## March 30, 2011

### 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:

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.

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.

Posted at March 30, 2011 7:34 PM UTC

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### 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:

We note that the universal property here is not the one we will use in this paper. In particular, we do not require the modification axioms as an extra condition. We circumvent this requirement by defining the pullback as a weighted strict 2-limit rather than a pseudo 2-limit.

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.

Posted by: Mike Shulman on March 31, 2011 5:16 AM | Permalink | PGP Sig | Reply to this

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

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?

Yeah, what he said….

(ok I’m looking at just above definition 23, in the section called ‘left unitor’)

Why can’t you specify the pullback of the cospan $B \xrightarrow{1_B} B \xleftarrow{q} S$ to be the span $B \xleftarrow{q} S \xrightarrow{1_S} S$? That is, the composition of $B \xleftarrow{1_B} B \xrightarrow{1_B} B$ and $B \xleftarrow{q} S \xrightarrow{p} A$ is $B \xleftarrow{1_B q} S \xrightarrow{p 1_S} A$.

I found in my anafunctors paper that if one makes the canonical choice of pullback $X\times_X Y = Y$ (pullback along the identity), then things become much nicer. Or do you need the universal property of the isocomma object in the definition of composition in defining stuff later?

Perhaps what I was looking at was too nice a situation to work for you, though.

Posted by: David Roberts on March 31, 2011 6:17 AM | Permalink | Reply to this

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

No, I don’t think the difference is very substantial David. In particular, I think you are right that choosing nicer pullbacks along identities should make the construction nicer.

The main reason I didn’t do this to begin with is that I wanted to see what structure came from the most naive construction of a span tricategory. Since I had often heard people suggest that spans in a bicategory give a monoidal tricategory, I wanted to see how straightforward this idea really was.

Now, going back in, I think I should be able to clean up big parts of the paper (and hopefully make it shorter) by making better choices as you suggest.

Posted by: Alex Hoffnung on March 31, 2011 7:17 AM | Permalink | Reply to this

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

Actually, now that I have tried choosing various nicer composites along identity morphisms in order to clean up the unitors, I feel that I don’t have any idea how to do this without destroying the universal property or getting around employing it at the later stages. I should take a look at your anafunctors paper and maybe I can get some idea that I havent thought of yet.

Posted by: Alex Hoffnung on May 27, 2011 8:32 AM | Permalink | Reply to this

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

So, first I have a big job of going in and cleaning up the terminology, some of which I created on the fly as I realized exactly how strict things were turning out to be. I will probably actually remove all (bi)adjoint (bi)equivalences, since I do not really need either of these notions at this point. I was just curious about the Gurski’s algebraic definition of tricategory, and also thought it would be helpful to understand what Todd meant by triequivalence in his definition by using biadjoint biequivalences in place of tritransformations.

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

An “adjoint isomorphism” is not in any significant way different than an isomophism, except that maybe we should specify, in this case, that by isomorphism, we mean $1$-isomorphism inside the functor $2$-category of strict $2$-functors, strict transformations and modifications.

Mike said: 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?

That is a good question. Nick Gurski made a similar suggestion about adjusting the composition a short time ago. I think it is likely that this will work, but I haven’t done it yet.

Do you have a specific notion of enriched bicategory in mind?

Mike said: 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?

I will need to get a little sleep before figuring out what went wrong here. I certainly agree that the usual definition of pullback, say for groupoids, requires this condition. We can say the pullback of a cospan of groupoids: $\begin{matrix} A&&&&B\\ &\searrow_{f} && \swarrow_{g} &\\ && C&& \end{matrix}$ is a groupoid consisting of objects which are triples $(a,\mu,\b)$, such that $\mu:f(a)\rightarrow g(b)$ is an isomorphism in $C$, and morphisms $(\sigma,\tau): (a,\mu,b)\rightarrow (a',\mu;,b')$, where $\sigma:a\rightarrow a' and \tau:b\rightarrow b'$ are morphisms in $A$ and $B$ such that $\begin{matrix} f(a)\stackrel{\mu}\rightarrow & g(b)\\ \downarrow^{f(\sigma)}&\downarrow^{g(\tau)}\\ f(a')\rightarrow_{\mu'} & g(b')\\ \end{matrix}$ commutes.

