Since coming back from Montreal, I’ve given an exam for a very large linear algebra class, but before I forget, I’d like to make a few notes about some of the talks.

The first day, Saturday, October 4, was a long day of mostly half-hour talks, and some 20-min talks, including my late-registering contribution. It was about the 2-linearization of spans of groupoids which I’ve talked about before, but with a problem fixed. I’ll say more about that soon.

It was interesting to see the range of talks – category theory spans a few areas of mathematics, after all. To start off the day, there was a session in which Michael Makkai and Victor Harnik both gave talks about higher-dimensional categories in one form or another.

Makkai’s was about “revisiting coherence in bicategories and tricategories”. Coherence is an issue that comes up once you get into higher categories – that is, looking at things bearing more complicated relationships than “equal” and “not-equal”, such as “isomorphic”, or “equivalent”. Or “biequivalent”, I suppose – Makkai covered some work of Nick Gurski and Steve Lack about how bicategories and tricategories are (or are not) equivalent to strict versions of themselves. More precisely, that there’s a biequivalence between (the strict form) and (the weak form). Whereas there is no triequivalence between (strict) and (weak) . There is a triequivalence between and – something intermediate between strict and weak. He also explained how these equivalences pass through a relationship with the category of graphs. (An equivalence is a pair of adjoint functors – the equivalence between and factors through pairs of adjoint functors between each of these and ). There was more to the talk, but it was somewhat over my head.

Harnik’s talk, “Placed composition in higher dimensional categories”, was about a recursive way of defining partial composition operations in higher dimensions. Here, the point is that it’s easy and obvious how to compose one-dimensional arrows: you stick them tip-to-tail. Higher-dimensional morphisms need more complicated rules telling how to stick them together along various numbers of shared faces. (A line-segment arrow has only two faces, both points with no sub-faces). Harnik described how to generate an -category recursively: generate faces of dimension by freely adjoining some indeterminate cells, which need all these operations telling how they can be stuck together. Then you have to impose some algebraic relations – certain composites are the same. This is like a problem of presenting groups in terms of generators and relations: it can be hard to tell whether two elements are equal or not – two elements being declared equal if they can be proved so in some algebraic system (not an easy question to test, usually).

In fact, questions about computability came up a lot, since there is a lot of interaction between category theory and computer science. We saw several talks that touched on that in the afternoon: B. Redmond gave a talk, “Safe Recursion Revisited”, about a categorical point of view on defining recursion “safely” (i.e. keeping algorithms in polynomial time); G. Lukacs described “A cartesian closed category that might be useful for higher-type computation” – higher types being apparently the type-theory correlate of higher categories. We had heard about this earlier – M. Warren talked on “types and groupoids”, showing how to use -groupoids to look at types, variables of those types (objects), and terms or “elements of proofs” (as morphisms), and so on for “higher types”. A different take on the intersection between computing and categories was N. Yanofsky’s talk “On the algorithmic informational content of categories”, which applied Kolmogorov complexity (the size of a turing machine required to produce a given output) to productions describing categories. Productions like the one that takes a simpler description – of the category of topological spaces, say – and turns it into a more complex one, like the category of pointed topological spaces. Or from vector spaces to Banach spaces, or what-have-you. He described a little language that can be used to specify (some, not all) categories by such operations, starting with a few building blocks – which then allows you to ask about the Kolmogorov complexity of the category itself.

On a different vein, there was also a reasonable cross-section of topological ideas going around. Certainly any time -groupoids come up, it also comes up that they classify homotopy types of spaces. But much more detailed geometric pictures also come up. Walter Tholen talked about the Gromov metric on the category of metric spaces: the distance between two metric spaces is defined as a minimum over all possible isometric embeddings into a common space, of a certain maximum separation between the spaces. One can then talk about Cauchy sequences of metric spaces, and the fact that (for example), the category of complete metric spaces is itself complete.

Dorette Pronk also brought in some geometry when she talked about “Transformation groupoids and orbifolt homotopy theory”. I’m quite interested in transformation groupoids, which show up when a set is acted on by a group. The example I’ve talked about is from gauge theory, where there is a group of gauge transformations acting on the moduli space of configurations (i.e. connections). This was one of the examples she gave for where these sorts of things come from. Then she got into the connections between these sorts of groupoids and the homotopy theory of orbifolds. Orbifolds are like manifolds, except that their neighborhoods have isomorphisms to , where is an open set in , and is a finite group (a nontrivial group action distinguishes orbifolds from mere manifolds). Most can be said in the case where the orbifold is just where is a manifold and is a Lie group, acting globally. Orbifolds like this are called representable.

