This is a summary of talks at the conference in Lisbon, continuing from the previous post. The ones I classified under “Field Theory” were a subjective choice, and the categories I list here are even more so, but I think they roughly summarize some of the big themes. I’m hoping to get back to posting here somewhat often, maybe with a wider variety of topics – but for now, this seems like a good start.

### Infinity-Categorical Structures

Simona Paoli gave an overview of infinity-categories generallycalled “Segal-Type Models of Higher Categories“, which was based on her recent monograph of the same title. The talk, which basically summarizes the first chapter in the monograph that lays out the groundwork, described the state of the art on various kinds of higher categories constructed by simplicial methods (like the definitions of Tamsamani and Simpson, etcetera). Since I discussed this at some length back when I was describing the seminar on this subject we did at Hamburg, I’ll just say that Simona’s talk was a nice summary of the big themes we looked at there.

The first talk of the conference was by Dave Carchedi, called “Higher Orbifolds as Structured Infinity-Topoi” (it was a board talk, so there are no slides, but it appears to be based on this paper). There was some background about what “higher orbifolds” might be – to begin with, looking at the link between orbifolds and toposes. The basic idea, as I understand it, being that you can think of nice toposes as categories of sheaves over groupoids, and if the toposes have some good properties and extra structure – like a commutative ring object – you can think of them as being like the sheaves of functions over an orbifold. The commutative ring object is then the structure sheaf for the orbifold, thought of as a ringed space. In fact, you may as well just say such a topos with this structure is exactly what you mean by an orbifold, since there’s a simple correspondence. The way to say this is that orbifolds “are” just Étale stacks. (“Étale”, of a groupoid, means that the source map from morphisms to objects is a local homeomorphism – basically, a sheeted cover. An Étale stack is one presented by an Étale groupoid.)

So then the idea is that a “higher orbifold” should be a gadget that has a similar relation to higher toposes: Étale $\infty$-stacks. One interesting thing about $\infty$-toposes is that the totality of them forms an $\infty$-topos itself. The novel part here is showing that the $\infty$-topos of higher orbifolds also, itself, has the same properties – in particular, a universal structure sheaf called $\theta_U$. This means that it is, in itself, an orbifold! (Someone raised size concerns here: obviously, a category which is one of its own objects presents foundational issues. So you do have to restrict to orbifolds in a universe of some given maximum size, and then you get that the $\infty$-category of them is itself an orbifold – but in a larger universe. However, the main issue is that there’s a universal structure sheaf which gives the corresponding structure to the category of all such objects.)

### Categorification

There was one by Imma Galvez-Carillo, called “Restriction Species“, which talks about categorifying linear algebra, and in particular coalgebra. The idea of using combinatorial species for this purpose has been around for a while – it takes the ideas of Baez and Dolan on groupoid cardinality and linear functors (which I’ve written about plenty in this blog and elsewhere). Here, the move beyond that is to the world of $\infty$-groupoids, using simplicial models. A lot of the same ideas appear: slice categories $\mathbf{Gpd}/B$ over the groupoid of finite sets and bijections $B \in \mathbf{Gpd}$ can be seen as generalizing vector spaces with a specified basis (consisting of the cardinalities 0, 1, 2, … of the finite sets). Individual maps into $B$ are “species”, and play the role of vectors. The whole apparatus of groupoidification gives a way to understand this: the groupoid cardinality of the fibre over each $n$ becomes the component of a vector, and so on. Spans are then linear maps, since composition using the fibre product has the same structure as matrix multiplication. This talk considered an $\infty$-groupoid version of this idea – groupoid cardinality generalizes to a kind of Euler characteristic – and talked about what incidence coalgebras look like in such a context. Another generalization is that decomposition spaces – related to restriction species, which are presheaves on $I$, the category (no longer a groupoid) of finite sets and injections, which carry information about how structures “restrict” along injections. The talk discussed how this gives rise to coalgebras. An example of this would be the Connes-Kreimer bialgebra, whose elements are forests. It turns out this talk just touched on one part of a large project with Joachim Kock and Andrew Tonks – the most obviously relevant references here being this, on the categorified concept called “homotopy linear algebra” involved, and this, about restriction species and decomposition spaces.