Then it is easy to see where that condition is needed. We can specify the components of $\gamma$, but to prove the components are morphisms in the pullback groupoid, we need the above square to commute, and this is precisely the equation you are asking for.

At some point it seemed that one less equation was required in the definition I was giving in the paper, but I must have made a mistake in translating into the more explicit definition. I will get back in there and figure out how I ended up losing that condition.

Mike said: Do you mean that more people are giving up on the “as strict as possible but not stricter” approach and just using bilimits?

No, I definitely did not intend to imply this. I don’t know if this is true or not. I would guess not, since I think the “as strict as possible but not stricter” approach is more or less the idea of coming up with coherence theorems, right?

Mike continued: I like flexible limits… although I think I like PIE-limits even better.

This captures the spirit in which I intended the statement to read. I was attempting to avoid introducing more classes of limits, since I wasn’t really saying anything about substantial about them. But my meaning was that people usually work with PIE-limits rather than flexible limits.

I had initially intended to write a much longer section on limits, since limits are confusing, and I wanted to get a good feel for the different classes and types of limits. At some point I decided that I should stop writing about limits and finish writing down the span construction, but I hope to continue the exposition on limits, at least for my own personal edification.

Posted by: Alex Hoffnung on March 31, 2011 7:08 AM | Permalink | Reply to this

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

That all makes sense.

Do you have a specific notion of enriched bicategory in mind?

There’s really only one notion, isn’t there (aside from variations in how strict the monoidal bicategory that one is enriching in is assumed to be)? Although it seems like a lot of people have written it down independently in the course of doing other things.

Posted by: Mike Shulman on March 31, 2011 7:53 AM | Permalink | Reply to this

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

Yes, that seems right. Do we really need to mess with the composition to see that the tricategory is an enriched bicategory though? It seems that asking for the unitors to be isomorphisms is just asking that the unit modifications be identities (the counit modifications are already identities).

If we are not working with Gurski’s algebraic definition than the equivalence property of the unitors is just that - a property.

So, the unit modifications would not show up as extra structure. In this case, it seemed to me that we should be able to enrich over something like $2$Gray to see the span tricategory as an enriched bicategory without changing the composition functor.

Posted by: Alex Hoffnung on March 31, 2011 8:13 AM | Permalink | Reply to this

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

It seems that asking for the unitors to be isomorphisms is just asking that the unit modifications be identities (the counit modifications are already identities).

Yes, that’s certainly right.

So, the unit modifications would not show up as extra structure.

That’s true, but the unit transformations would still not have the property of being isomorphisms. But you could get something like an “enriched lax bicategory” whose associators are isomorphisms and whose unit transformations are equivalences. That doesn’t sound like too nasty of a structure; certainly less scary-looking to me than a one-object tetracategory.

Posted by: Mike Shulman on March 31, 2011 7:54 PM | Permalink | Reply to this

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

Right. For some reason. I got mixed up and forgot that enriched bicategories have invertible structure 2-cells.

I will post a definition of enriched bicategory at the nLab, since I don’t think it is widely available. I guess a link would be best, so that people can see the diagrams.

This should complement the discussion on enriched category theory as a tool in higher category theory well I think.

Posted by: Alex Hoffnung on March 31, 2011 11:12 PM | Permalink | Reply to this

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

I will post a definition of enriched bicategory at the nLab

Excellent, thank you! I had the same thought that such a thing would be helpful.

I have to say, though, that I am not fond of the term “enriched bicategory” for this sort of thing. It seems to me more appropriate to call it merely an enriched category, but one where the thing we enrich over happens to be a monoidal bicategory, and thus the notion of “enrichment” is appropriately weakened. (One could say a pseudo enriched category, for instance.) In particular, if our monoidal bicategory happens to be a monoidal 1-category, regarded as a locally discrete bicategory, then when we enrich over it in this sense we just get the usual notion of enriched category.

By contrast, what I would want to call a “$V$-enriched bicategory” is a “pseudo ($V Cat$)-enriched category”, i.e. a bicategory where the sets of 2-cells are replaced by objects of $V$. These of course do also arise naturally, such as $V Cat$ itself, or $V Prof$.

