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Joshua Meyers (Feb 21 2024 at 07:37):Hi,
It's standard to speak of presenting structures by generators and relations (https://ncatlab.org/nlab/show/generators+and+relations). However I find scarcely any mention in the literature of morphisms between such presentations (rather than morphisms between the structures they generate). I found it surprising that category theorists would discuss a kind of thing but then refrain from discussing morphisms between things of that kind. Is this in the literature and I'm just not finding it? Or is there some deeper reason that morphisms of presentations are not discussed?
Here are the mentions/non-mentions I am aware of:
Also relevant would be literature about morphisms between presentations of categories by generators or relations, or between sketches, or insight into why there is not such literature.
For what it's worth, the reason I am inclined to discuss morphisms of presentations is that in computing, the presentation is the data that is stored, so if you want to describe morphisms, you must use presentations as objects -- you do not have access to the "object itself" generated by the presentation.
P.S. I also sent this to the categories mailing list but it was not approved (yet?) So if it is, then the post in #community: mailing list mirror should be cross-referenced to this
Ivan Di Liberti (Feb 21 2024 at 09:17):See Sec. 2 in Monads with arities and their associated theories by Mellies et al. As a general statement, the notion of arity is what you are looking for. The notion of accessible functor is also relevant. See also On finitary functors and their presentations by Adamek et al.
Nathanael Arkor (Feb 21 2024 at 09:30):There's a notion of translation between presentations of second-order algebraic theories in the paper by Fiore and Mahmoud, where they cite Fujiwara's two papers "On Mappings between Algebraic Systems" and "On Mappings between Algebraic Systems, II" for the original study in the algebraic setting.
Nathanael Arkor (Feb 21 2024 at 09:32):Ivan Di Liberti said:
See Sec. 2 in Monads with arities and their associated theories by Mellies et al. As a general statement, the notion of arity is what you are looking for. The notion of accessible functor is also relevant.
These concepts don't seem relevant to the question, as they do not regard syntactic presentations. (Perhaps you are referring to density presentations, but this is a different concept.)
Nathanael Arkor (Feb 21 2024 at 09:34):I also don't see that "On finitary functors and their presentations" contains any consideration of morphisms of presentations.
Nathanael Arkor (Feb 21 2024 at 10:38):Joshua Meyers said:
It's standard to speak of presenting structures by generators and relations (https://ncatlab.org/nlab/show/generators+and+relations).
I think it would be helpful to clarify, though, as perhaps I have misinterpreted your question. The word "presentation" is typically used in universal algebra to mean a signature together with a set of equations between terms derived from the signature. However, the nLab page for "generators and relations" describes a more abstract notion. The examples you give suggest you are after a description in the former style, but linking the nLab page suggests you are after a more abstract description. Which are you referring to?
Joshua Meyers (Feb 21 2024 at 14:55):I am mainly talking about the former style, but references to morphisms of presentations in the latter style would also be appreciated.