So I recently got back from a trip to the UK – most of the time was spent in Cardiff, at a workshop on TQFT and categorification at the University of Cardiff.  There were two days of talks, which had a fair amount of overlap with our workshop in Lisbon, so, being a little worn out on the topic, I’ll refrain from summarizing them all, except to mention a really nice one by Jeff Giansiracusa (who hadn’t been in Lisbon) which related (open/closed) TQFT’s and cohomology theories via a discussion of how categories of cobordisms with various kinds of structure correspond to various sorts of operads.  For example, the “little disks” operad, which describes the structure of how to compose disks with little holes, by pasting new disks into the holes of the old ones, corresponds to the usual cobordism category.

This workshop was part of a semester-long program they’ve been having, sponsored by an EU network on noncommutative geometry.  After the workshop was done, Tim Porter and I stayed on for the rest of the week to give some informal seminars and talk to the various grad students who were visiting at the time.  The seminars started off being directed by questions, but ended up talking about TQFT’s and their relations to various kinds of algebras and higher categorical structures, via classifying spaces.  We also had some interesting discussions outside these, for example with Jennifer Maier, who’s been working with Thomas Nicklaus on equivariant Dijkgraaf-Witten theory; with Grace Kennedy, about planar algebras and their relationships to TQFT‘s. I’d also like to give some credit to Makoto Yamashita, who’s interested in noncommutative geometry (viz) and pointed out to me a paper of Alain Connes which gives an account of integration on groupoids, and what corresponds to measures in that setting, which thankfully agrees with what little of it I’d been able to work out on my own.

However, what I’d like to take the time to write up was from the earlier part of my trip, where I visited with Jamie Vicary at Oxford. While I was there, I gave a little lunch seminar about the bicategory $Span(Gpd)$ (actually a tricategory), and some of the physics- and TQFT-related uses for it. That turned out to be very apropos, because they also had another visitor at the same time, namely Jean Benabou, the fellow who invented bicategories, and introduced the idea of bicategories of spans as one of the first examples.  He gave a talk while I was there which was about the relationship between spans and what he calls “distributors” (which are often called “profunctors“, but since he was anyway the one who introduced them and gave them that name in the first place, and since he has since decided that “profunctors” should refer to only a special class of these entities, I’ll follow his terminology).

(Edit: Thanks to Thomas Streicher for passing on a reference to lecture notes he prepared from lecture by Benabou on the same general topic.)

The question to answer is: what is the relation between spans of categories and distributors?

This is related to a slightly lower-grade question about the relationship between spans of sets, and relations, although the answer turns out to be more complicated.  So, remember that a span from a set $A$ to a set $B$ is just a diagram like this: $A \leftarrow X \rightarrow B$.  They can be composed together – so that given a span from $A$ to $B$, and from $B$ to $C$, we can take fibre products over $B$ and get a span from $A$ to $C$, consisting of pairs of elements from the $X$ sets which map down to the same $b \in B$.  We can do the same thing in any category with pullbacks, not just ${Sets}$.

A span $A \leftarrow S \rightarrow B$ is a relation if the pair of arrows is “jointly monic”, which is to say that as a map $S \rightarrow A \times B$, it is a monomorphism – which, since we’re talking about sets, essentially means “a subset”.  That is, up to isomorphism of spans, $S$ just picks out a bunch of pairs $(a,b) \in A \times B$, which are the “related” pairs in this relation.  So there is an inclusion ${Rel} \hookrightarrow Span({Sets})$.  What’s more  the inclusion has a left adjoin, which turns a span into a corresponding relation.  It follows from the fact that $Sets$ has an “epi-mono factorization”: namely, the map $f: S \rightarrow A \times B$ that comes from the span (and the definition of product) will factor through the image.  That is, it is the composite $S \rightarrow Im(f) \rightarrow A \times B$, where the first part is surjective, and the second part is injective.  Then the inclusion $r(f) : Im(f) \hookrightarrow A \times B$ is a relation.  So we say the inclusion of $Rel$ into $Span(Set)$ is a reflection.  (This is a slightly misleading term: there’s an adjoint to the inclusion, but it’s not an adjoint equivalence.  “Reflecting” twice may not get you back where you started, or anywhere isomorphic to it.)

(Edit: Actually, this is a bit wrong.  See the comments below.  What’s true is that the hom-categories of $Rel$ have reflective inclusions into the hom-categories of $Span(Set)$.  Here, we think of $Rel$ as a 2-category because it’s naturally enriched in posets.  Then these reflective inclusions of hom-categories can be used to build  a lax functor from $Span(Set)$ to $Rel$ – but not an actual functor.)