However, it may be too late to change the world on this one. (-:

Posted by: Mike Shulman on April 1, 2011 5:40 AM | Permalink | Reply to this

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

I typed up a short note with the definition of enriched category followed by enriched bicategory. I included both since the first is easy to write down and its nice to see them together.

I am trying to decide whether to start a new nLab page for enriched bicategories or to link from the existing page on enrichment. I guess I’ll start with the latter.

Any mistakes in the definition are mine. Let me know if anyone spots one.

Posted by: Alex Hoffnung on April 2, 2011 7:57 AM | Permalink | Reply to this

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

I am trying to decide whether to start a new nLab page for enriched bicategories or to link from the existing page on enrichment.

I suggest to use the page enriched bicategory.

Posted by: Urs Schreiber on April 2, 2011 9:17 AM | Permalink | Reply to this

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

:) That looks good.

Posted by: Alex Hoffnung on April 2, 2011 9:19 AM | Permalink | Reply to this

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

There is discussion of symmetric monoidal $(\infty,n)$-categories of higher spans of $\infty$-groupoids in the literature, for all $n$. Also a little bit on spans of $(\infty,n)$-categories with duals. Some references are collected at $(\infty,n)$-category of spans.

I believe several people know how to regard tricatgories as $(\infty,3)$-categories, but I guess it has not been written out yet.

By the way, in your def. 6 of morphisms of spans in a 2-category you have

$\array{ && Z \\ & \swarrow &\downarrow& \searrow \\ X &&\downarrow&& Y \\ & \nwarrow &\downarrow& \nearrow \\ && Z' }$

(notationally suppressing the 2-morphisms). More generally you could consider spans-of-spans proper, of the form

$\array{ && Z \\ & \swarrow &\uparrow& \searrow \\ X &&Q&& Y \\ & \nwarrow &\downarrow& \nearrow \\ && Z' } \,.$

Is this restriction to make the problem simpler (which would be good rason enough, I guess), or is there a deeper reason?

Posted by: Urs Schreiber on March 31, 2011 8:27 AM | Permalink | Reply to this

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

Thanks for pointing out the nLab page. I will take a closer look in the morning, since it is getting late here.

The definition of maps of spans was meant in part to be a categorified version of intertwining operator, so I guess the reason was mostly that I haven’t yet needed the more general notion of map of spans.

I would be really interested to see how (\infty,n)-categories help side-step the huge number of calculations that I had to do for this paper. Again I will look at some of the nLab links in the morning.

I didn’t really include any calculations in this draft. Many were in earlier drafts, but over time they were all taken out. So the paper becomes twice as long if they are reinserted, but since all structures are defined by universal properties, after verifying a few calculations, it becomes reasonably easy to believe that the calculations can be recovered without much trouble.

Posted by: Alex Hoffnung on March 31, 2011 8:50 AM | Permalink | Reply to this

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

The span-of-spans is the approach that I’ve looked at, because it seems to be the best way to reflect the structure of dualities in Span(I). There’s actually a bit of a fancy story involved. In particular, as well as being monoidal, there is something like a star-structure here. You mention it at the top, but don’t include it in the structure of what kind of (tetra/tri)-category you’re talking about. Every span has a dual, which just reverses the sense of source and target. This gives a duality for morphisms.

Now, for a category C with pullbacks, Span(C) is the universal bicategory containing C, but with a biadjoint for every morphism, which gives a precise sense of the above “dual”, with some properties. The kind of 2-morphism which makes this true is the map-of-spans. (See e.g. this describing work in preparation by Kenney and Pronk). This is the same kind of construction you’ve got. So in some sense this is the most natural way to go.

This is a question I have, though: what if you want morphisms of all orders to have duals? This comes partly from the groupoidification program. In groupoidification, the “dual” of a span of groupoids gives the linear map between Hilbert spaces which is adjoint to the one that comes from the original span. In the generalization to what I called “2-linearization”, it gives a 2-linear map which is biadjoint to that coming from the original span. (This biadjointness comes from the adjunction between induced and restricted representation functors). Moreover, the unit and counit from opposite adjunctions are (componentwise) Hilbert space adjoints. So there’s a “fully dualizable” aspect to the image of the span categories. One would like the span categories to have them originally.

