Pullbacks are limits of diagrams of the following shape:$\begin{matrix}S&&&&R\\&\searrow&&\swarrow&\\&&B&&\\\end{matrix}$called a `cospan’.

The big idea here is that we can form a bicategory $Span(\mathcal{C})$ with spans as $1$-morphisms from any category $\mathcal{C}$ with pullbacks. If $\mathcal{C}$ also has finite products (really, just adding a terminal object to a category with pullbacks is enough), then $Span(\mathcal{C})$ can also be given a monoidal structure.

The *span construction* is very well-known, but the seemingly minor nuisance of having non-associative composition, can be more troublesome than it might first appear.

It is quite common for mathematicians to work with spaces, which are themselves categories, or at least have, in addition to a notion of maps, a notion of maps between maps.

So, given a $2$-category $\mathcal{B}$ with pullbacks, what kind of structure is $Span(\mathcal{B})$?

The answer, which probably belongs to the realm of `folk theorems’, is *a tricategory*. This is the beginning of a pattern that, while nice, makes the span construction rather difficult to describe functorially. This is:

- Given a category $\mathcal{C}$ with products and pullbacks, there is a monoidal bicategory $Span(\mathcal{C})$.
- Given a bicategory $\mathcal{B}$ with products and pullbacks, there is a monoidal tricategory $Span(\mathcal{B})$.

But,$\textstyle{What is a monoidal tricategory?}$

Monoidal structures on $n$-categories have a very nice description as one-object $(n+1)$-categories, at least for very small values of $n$. The pattern is as follows:

- A monoid is a one-object category.
- A monoidal category is a one-object bicategory.
- A monoidal bicategory is a one-object tricategory.

Tricategories were defined by Gordon, Power, and Street in their manuscript *Coherence for Tricategories*.

We want to go a step further. In 1995, Todd Trimble explicitly defined tetracategories at the request of Ross Street. The definition, which sprawls over 51 pages, is available on John Baez’s website. We can naively proceed with the above pattern, and say:

- A monoidal tricategory is a one-object tetracategory.

If you have ever wanted to see an example of a tetracategory, take a look at this draft of the paper Spans in $2$-Categories: A one-object tetracategory.

Then if anyone can point the way to other examples of tetracategories, I would be grateful.

So, what is the point of constructing this monoidal tricategory of spans? Well, as pointed out above, spans are everywhere.

One place that spans can always be found lurking about is in the groupoidification program. This is an approach to categorification that is meant to both make new connections and clarify old connections across a broad range of mathematical ideas. An essential structure in understanding groupoidification as a type of categorification is the degroupoidification functor, which takes groupoids to vector spaces and spans of groupoids to linear maps. See HDA7: Groupoidification and The Hecke Bicategory for details.

In the above papers, we worked explicitly with a bicategory of spans of groupoids. You might object and argue that groupoids form a $2$-category, so according to the above discussion, we should have been working with a tricategory of spans of groupoids consisting of:

- groupoids as objects,
- spans of groupoids as $1$-morphisms,
- maps of spans of groupoids as $2$-morphisms, and
- maps of maps of spans of groupoids as $3$-morphisms,

and you would have a good point.

So what is going on? Well, we just did not have a pressing need for the full tricategory structure. We defined a notion of isomorphism of maps of spans, and defined the $2$-morphisms to be isomorphism classes of maps of spans, effectively killing off the tricategory structure. However, constructing this bicategory is not really any easier than constructing the tricategory. We avoid checking some imposing tricategory structure, but this headache, which is at least interesting, is replaced with checking equivalence class equations throughout the construction. On the other hand, checking that the span construction yields a one-object tetracategory does add a significant amount of work.

Is our example of a tricategory of spans useful? Well, it should be useful for at least two reasons. The first is that it provides insight into the span construction. Earlier we mentioned the intractability of the span construction caused, for the most part, by the fact that $T(SR)$ $\neq$ $(TS)R$, but rather $T(SR)$ $\cong$ $(TS)R$ (in the case of sets, for example). From the low-dimensional cases, it makes sense to conjecture that the span construction continues to push us towards higher categories. However, the tricategory of spans we construct is what we call a `semi-strict cubical tricategory’, meaning, in part, that the span construction does not yield a fully weak tricategory. So, does the construction stabilize at some point, at least for all intents and purposes?

Looking at the title of the paper, *Spans in $2$-Categories*, you might again raise an objection, arguing that if we worked with weak $2$-categories, rather than strict $2$-categories, and bicategorical limits, rather than pseudo limits, the resulting structure would not be as strict. This is, of course, the case, but brings us to the next reason for the construction: to study coherence for tricategories (and eventually coherence for tetracategories).

A theorem of Power describes coherence for bicategorical limits (bilimits). The theorem is an extension of the coherence theorem for bicategories. The theorem states:

$\textstyle{Every bicategory with finite bilimits is biequivalent to a strict\: 2-category with finite flexible limits.}$

It is then reasonable to conjecture that given a bicategory $\mathcal{B}$ with finite bilimits and the biequivalent $2$-category $\mathcal{B}'$ with finite flexible limits, then $Span(\mathcal{B})$ is equivalent to $Span(\mathcal{B}')$.

What type of equivalence are we suggesting? Well, certainly triequivalence, but also monoidal equivalence. However, not having a definition of tetraequivalence on hand, this would be more difficult to verify. Remember though that we are working only with one-object tetracategories, so a notion of monoidal equivalence could probably be written down without too much difficulty.