So a slightly more general question is: if $\mathbb{V}$ is a monoidal category, and $\mathbb{V}' \subset \mathbb{V}$ is a  “reflective subcategory“, can we make $\mathbb{V}'$ into a monoidal category just by defining $A' \otimes ' B'$ (the product in $\mathbb{V}'$) to be the reflection $r(A' \otimes B')$ of the original product?   This is the one-object version of a question about bicategories.  Namely, say that $\mathbb{S}$ is a bicategory, and $\mathbb{S}'$ is a sub-bicategory such that every pair of objects gives a reflective subcategory: $\mathbb{S}' (A,B) \subset \mathbb{S}(A,B)$ has a reflection.  Then can we “pull” the composition of morphisms in $\mathbb{S}$ back to $\mathbb{S}'$?

The answer is no: this just doesn’t work in general.  For spans of sets, and relations, it works: composing spans essentially “counts paths” which relate elements $A$ and $B$, whereas composing relations only keeps track of whether or not there is a path.  However, composing spans which come from relations, and then squashing them back down to relations again, agrees with the composite in $Rel$ (the squashing just tells whether the set of paths from $A$ to $B$ by a sequence of relations is empty or not).  But in the case of $Span(Cat)$ and some reflective subcategory – among other possible examples – associativity and unit axioms will break, unless the reflections $r_{A,B}$ are specially tuned.  This isn’t to say that we can’t make $\mathbb{V}'$ a monoidal category (or $\mathbb{S}'$ a bicategory).  It just means that pulling back $\otimes$ or $\circ$ along the reflection won’t work.  But there is a theorem that says we can always promote such an inclusion into one where this works.

So what’s an instance of all this?  A distributor (again, often called “profunctor”) $\Phi : \mathbb{A} \nrightarrow \mathbb{B}$ from a category $\mathbb{A}$ to $\mathbb{B}$ is actually a functor $\phi : \mathbb{B}^{op} \times \mathbb{A} \rightarrow Sets$.  Then there’s a bicategory $Dist$, where for each objects there’s a category $Dist(\mathbb{A},\mathbb{B})$.  Distributors represent, in some sense, a categorification of relations. (This observation follows the periodic table of category theory, in which a 1-category is a category, a 0-category is a set, and a (-1)-category is a truth value.  There’s a 1-category of relations, with hom-sets $Rel(A,B)$, and each one is a map from $B \times A$ into truth values, specifying whether a pair $(b,a)$ is related.)

The most elementary example of a distributor is the “hom-set” construction, where $\Phi (\mathbb{A},\mathbb{B}) = hom(\mathbb{A},\mathbb{B})$, which is indeed covariant in $\mathbb{A}$ and contravariant in $\mathbb{B}$.  A way to see the general case in that $\Phi$ obviously determines a functor from $\mathbb{A}$ into presheaves on $\mathbb{B}$: $\Phi : \mathbb{A} \rightarrow \hat{\mathbb{B}}$, where $\hat{\mathbb{B}} = Psh(\mathbb{B})$ is the category $hom(\mathbb{B},Sets)$.

In fact, given a functor $F : \mathbb{A} \rightarrow \mathbb{B}$, we can define two different distributors:

$\Phi^F : \mathbb{B} \nrightarrow \mathbb{A}$ with $\Phi^F(A,B) = Hom_{\mathbb{B}}(FA,B)$

and

$\Phi_F : \mathbb{A} \nrightarrow \mathbb{B}$ with $\Phi_F(B,A) = Hom_{\mathbb{B}}(B,FA)$

(Remember, these $\Phi$ are functors from the product into $Sets$: so they are just taking hom-sets here in $\mathbb{B}$ in one direction or the other.)  This much is a tautology: putting a value in $\mathbb{A}$ in leaves a free variable, but the point is that $\hat{\mathbb{B}}$ can be interpreted as a category of “big objects of $\mathbb{B}$“.  This is since the Yoneda embedding $Y : B \hookrightarrow \mathbb{B}$ embeds $\mathbb{B}$ by taking each object $b \in B$ to the presentable presheaf $hom_B(-,b)$ which assigns each object the set of morphisms into $b$, so $\hat{\mathbb{B}}$ has “extended” objects of $\mathbb{B}$.

So distributors like $\Phi$ are “generalized functors” into $\mathbb{B}$ – and the idea is that this is in roughly the same way that “distributions” are to be seen as “generalized functions”, hence the name.  (Benabou now prefers to use the name “profunctor” to refer only to those distributors which map to “pro-objects” in $\hat{\mathbb{B}}$, which are just special presheaves, namely the “flat” ones.)