This (totally ad hoc) reason was why I originally wanted to use spans-of-spans: the top level of morphisms is forced to be spans of span maps. Then there’s a hitch (because we’ve just thrown away the nice universal property), which is that we have to do one of two bad things. First, we can give up the good composition properties such as associativity by discarding maps-of-spans-of-span-maps altogether (3-morphisms). Second, we can brute-force the matter by taking the homotopy category (i.e. taking spans-of-spans up to equivalence).

The story for n-fold spans seems to be the same - you always run into this issue at the top level. It’s familiar from the 1-morphisms in the span category used in the groupoidification program.

This ad-hoc reason - wanting to have the adjointness of unit and counit for a biadjunction to be “adjoint” in the Hilbert space sense be a reflection of a structure in the span-category- arises since 2Hilb is Hilb-enriched. That is, the top-level hom-sets are actually objects in a star-category. The star-structure for the lower-dimension morphisms can be described as “categorical” adjoints - the “Hilbert” adjoint has no description in terms of lower-dimension morphisms. So it’s harder to represent naturally in the span category.

Posted by: Jeffrey C Morton on April 1, 2011 2:38 AM | Permalink | Reply to this

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

The span-of-spans is the approach that I’ve looked at, because it seems to be the best way to reflect the structure of dualities

[…]

So there’s a “fully dualizable” aspect to the image of the span categories. One would like the span categories to have them originally.

I think that’s exactly right: the point of having spans of spans of spans… is that this gives an $n$-category with all duals.

This is the claim, anyway: that $Span_n(\infty Grpd)$ (the $(\infty,n)$-category of order-$n$ spans of $\infty$-groupoids) is a symmetric monoidal $(\infty,n)$-category with duals.

And if $C$ is any other symmetric monoidal $(\infty,n)$-category with duals, then so is the $(\infty,n)$-category $Span_n(\infty Grpd, C)$ of $n$-fold spans over $C$.

Posted by: Urs Schreiber on April 1, 2011 8:25 AM | Permalink | Reply to this

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

Urs: this statement is just the idea I had in mind, though I’ve yet to really absorb the $(\infty,n)$ machinery here. A couple of things come to mind. First: why does $C$ need to have duals? It seems like the operation of taking spans provides duals even for morphisms that don’t already have them. Or is there some reason why the span with reversed orientation will fail to have the property of a dual if $C$ doesn’t already have them? When $n=1$, we only require that $C$ have pullbacks for $Span(C)$ to have duals.

Second: is $Span_n(C)$ in the sense with spans-of-spans as the top level a universal symmetric monoidal $(\infty,n)$-category with duals which contains $C$? I guess the results Kenney and Pronk have shown imply something like this. Kenney’s slides which I linked to only go to $n=2$, but I think you could perhaps extend it by applying the argument to hom-bicategories. The top-level morphisms would just be span maps, which is how they build the $Span$ monad, however. Perhaps one could just go one dimension higher and then take the homotopy category, but that seems likely to cause a problem.

This goes back to my previous comment that the duality at the top level, though it satisfies the dagger-category axioms, can’t naturally be anything like a “biadjoint”, with good universal properties. Or is that not necessary if we just define a formal dual?

Posted by: Jeffrey C Morton on April 1, 2011 2:51 PM | Permalink | Reply to this

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

why does $C$ need to have duals?

Notice that the $C$ in my comment denoted the category over which we consider spans, not in which we consider spans, as you are assuming here.

What I wrote $Span_n(\infty Grpd, C)$ is supposed to be the $(\infty,n)$-category whose

• objects are $\infty$-groupoids $X$ equipped with an $\infty$-functor $X \to C$

• morphisms are diagrams of $(\infty,n)$-categories of the form

$\array{ && Z \\ & \swarrow && \searrow \\ X &&\swArrow&& Y \\ & \searrow && \swarrow \\ && C }$

• “and so on” .

For $C = *$ this is (supposed to be) the $(\infty,n)$-category of “$n$-fold spans in $\infty$-groupoids”.

