### Tensor Categories in Fredericton

#### Posted by Mike Shulman

While other people are having fun at QPL in Oxford, a bunch of us have been having our own fun on the other side of the pond, at a special session on “Tensor categories” at the summer meeting of the Canadian Mathematical Society in Fredericton, New Brunswick. I thought I’d share a few of the highlights that might be of most interest to you all.

First, in old news for cafe regulars, Geoff Cruttwell talked about our work on generalized multicategories, and I talked about extraordinary multicategories (you can see my slides here). Claudio Hermida was kind enough to point out that in general, the good notion of “partial category” is a paracategory, where the partially defined composition is “unbiased” (although it doesn’t make a whole lot of difference in the case I needed for extraordinary multicategories).

Jeff Egger proposed an equivalent definition of dagger category and dagger compact closed category which is “non-evil”, in that it can be expressed as structure on a category without needing to talk about equality of objects. I didn’t quite follow the details, but it had something to do with the covariant “conjugation” endofunctor of Hilb.

Micah McCurdy described a new version of string diagram calculus for monoidal functors. The more traditional notation involves putting a box around things when you apply a functor to them, whereas his new notation involves “bathing” things in a conduit of color. There were lots of slides with pretty pictures; I hope these will be online sometime.

Susan Niefield described a generalization of the “gluing construction” to double categories, including a double category of topological spaces which may never have been considered before. The idea is this: suppose $D$ is a double category, $C$ is an ordinary category, and $f:C\to V D$ is a normal lax 2-functor from $C$ to the vertical 2-category of $D$. Here the vertical 2-category is the “proarrow” direction, so $f$ is landing in spans, profunctors, relations, etc. Then we can ask about a double-categorical “lax colimit” $\Gamma f$ of $f$, which has a universal property relative to

*horizontal*arrows and 2-cells; call this the*gluing*of $f$.When $D$ consists of categories, functors, and profunctors and $C$ is the interval category, then this gluing $\Gamma f$ is precisely the collage of a profunctor. For this $D$ and other $C$, it is a suitable generalization. In fact, however, it is known that for any category $C$, normal lax functors from $C$ to the bicategory $Prof$ are the same as arbitrary categories over $C$; this generalizes the Grothendieck construction which identifies

*pseudo*functors landing in $Cat$ with (op)fibrations over $C$. (Pseudo functors landing in $Prof$ can be identified with Conduche functors.) Moreover, in this case $C$ itself is $\Gamma$ of the functor constant at the terminal object; thus we have $Lax(C,V D) \simeq H D / \Gamma 1$.Now consider a different $D$, whose objects are topological spaces, whose horizontal arrows are continuous maps, and whose vertical arrows $X\to Y$ are finite-meet-preserving functions $O(X) \to O(Y)$. The versions of this double category for toposes and locales are well-known, and presumably there is a generalization to ionads. Here the gluing construction is a version of the well-known construction for toposes, and it turns out to also induce an equivalence $Lax(C,V D) \simeq H D / \Gamma 1$ as long as $C$ is a finite poset (and also a bit more generally). In this case $\Gamma 1$ is the Alexandrov topology of $C$. In particular, if $C$ is the interval category then $\Gamma 1$ is the Sierpinski space $\mathbb{S}$ (the two-point space with one open point), and a space over $\mathbb{S}$ is just a space with a specified open subset. In this case we recover the “classical” gluing construction which reconstructs a space from an open subspace, its complementary closed subspace, and “gluing data” consisting of a left-exact functor between their posets of opens (or equivalently their categories of sheaves). This double category also has other nice properties; for instance, the “Cauchy complete objects” are precisely the sober spaces.

Since all of these double categories arise from codiscrete cofibrations in some 2-category, I wonder whether there is some underlying general principle at work. Susan was motivated by seeing connections to exponentiability in both cases as well.

Finally, Michael Makkai proposed a definition of

*semistrict $\omega$-category*. The idea is quite simple really. Start with a definition of strict $\omega$-category which takes*whiskering*as a basic operation. I believe this is described in his paper The word problem for computads. There is one basic operation $\cdot$ which acts on two cells of dimensions $m$ and $n$, which share a boundary of dimension $min(m,n)-1$. Thus we can compose two 2-cells along a 1-cell, but not along a 0-cell, and we can whisker a 3-cell with a 2-cell along a 1-cell or a 3-cell with a 1-cell along a 0-cell. The axioms are quite pretty too: there is an*associativity*axiom $a\cdot (b\cdot c) = (a\cdot b)\cdot c$ whenever both sides parse, a*distributivity*axiom $a\cdot (b\cdot c) = (a\cdot b) \cdot (a\cdot c)$ also whenever both sides parse (these conditions being disjoint from those which make associativity parse), and a*commutativity*axiom which essentially states the interchange law.Now, we simply remove the commutativity axiom; this gives a structure which he called an $\omega^-$-category, which is a higher version of a “sesquicategory.” Then you can add, inductively, a sequence of new operations which specify mediating cells where the commutativity axiom would have been, and higher coherences for this; the general pattern of these is apparently not difficult to guess. It seems very intuitively appealing as a “higher sort of Gray-category”, but of course the question is completely open of whether it is actually “semistrict” in the sense of being the target of a semistrictification theorem.

There is also a question of whether the new mediating cells can be required to be

*isomorphisms*(which is what he originally proposed in the talk), or*equivalences*(in which case you would really need to specify their adjoint inverses too, and lots more axioms, at least if you want it to be fully algebraic). My gut feeling is that isomorphisms will be too strict, but I can’t think of any actual evidence. I asked this on MO last night, and it had 10 upvotes by the morning, but no answers. Any takers?

## Re: Tensor Categories in Fredericton

The full program is here.