One by Vanessa Miemietz on 2-Representations of finitary 2-Categories also tied into the question of algebraic categorification (Miemietz is a collaborator of Mazorchuk, who wrote these notes on that topic). The idea here is to describe monoidal categories which are sufficiently algebra-like to carry an interesting representation theory that’s a good 2-categorical analog for the usual kind for Lie algebras, and then develop that theory. These turn out to be “FIAT” categories (an acronym for “finitary, involution, adjunction, two-category”, which summarizes the kind of structures they need to have). This talk developed some of this theory, including an analog of the Artin-Wedderburn Theorem (which says that all reasonably nice rings are essentially just sums of matrix rings over division algebras), and used that to talk about the representation theory of FIAT categories.

Christian Blohmann spoke about “Morita Equvialence of Higher Geometric Groupoids”. The basic idea was to generalize, to $\infty$-stacks, the correspondence between three different definitions of principle $G$-bundles: in terms of local trivializations and gluing functions; in terms of a bundle over $X$ with a free, proper action of $G$; and in terms of classifying maps $X \rightarrow BG$. The first corresponds to a picture involving anafunctors and a complex of $n$-fold intersections $U_i \cap U_j \cap U_k$ and so on; the third generalizes naturally by simply taking $BG$ to by any space, not just a homotopy 1-type. The talk concentrated on the middle term of the three. A big part of it amounted to coming up with a suitable definition for a “principal” action of an $\infty$-group: it’s this which turns out to involve Morita equivalences, since one doesn’t necessarily want to insist that the action gives strict isomorphisms, but only Morita equivalence.

A talk I didn’t follow very well, but seemed potentially pretty interesting, was Charles Cascauberta’s “Homotopy Algebras vs. Algebras up to Homotopy“. This involved the relation between the operations of taking algebras of a monad $T$ in a model category $M$, and taking the homotopy category. The question has to do with the relation between the two possible orders in which this can be done, and in particular the fact that the two orders give different results.

### Topology and Geometry

Ronnie Brown gave a talk called “Homotopical Excision“, which surveyed some of the ways crossed modules and higher structures can be used in topology. As with a lot of Ronnie Brown’s surveys, this starts with the groupoid version of the van Kampen theorem, but grows from there. Excision is about relative homotopy groups of spaces $X$ with distinguished subspaces $A$. In particular, this talks about unions of those spaces. As one starts taking unions, crossed modules come into the picture, and then higher crossed structures: crossed modules OF crossed modules (which are squares of groups satisfying a bunch of properties), and analogous structures that take the shape of $n$-cubes. There’s a lot of background, so check out the slides, or other work by Brown and others if you’re interested.

Manuel Barenz gave a talk called “Extending the Crane-Yetter Model” which talked about manifold invariants. There’s a lot of categorical machinery used in building these invariants, which uses various kinds of string diagrams: particular sorts of monoidal categories with some nice properties let you interpret knot-like diagrams as morphisms. For example, you need to be able to interpret a bend in which an upward-oriented strand turns around and becomes downward-oriented. There is a whole zoo of different kinds of monoidal category, each giving rise to a language of string diagrams with its own allowed operations. In this example, several different properties come up, but the essential one is pivotality, which says that there is a kind of duality for which this bend is interpreted as the morphism which pairs an object with its dual. If your category is enriched in vector spaces, a knot or link ends up giving you a complex number to compute (a morphism from the identity object to itself). The “string net space” for a manifold amounts to a space spanned by all the ways of embedding this type of graph in the manifold. Part of what this talk speaks to is the idea that such a construction can give the same state space as the Crane-Yetter (originally constructed as a state-sum invariant, based on a totally different construction).

