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Maths Problems

A photo of a tree. Here is a list of problems that I'm interested in. They might have existing answers but I don't know them. If you have questions/answers regarding any of these and happen to be reading this, please send me an email!

Bootstrapping Additivity

Homotopy theorists will be familiar with the Dunn additivity theorem, sometimes referred to as the Dunn--Lurie additivity theorem due to Lurie's statement and proof of the theorem as an equivalence $E_n \otimes E_m \simeq E_{n + m}$ of $\infty$-operads. I've been told it's awkward to prove.

For some reason, I wanted a variant of this but with the Swiss-cheese operads. This appears as Theorem 3.1 of Carmona and Švraka's preprint "Additivity of constructible factorization algebras over manifolds with corners". That's great! The proof does involve digging into the details a bit, and I think it would be useful to have a result from which we can directly deduce similar additivity theorems for "module-type" variants of the $E_n$ operads. Here's how I think this could go through.

Using the theory of displayed categories which generalises the Grothendieck construction, every functor $F : E \to C$ can equivalently be encoded as a pseudofunctor $C \to \text{Span}$, where $\text{Span}$ denotes the bicategory of spans. I expect that there is a generalisation of this formalism to $\infty$-operads, and that "module-type" operads $M \to O$ can be encoded as pseudofunctors $S : O^{\otimes} \to \text{Span}^{\otimes}$ (in the Swiss-cheese case we can take $\text{Span}$ to be $\text{Rel}$ instead!). Let's call $M$ the total operad of $S$. The image of $\langle 1 \rangle$ is the desired set of colours of $M$, with the apex of the span encoding which signatures an operation $f$ is allowed to have in $M$, and the 2-cells encoding the composition in $M$. One would hope that forming the total operad commutes with taking the tensor product, so that given another pseudofunctor $T : P^{\otimes} \to \text{Span}^{\otimes}$ encoding $N \to P$, we have $$\int(S \otimes T) \simeq \int S \otimes \int T = M \otimes N,$$ where $S \otimes T$ is represented by the bifunctor $$O^{\otimes} \times P^{\otimes} \xrightarrow{S \times T} \text{Span}^{\otimes} \times \text{Span}^{\otimes} \to \text{Span}^{\otimes}$$ of operads.

Now suppose we already have an equivalence $O \otimes P \simeq Q$ and a candidate $L \to Q$ (which we believe is equivalent to $M \otimes N$) encoded as a pseudofunctor $U :Q^{\otimes} \to \text{Span}^{\otimes}$. Then we can restrict $U$ along the bifunctor $O^{\otimes} \times P^{\otimes} \to Q^{\otimes}$ witnessing the equivalence, and if this agrees with $S \otimes T$, then the theory gives us an equivalence $L \simeq M \otimes N$ automatically. Checking that the pseudofunctors agree is probably not that hard in cases of interest, where the $\infty$-operads come from explicit constructions with topological spaces, and one can probably use a 1-categorical analogue of some of the theory to make these checks.

Comparing Proofs of the Deligne Conjecture

The Deligne conjecture, now a theorem, states that the chain complex computing Hochschild cohomology of an associative algebra carries an $E_2$ structure. Hochschild cohomology can be thought of as the derived centre of an algebra, and just as the centre of a ring is commutative, the derived centre has an $E_2$ structure (a derived analogue of commutativity).

This, and related results, admit a bunch of proofs (I'll add more examples of proofs to this list over time).

I would expect that the $E_2$ structures are all equivalent (they all come from the same sort of observation), but know of no reason why this is true a priori, and it would be satisfying to have a survey paper which compares the different approaches.

Work of Alexandre Quesney involves the construction of a Swiss-Cheese structure on the pair $(ZA, A)$ extending the $E_2$ structure on the McClure-Smith centre of $A$, which gives a comparison map from this to the universal centre. An $E_n$ variant of this appeared in a preprint of Florian de Leger.

Bundles and Classifying Objects

There are several results of the following form: there is a space $B$ such that isomorphism classes of bundles $E \to X$ correspond to homotopy classes of maps $X \to B$. For instance:

etc., and you can probably think of more examples of this behaviour in your field. In each example, a "bundle theory" has an associated classifying object. The problem is: what is a "bundle theory", and is there a way to reliably construct an associated classifying object? I think that abstracting this is probably quite tricky. Note that, while $BO(n)$ classifies bundles in the category of smooth manifolds and $\text{Cat}$ classifies bundles in the category of categories, $BO(n)$ is not a smooth manifold (in the traditional sense) and $\text{Cat}$ is not a 1-category.

One potential starting point is Cockett-Cruttwell's work on tangent categories (see e.g. this paper and Bauer-Burke-Ching's homotopical extension of this (see this paper). The latter gives a common generalisation of "vector bundle" and "bundle of stable $\infty$-categories", so the theory might lead to a reliable construction of a classifying object.

If I remember correctly, my original motivation for this was to see if there's a version of the Pontrjagin-Thom construction for various kinds of stratified spaces in the sense of Ayala-Francis-Tanaka, giving rise to bordism spectra for manifolds with singularities, possibly also coming equipped with $E_n$ structures if certain criteria are met.

Duality of Traces and Cotraces

Callum Reader has a really cool PhD thesis where he internalises the construction of the cotrace of an endomorphism in a monoidal bicategory. Morally speaking (and in a ton of examples) the trace and cotrace are dual, and Callum's thesis tells us that they are both given by an endomorphism of the monoidal unit, placing them in the same context. For example:

At the end of his thesis, Callum asks if there is "some sense in which the cotrace is formally dual to the trace". I'd like to know the answer to this too.