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Stream: theory: category theory

Topic: category of presentations

Sacha Ikonicoff (Nov 24 2025 at 19:11):

Hello hiveminds! I was looking at algebras constructions on groups that depend on a given presentation of said groups, and found myself thinking about the category of presentations of groups and presentation morphisms. My thought was, this must have been studied in depth, but I can't find much. I can find a couple of article of Ivanov--Mikhailov, one of Emmanouil--Mikhailov that looks at presentations of a given group, but that's it. A lot of classic sources on combinatorial group theory, Fox's books, or Lyndon--Schupp, seem to avoid the categorical point of view. Please tell me I missed an obvious reference?

It's a fun category with all coproducts but not much if any products, if someone ever wanted an example of such.

Adrian Clough (Nov 24 2025 at 21:06):

There is this https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Group%20theory.pdf

Adrian Clough (Nov 24 2025 at 21:07):

It is written tongue-in-cheek but I think there is a chance it contains information which might be helpful.

Sacha Ikonicoff (Nov 25 2025 at 07:33):

Ah, that's a good reference. I realise that there are different constructions for this category: here they define morphisms $\langle S|R\rangle\to \langle S'|R'\rangle$ as maps $S\to S'$ sending $R\to R'$, while I was looking at morphisms that induce group morphisms on the quotient group, so maps $S\to F(S')$ sending $R$ to the normal closure of $R'$ in $F(S')$, and then I'm surprised they would call theirs the category of groups, since they don't obtain much of the group morphisms - but this is still a good reference to understand the Quillen model structure though.

Amar Hadzihasanovic (Nov 25 2025 at 08:08):

They get the group morphisms as morphisms between fibrant objects, which are the ones whose generators and relations are already suitably "saturated", so that there is a generator for every element of the presented group and a relation for every equality between expressions in the free monoid.

This is a typical pattern in cofibrantly generated model categories. For example, the boundaries of n-simplices are "presentations" of (n-1)-spheres in the Kan--Quillen model structure on simplicial sets, but their morphisms as simplicial sets do not classify any non-trivial generators of the homotopy groups of spheres; one has to pass to equivalent Kan complexes (which are the fibrant objects).

Amar Hadzihasanovic (Nov 25 2025 at 08:14):

Some version of a model structure is a very nice way to capture some of the aspects of the use of "presentations", such as the fact that typically one wants to only consider maps whose domain is a presentation; similarly, for the purposes of computing the homotopy theory presented by a model category, it suffices to consider only morphisms whose domain is cofibrant, and codomain is fibrant.

Sacha Ikonicoff (Nov 25 2025 at 10:31):

Hadn't considered that you could indeed saturate your presentation to get all morphisms. Clever.