For 4-manifolds, the idea is then that you can produce diagrams like this using the Kirby calculus, which is a way of summarizing a decomposition of the manifold into handle-bodies (the diagrams arise from marking where handles are attached to a 3-sphere boundary of a 4-disk). These diagrams can be transformed, because different handle-body decompositions can be deformed into each other by handle-slides and so forth. So part of the issue in creating an invariant is to identify just what kind of monoidal categories, and what kind of labellings, has just the kind of allowable moves to get along with the moves allowed in the Kirby calculus, and so ensure that the resulting diagrams actually give the same value. This type of category will then naturally be what you want to describe 4-manifold invariants.

### Other talks:

Here are a few talks which, I must admit, went either above my head, or are outside my range of skill to summarize adequately, or in some cases just went by too quickly for me to take adequate notes on, but which some readers might be interested to know about…

Some physics-related talks:

Christian Saemann gave an interesting talk about the relation between the self-dual string in string theory and higher gauge theory using the string 2-group, which is a sort of natural 2-group analog of the spin group $Spin(n)$, and for that reason is surely bound to be at least mathematically important. Martin Wolf’s talk, “Super Yang-Mills Theory from Higher Chern-Simons Theory“, which relates the particular 6-dimensional chiral, superconformal field theory SYM to some combination of twistor geometry with the geometry of gerbes (or “categorified principal bundles”). Branislav Jurco spoke about “Homological Perturbation, Minimal Models, and Effective Actions”, which involved effective actions in the Lagrangian formulation of quantum theories to some higher-algebraic gadgets, such as homotopy algebras.

Domenico Fiorenza’s talk on $T$-duality in rational homotopy theory seemed interesting (in particular, it touched on the Fourier transform as a special case of the “pull-push” construction which I’m very interested in), but I will have to think about this way of talking about it before I could give a good summary. Perhaps reading the associated paper would be a good start.

Operads aren’t really my specialty. The general idea is that they formalize the situation of having operations taking variable numbers of inputs to a given output and describe the structure of the situation. There are many variations which describe possible conditions which can apply, and “algebras” for an operad are actual implementations of such a structure on particular spaces with particular operations. The current theory is rather more advanced, though, and in particular $\infty$-operads seem to be under lots of development right now.

There was a talk by Hongyi Chu on “Enriched Infinity-Operads“, which described how to give a categorification of the notion of “operad” to something which is only homotopy-coherent. Philip Hackney’s talk, “Homotopy Theory of Segal Cyclic Operads” likewise used simplicial presheaves to talk about operads having the property, “cyclic”, which allows inputs to be “rotated” into outputs and vice versa in a particular way.

Other

Andrew Tonks’ talk “Tilings, Trees, DG2A’s and $B_{\infty}$-algebras” (DG2A’s stands for “differential graded 2-algebras”) was quite interesting to me, partly because of the nice combinatorial correspondence it used between certain special kinds of tile-arrangements and particular kinds of trees with coloured nodes. These form elements of those 2-algebras in question, and a lot of it involved describing the combinatorial operations that correspond to composition, the differential, and other algebraic structures.

Johannes Hubschmann’s talk, “Multi-derivative Maurer-Cartan Algebras and Lie-Reinhart Algebras” used a lot of algebraic machinery I’m not familiar with, but essentially the idea is that these are algebras with derivations on them, and a “higher structure” version of such an algebra will have several different derivatives in a coherent way.

Ahmad al-Yasry spoke on “Graph Homologies and Functoriality“, which talked about some work which seems to be closely related to span constructions I’m interested in, and bicategories. In this case, the spans are of manifolds with embedded graphs, and they need to be special branched coverings. This importantly geometric setup is probably one reason I’m less comfortable with this than I’d like to be (considering that Masoud Khalkhali and I spent some time discussing a related paper back when I was at U of Western Ontario. This feeds somehow into the idea – popular in noncommutative geometry – of the Tomita flow, a kind of time-evolution that naturally appears on certain algebras. In this case, there’s a bicategory, and correspondingly two different time evolutions – horizontal and vertical.

So those are the talks at HSL-2017, as filtered through my point of view. I hope to come back in a while with more new posts.