Posted by: Urs Schreiber on April 1, 2011 8:10 PM | Permalink | Reply to this

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

is $Span_n(C)$ in the sense with spans-of-spans as the top level a universal symmetric monoidal (∞,n)-category with duals which contains $C$?

Just to avoid the above confusion of symbols, let me restate this question as

is $Span_n(\mathcal{B})$ [the $(\infty,n)$-category of $n$-fold spans in $\mathcal{B}$] a universal symmetric monoidal (∞,n)-category with duals which contains $\mathcal{B}$?

I see where you are coming from, but maybe one has to be careful with this.

We know from the cobordism hypothesis that $Span_n(\mathcal{B})$ can not be the free symmetric monoidal $(\infty,n)$-category with duals on $\mathcal{B}$. For consider $\mathcal{B} = *$, then

$Span_n(*) \simeq *$

but the free symmetric monoidal gadget with duals on the point is something like

$Bord_n^{fr}$

But something along these lines ought to be true. I am not sure. What is true is that for $\mathcal{B} = \infty Grpd$ we have that every object of $\mathcal{B}$ becomes fully dualizable in $Span_n(\mathcal{B})$. That is part of this claim which asserts that $Span_n(\infty Grpd)$ has the property that the following are equivalent

1. symmetric monoidal $(\infty,n)$-functors

$Bord_n \to Span_n(\infty Grpd)$

2. Spaces ($\infty$-groupoids) $X$ equipped with a rank $n$-vector bundle with inner product.

There must be a better statement along the lines that you are looking for lurking here, but I am not into that stuff enough to know. Maybe we are lucky and some expert chimes in and helps us out.

Posted by: Urs Schreiber on April 1, 2011 9:15 PM | Permalink | Reply to this

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

I would be really interested to see how $(\infty,n)$-categories help side-step the huge number of calculations that I had to do for this paper. Again I will look at some of the nLab links in the morning.

Currently the $n$Lab-pages on $(\infty,n)$-categorical matters for $n \geq 2$ do not contain a whole lot of technical details, but mostly just try to lay out the existing notions and point to references. The references, in turn, are a bit thin currently on the theory on symmetric monoidal $(\infty,n)$-categories for $n \geq 2$. So I am not sure if that will help you much with what you are trying to do, but I think it is good to know that this road exists.

Generally, experience (for instance with extended cobordisms) suggests that

• the geometric definition of symmetric monoidal $(\infty,n)$-categories lends itself to the study of general abstract statements;

• the bi-, tri-, tetra-category incarnation lends itself to explicit constructions of examples in terms of generators and relations. (That’s after all, what these coherence conditions are: the relations imposed on a particular choice of cells in an $(\infty,n)$-category).

For instance in the first case it seems to be possible to readily say: the $(\infty,n)$-category of $n$-fold spans is symmetric monoidal, Saying this in the second case however is a pain and not very illuminating. But if one needs explicit generators-and-relations presentations of the monoidal structure, then I guess that’s essentially tantamount to the second path.

Posted by: Urs Schreiber on March 31, 2011 11:21 AM | Permalink | Reply to this

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

I would expect that using spans-of-spans as the 2-morphisms would push you up into the world of a monoidal 4-category, unless you truncated somehow. Morphisms of spans and spans of spans are of course different things, and which of them one wants in a particular case should vary by application.

Posted by: Mike Shulman on March 31, 2011 7:56 PM | Permalink | Reply to this

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

I realized that I haven’t thought enough about spans of spans. Is a span of spans … of spans always an object with a cone over the domain and a cone over the codomain with all faces filled in with 2-cells? (I am just thinking of the span construction on a 2-category).

So this would give the following morphisms:

• spans as $1$-morphisms ($2$ one-cells, $0$ two-cells)
• spans of spans $2$-morphisms ($4$ one-cells, $4$ two-cells)
• spans of spans of spans $3$-morphisms ($10$ one-cells, $16$ two-cells)

and so on…

Posted by: Alex Hoffnung on March 31, 2011 11:23 PM | Permalink | Reply to this
Read the post Spans in 2-Categories: A Monoidal Tricategory
Weblog: The n-Category Café
Excerpt: A paper constructing a monoidal tricategory of spans.
Tracked: December 16, 2011 6:32 AM

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