Now, orbifolds have groupoids associated to them (in various ways), and Dorette Pronk’s talk dealt with the fact that the orbifolds being representable (i.e. arising from a global group action) is equivalent to the associated groupoid being Morita equivalent to a transformation groupoid (i.e. one arising from a global group action). Morita equivalence for groupoids and turns out to be the same as having a nice enough SPAN of groupoids

So in fact here are spans of groupoids again – just the sort of thing I was there to talk about, and should have more to say on here shortly. So that was interesting. This situation of having a span of groupoids seems to show up in several different guises.

There were some other talks I’ve missed, but it’s taken me a while to get to this, and some of them have faded a bit, so I’ll cut this short there.

October 14, 2008 at 7:43 am

Thanks for the useful report!

Interesting coincidence. I am in the middle of thinking about something related to this. I am mostly following the nice work by Francois Metayer on this topic.

In Resolutions by polygraphs he discusses a procedure to obtain omega-categories which are “degreewise freely generated” from a globular set.

This is pretty much the familiar generators-and-relations description of 1-categories, only that now each relation of generators of k-morphisms is itself a generator of (k+1)-morphisms and subject to further relations in degree (k+2). And so on.

In Cofibrant complexes are free Metayer shows that the omega-categories obtained this way are precisely the cofibrant ones in the folk model structure on omega-cat. This is a nice consistency check on the construction.

Did Harnik mention Metayer’s work? Does he have an article about his work that I can look at?

October 14, 2008 at 7:50 am

Wait, what precisely do you mean by “omega-groupoid” here? Usually the statement is that these do not classify homotopy types.

We recently talked about that in Semistrict Infinity-Categories.

The omega-groupoids mentioned there are the entirely strict ones: all composition is strictly associative and all k-morphisms have strict inverses for all k. These fail to capture all homotopy types from degree 3 on:

While strict 3-groupoids are insufficient to model all 3-types, Gray-groupoids are sufficients. This is supposed to be just another incarnation of the fact you mention above, that every tricategory is at best equivalent to a Gray category, but not in general to a strict 3-category.

But I know that sometimes people say “omega-groupoid” for something weaker than the “true” omega-groupoids. Which definition precisely do you have in mind?

October 15, 2008 at 5:40 pm

Hi Urs:

I can’t find a paper by Harnik to go with that talk, but I imagine there is one somewhere.

As for omega-groupoids, I don’t know that I’m thinking of one particular definition, except that my default assumption is that an n-something or omega-something should be as weak as necessary for the greatest possible generality (that way, you can add adjectives to denote strict ones). So I guess I’m thinking of something like weak (infinity,1)-categories?

October 15, 2008 at 7:12 pm

Okay, I see. I am aware that some people use the “omega” this way, but I find it a bit odd, since my impression had been (possibly incorrectly) that the use of “omega” over “infinity” has originally (and still so in the majority of publications) precisely been to emphasize that one refers to the strict version, because the “omega” alludes the the fact that we reach omega-categories by taking the limit of the iterated enrichment

(n+1)Cat = nCat-Cat

starting with 0Cat = Set, which produces strict n-categories at each level.

In any case, why should one otherwise say “omega-category” instead of “infinity-category”?

Well, sorry for the terminological nitpicking. It’s not really important. Feel free to ignore this comment of mine.

October 15, 2008 at 9:55 pm

Hmm.. Okay, I wasn’t aware that people distinguish the two terms that way. It’s yet another non-systematic usage – which is a little annoying, like the way people use hard-to-generalize latinate terms like “bicategory” to mean what ought to be called a 2-category (as opposed to a *strict* 2-category, which is what that term tends to mean). Still, I’d rather know what people will think I mean when I use a word, so that’s good to know…

July 2, 2009 at 9:09 pm

[…] Warren’s talk, “Martin-Löf complexes” (I reported on a similar talk he gave at Octoberfest last year). The idea here is to start by defining a globular set, given a theory and type – […]