The n-Category Café

Nice slides! I’m curious what you said to accompany slides 3 and 4.

Thanks! I just said that although I wasn’t going to assume much category theory, when I did, I’d be proud (like the little boy) rather than being apologetic and blaming John. I also said that category theory isn’t just a language or a way of organizing things, but also a guide as to what to do next.

It would be nice to write up the work on magnitude homology

Definitely! There were lots of requests for a write-up. This community would certainly be interested.

Re the open problems, I haven’t really given them much thought in the last year.

The one exception is the question of what happens if you take coefficients in some other functor $A: [0, \infty] \to Ab$, and my “progress” there has been negative/puzzling. I’m failing to see what interest there is in taking $A$ to be anything other than the “Dirac deltas” called $A_\ell$ in my slides. There’s a structure theorem for functors $[0, \infty] \to Ab$ (or at least $[0, \infty] \to Vect$), which maybe I’ll write about another day, but it puts limits on what such functors there are.

I’d be interested to hear anything you can say about why non-Dirac-delta coefficients seem uninteresting. Is it at all related to my observation last year that for magnitude homology with the obvious interval-parametrized coefficients $A_J$ (with $A_J(\ell)=\mathbb{Z}$ iff $\ell\in J$) the blurring is “non-uniform” in that points near the middle of $J$ are blurred more than the ends?

It’s amazing how many benefits there are to writing one’s slides at the last minute…

In question time, someone raised the possibility mentioned in your second paragraph: homology with coefficients in homology. We had a bit of a laugh about it. But it occurred to me that it might be exactly as wacky and way-out an idea as considering categories enriched in Cat: how crazy would that be!

exactly as wacky and way-out an idea as considering categories enriched in Cat: how crazy would that be!

Or maybe a spectral sequence that converges to the first page of another spectral sequence, or chain complexes with values in (a certain subcategory of) a triangulated category? (-:

Very good; thank you!

Decent photos of Mike were hard to find. In the end I used the $64 \times 64$ pixel one from the front page of this blog, so he comes out looking noticeably blurrier than everyone else, which isn’t true in real life.

cohomology with coefficients in homology is a staple of sheaf theory…

Yeah. And even for someone who doesn’t know about sheaves, there’s the $E_2$ page of the Serre spectral sequence.

I bet you’re thinking about this:

• Charles Rezk, A model for the homotopy theory of homotopy theory.

Abstract. We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy theory”, or more precisely that the category of such models has a well-behaved internal hom-object.

Yes indeed, we should definitely look at magnitude homology for other values of $\mathcal{V}$!

But I thought non-semicartesian $\mathcal{V}$s held no fear for you? In a previous discussion of magnitude homology, you seemed happy to drop the semicartesian hypothesis and replace it by a 2-sided thingy involving a richer system of coefficients. I confess I haven’t got to proper grips with that construction.

Re the result on projective indecomposables, contrary to what we originally thought, the Koszul hypothesis can be dropped. Joe Chuang, Alastair King and I wrote it up as a little paper here. Algebraists refer to the magnitude of this linear category as the “Euler form” of a certain module (called $S$ there).

So, the question is whether the Euler form is the Euler characteristic of some homology theory. Presumably yes! But I don’t know.

Ultrametric spaces: I wrote in the dial post that Mark Meckes’s working-out of “ultramagnitude” was unpublished. It later appeared as section 8 of this paper. But I haven’t thought about the homology.

But I thought non-semicartesian $\mathcal{V}$s held no fear for you? In a previous discussion of magnitude homology, you seemed happy to drop the semicartesian hypothesis and replace it by a 2-sided thingy involving a richer system of coefficients.

Yes; what I should have said is that I’m not sure how to go about choosing the coefficients for a magnitude homology of $FDVect$-categories so as to make it decategorify to magnitude. In general the type of those richer coefficients depends on the particular category we are taking the homology of (like how group homology takes coefficients in a $G$-module), so we can’t just define them once and for all; we would at least need a way to construct such a coefficient system for the category of indecomposable projective $A$-modules for any $A$.

Okay, here is something of an answer to the indecomposable projectives case. In general, the extra coefficients for magnitude homology of a category $X$ consist of a functor $X^{op} \otimes X \to Ab$ (or chain complexes, etc.). Now the category $IP(A)$ of indecomposable projectives in $A$ is Morita equivalent to $A$, so such a functor is essentially determined by an $A$-$A$-bimodule $P$, in which case the magnitude homology should be the Hochschild homology $HH_\ast^K(A;P)$ (where $K$ is the ground field). The appearance of $Ext$ in your formula, however, suggests that we should instead look at magnitude cohomology (which we have noted the existence of, but heretofor otherwise ignored), which in this case should dually give the Hochschild cohomology $HH^\ast_K(A;P)$.

Now, for any two left $A$-modules $M$ and $N$, we have an $A$-$A$-bimodule structure on $Hom_K(M,N)$, and my reference for homological algebra (Weibel) tells me that we have $HH^\ast_K(A;Hom_K(M,N)) = Ext^\ast_{A/K}(M,N)$. And while I don’t see this explicitly in Weibel, I think that because $K$ is a field, the “relative Ext” should coincide with the absolute one $Ext^\ast_{A}(M,N)$. Thus, assuming I didn’t mess something up, the magnitude cohomology of $IP(A)$ with coefficients in $Hom_K(S,S)$ categorifies its magnitude.

This is admittedly a bit unsatisfying; the coefficients $Hom_K(S,S)$ seem chosen somewhat tautologically to make this true. I’d like to understand better what the $IP(A)$-bimodule corresponding to $Hom_K(S,S)$ looks like; maybe it does look like some kind of replacement for a “constant functor at $1$”?

It’s also interesting because the same argument would work with $IP(A)$ replaced by any category Morita equivalent to it, such as the one-object category $A$ itself, or the category of all finitely generated projective modules. Indeed, in general I believe magnitude homology should be Morita invariant, whereas magnitude is not. I suspect that this difference arises from some condition in the presumptive general decategorification theorem that makes it apply to $IP(A)$ but not to $A$, or perhaps a difference in ways of summing a divergent series (in particular, the magnitude of $A$ itself is $1/dim(A)$, which can never arise as a finite alternating sum of dimensions of homology groups).

A different question is whether there are any coefficients for magnitude homology of $IP(A)$ that reproduce the magnitude. Is there any way to express the magnitude in terms of $Tor$ functors (or more generally Hochschild homology) instead of $Ext$?