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Stream: learning: questions

Topic: Cohomological characterization of monoidal categories

Joshua Meyers (Apr 04 2024 at 20:33):

I have just been studying Sinh's characterization of 2-groups through cohomology by reading:

• Higher-Dimensional Algebra V: 2-Groups, by John Baez

• Hoàng Xuân Sính’s Thesis: Categorifying Group Theory, by John Baez

• Lectures on n-Categories and Cohomology, talks by John Baez, notes by Michael Shulman

Briefly, a (weak) 2-group is a monoidal category $(G,\otimes,1)$ such that each object is weakly invertible. The problem is to characterize these up to weak equivalence. It turns out (see the first of the references above) that every 2-group is equivalent to a "special 2-group", which is a 2-group which is skeletal (so the objects form a group under $\otimes$), whose objects are strictly invertible, and whose unitors are identity.

A special 2-group is susceptible to cohomology: it has a 1-level Postnikov tower, which is simply its projection to the group of objects $G_0$:

$G \to G_0$

(We could also do this with a general 2-group, I think, by projecting down to the group of isomorphism classes of objects, but I haven't worked out the details.)

The "fiber" over $1\in G_0$ is then $A\coloneqq\text{Aut}_G(1)$, which is an abelian group by the Eckmann-Hilton argument. Cohomology says that projections $G\to G_0$ with fiber $A$ are characterized by how $G$ acts on this fiber:

• a 1-action $\rho: G\to \text{Aut}(A)$
• and a 3-action $\alpha:H^3_\rho(G,A)$

I am wondering how one might make a similar characterization in the more general case of monoidal categories $(M,\otimes, 1)$. First, how can we construct a Postnikov tower? Do we project to the monoid $M_0$ of isomorphism classes of objects? Or the monoid of connected components? Or do we truncate $M$ to a monoidal poset? Then I am aware that lacking invertibility, we will need not just the fiber of $1$, but all fibers. So we are perhaps talking about a (lax?) functor from $M_0$ to something. Has anyone worked this out already? Or have some pointers? I am very new to thinking this way.

Graham Manuell (Apr 08 2024 at 05:44):

I don't think this will be possible for general monoidal categories, but I believe you'll find a version of this by looking up about Schreier internal categories and crossed semimodules.