So when we ask about stabilization of the span construction, we are really asking about stabilization up to equivalence. This is then, in part, a question of coherence.

We should note that we have now slipped into a discussion of $2$-categorical limits, which deserves some attention. For example, flexible limits seem to be falling out of use in current $2$-category theory vocabulary, so we might want to restate the theorem. In the paper, before constructing the span tricategory, we give an expository discussion on limits in $2$-categories and a bit on weighted limits. This is to help as we move forward in proving a statement about functoriality of the span construction, and so that we can make clear what definition of pullback (really, iso-comma object) we use to define composition.

In addition to discussing limits, we attempt to characterize the span construction at both the $3$- and $4$-categorical levels. To this end, we provide definitions of maps between $2$-categories and maps between $3$-categories, including only the smallest amount of structure that captures our construction.

While this draft is still continually being rewritten, we can already begin to see the coherence issues that arise for the tricategory. For example, there are no non-trivial modification cells in the tricategory structure (except as counits and units of the adjoint equivalences). The tricategory has strict $2$-category hom-spaces, and locally strict composition.

How does $Span(\mathcal{B})$ fit into coherence for tricategories? Well, it is somewhat of a hybrid structure. Let’s recall the strongest coherence theorem for tricategories.

$\textstyle{Every tricategory is triequivalent to a Gray-category.}$

A Gray-category can be defined as a category enriched over the category of $2$-categories with the ‘Gray tensor product’ in place of the usual monoidal structure. An alternative description is as a strict cubical tricategory.

The cubical condition is mainly a property of the composition and unit functors. Cubical functors can be used to define the Gray tensor product, so the connection between these descriptions is fairly straightforward, although we won’t say anymore about it here. Complete details can be found in *Coherence for Tricategories*, Nick Gurski’s thesis, and relevant cited works of Gray within.

The condition that the tricategory be strict means there cannot be any non-trivial transformation or modification structure. This is immediate from the enriched category definition, since there is no room in the definition of enriched categories for these structures.

The span tricategory we construct is cubical, but only partially satisfies the strictness condition, since there are non-trivial transformation cells in the structure. This suggests a characterization of the span tricategory as a Gray-bicategory. That is, an enriched bicategory as defined in Carmodey’s thesis. Then what are we enriching over? We would need to extend the Gray tensor product to a Gray $2$-category which has $2$-categories as objects, $2$-functors as morphisms, and transformations as $2$-morphisms. One nice feature of the enriched setting is that we can use change of base functors as strictification functors in studying coherence.

The tetracategorical structure is also very strict. Very briefly, given a strict $2$-category $\mathcal{B}$ with pullbacks and finite products, the monoidal tricategory $Span(\mathcal{B})$ consists of:

$\bullet$ a semi-strict cubical tricategory consisting of:

- objects of $\mathcal{B}$ as objects,
- spans in $\mathcal{B}$ as $1$-morphisms
- maps of spans in $\mathcal{B}$ as $2$-morphisms
- maps of maps of spans in $\mathcal{B}$ as $3$-morphisms,

$\bullet$ for objects $A,B,C,D$,

- strict $2$-categories $Span(A,B)$ of morphisms,
- strict composition functors $c_{ABC}$ defined by pullbacks,
- strict unit functors $I_A$,
- associator adjoint equivalences $a_{ABCD}$, (pairs of strict transformations) with identity modification counits and units,
- left and right unitor adjoint equivalences $l_{B}$ and $r_{A}$ with identity counit modifications and invertible unit modifications,

$\bullet$ all satisfying axioms given by declaring all modification structure cells in the definition of tricategory to be identities, and

$\bullet$ a locally strict homomorphism of tricategories called the monoidal product,

$\bullet$ a strict homomorphism of tricategories called the monoidal unit,

$\bullet$ biadjoint biequivalences (pairs of tritransformations) with trimodification units and counits for monoidal associativity and monoidal left and right unitors,

$\bullet$ invertible trimodifications $\pi$, $l$, $m$ and $r$,

$\bullet$ all satisfying axioms given by declaring all perturbation structure cells in the definition of tetracategory to be identities.

## Re: A Tetracategory of Spans (or, What is a Monoidal Tricategory?)

Wow! I’m impressed that you had the patience to write down all of that. (-: A few thoughts after a brief skimming:

I am a little confused by the notion of an “adjoint isomorphism” — is it in any way different from a plain “isomorphism”?

It looks like the only thing that prevents your tricategory from being simply a bicategory enriched over the 2-category 2Cat is that the unitors are not isomorphisms. But strictifying units is usually a lot easier than strictifying associativity; would it be possible to just modify the composition operation in order to make the unitors isomorphisms and obtain a plain enriched bicategory?

I don’t understand what is being said by the penultimate paragraph on p22:

Perhaps this is related to the fact that the last part of Def. 4 on p26 seems to me to be missing a condition; don’t you need to require that $\kappa k . f\alpha = g\beta . \kappa h$ in order for $\gamma$ to exist?

Finally, why do you say that “flexible limits seem to be falling out of use in current 2-category theory vocabulary”? Do you mean that more people are giving up on the “as strict as possible but not stricter” approach and just using bilimits? I like flexible limits… although I think I like PIE-limits even better.