Now we have an idea that there is a bicategory $Dist$, whose hom-categories $Dist(\mathbb{A},\mathbb{B})$ consist of distributors (and natural transformations), and that the usual functors (which can be seen as distributors which only happen to land in the image of $\mathbb{B}$ under the Yoneda embedding) form a sub-bicategory: that is, post-composition with $Y$ turns a functor into a distributor.

But moreover, this operation has an adjoint: functors out of $\mathbb{B}$ can be “lifted” to functors out of $\hat{\mathbb{B}}$, just by taking the Kan extension of a functor $G : \mathbb{B} \rightarrow \mathbb{X}$ along $Y$.  This will work (pointwise, even), as long as $\mathbb{X}$ is cocomplete, so that we can basically “add up” contributions from the objects of $\mathbb{B}$ by taking colimits.  In the special case where $\mathbb{X} = \hat{\mathbb{C}}$ for some other category $\mathbb{C}$, then this tells us how to get composition of distributors $Dist(\mathbb{A},\mathbb{B}) \times Dist(\mathbb{B},\mathbb{C})\rightarrow Dist(\mathbb{A},\mathbb{C})$.

Now, for a functor $F$, there are straightforward unit and counit natural transformations which makes $\Phi^F$ (the image of $F$ under the embedding of $Cat$ into $Dist$) a left adjoint for $\Phi_F$.  So we’ve embedded $Cat$ into $Dist$ in such a way that every functor has a right adjoint.  What about $Span(Cat)$?  In general, given a bicategory $B$, we can construe $Span(B)$ as a tricategory, which contains $B$, in such a way that every morphism of $B$ has an ambidextrous adjoint (both left and right adjoint).  (There’s work on this by Toby Kenney and Dorette Pronk, and Alex Hoffnung has also been looking at this recently.)  So how does $Span(Cat)$ relate to $Dist$?

One statement is that a distributor $\Phi : \mathbb{A} \nrightarrow \mathbb{B}$ can be seen as a special kind of span, namely:

$\mathbb{A} \stackrel{q}{\longleftarrow} Elt(\Phi) \stackrel{p}{\longrightarrow} \mathbb{B}$

where $Elt(\Phi)$ consists of all the “elements of $\Phi$” (in particular, pasting together all the images in $Sets$ of pairs $(A,B)$ and the set maps that come from morphisms between them in $\mathbb{B}^{op} \times \mathbb{A}$).  (As an aside: Benabou also explained how a cospan, $\mathbb{A} \rightarrow C(\Phi) \leftarrow \mathbb{B}$ can be got from a distributor.  The objects of $C(\Phi)$ are just the disjoint union of those from $\mathbb{A}$ and $\mathbb{B}$, and the hom-sets are just taken from either $\mathbb{A}$, or $\mathbb{B}$, or as the sets given by $\Phi$, depending on the situation.  Then the span we just described completes a pullback square opposite this cospan – it’s a comma category.)

These spans $(Elt(\Phi),p,q)$ end up having some special properties that result from how they’re constructed.  In particular, $p$ will be an op-fibration and $q$ will be a fibration (this, for instance, is alifting property that let one lift morphisms – since the morphisms are found as the images of the original distributor, this makes sense).  Also, the fibres of $(p,q)$ are discrete (these are by definition the images of identity morphisms, so naturally they’re discrete categories).  Finally, these properties (fibration, op-fibration, and discrete fibres) are enough to guarantee that a given span is (isomorphic to) one that comes from a distributor.  So we have an embedding $i : Dist \rightarrow Span(Cat)$.

What’s more, it’s a reflective embedding, because we can always mangle any span to get a new one where these properties hold: it’s enough to force fibres to be discrete by taking their $\pi_0$ – the connected components.  The other properties will then follow.  But notice that this is a very nontrivial thing to do: in general, the fibres of $(p,q)$ could be any sort of category, and this operation turns them into sets (of isomorphism classes).  So there’s an adjunction between $i$ and $\pi_0$, and $Dist$ is a reflective sub-bicategory of $Span(Cat)$.  But the severity of $\pi_0$ ends up meaning that this doesn’t get along well with composition – the composition of distributors (described above) is not related to composition of spans (which works by weak pullback) via this reflection in a naive way.  However, the theorem mentioned above means that there will be SOME reflecction that makes the compositions get along.  It just may not be as nice as this one.

This is kind of surprising, and the ideal punchline to go here would be to say what that reflection is like, but I don’t know the answer to that question just now.  Anyone else know?

Thanks to Bob Coecke, here are some pictures of me, a few of the people from ComLab, and Jean Benabou at dinner at the Oxford University Club, with a variety of dopey expressions as Bob snapped the pictures unexpectedly.  Thanks Bob.