Definition of a profinite category When studying objects like profinite groups, profinite spaces and profinite rings, I have noticed that some properties just remain the same. For example they will always be inductive limits of some discrete finite spaces, or equivalently totally disconnected compact Hausdorff spaces.
I was therefore wondering, if there is some kind of a categorical generalization of these notions. Something I would call a profinite category.
I was thinking defining it either in terms of a subcategory of profinite spaces, where the objects would be profinite spaces and morphisms some subsets of continuous maps between two profinite spaces. (For profinite groups we'd require them to be morphisms of groups, for rings morphisms of rings and for modules, morphisms of modules etc...)
Other possible definition, I had in mind would be taking a subcategory of the category of finite sets with some properties that would make it possible to do projective limits and then define the objects in the profinite category as these projective limits.
Are there any references for these kinds of notions?
 A: There is a notion of pro-object in a general category $\mathbf{C}$, which generalises the usual profinite objects - profinite spaces are the case $\mathbf{C} = \mathbf{FinSet}$, profinite groups are $\mathbf{C} = \mathbf{FinGroup}$, etc. See this nLab article for details. Perhaps this is what you are looking for.
A: There are two natural definitions of a profinite category.  You can look at inverse limits of finite categories or you can look at topological categories whose underlying spaces are profinite (call these Stone categories).  Any profinite category is Stone and any Stone category with finitely many objects is profinite.  But there are Stone categories that are not profinite.  Many years ago I asked George Bergman for an example, and in typical Bergman fashion he immediately provided me one.  It's been many years, so I might be slightly off, but there is a well known example of a Stone lattice that is not a profinite lattice.  Any poset can be viewed as a category and I believe just used that poset, viewed as a category, as his example or something very close to that.
Stone categories come up very frequently in monoid theory via taking the free Stone category on a profinite graph.  This comes up when trying to understand semidirect product decompositions of monoids.
It is typical given a homomorphism of profinite monoids $f\colon M\to N$ to build the category of elements of $N$ viewed as a right $M$-set with $M$ acting on $N$ via $f$.  This is a profinite category and so can be generated by a profinite graph.  People then like two write it as a quotient of a free Stone category on that profinite graph to get profinite identity theory involved.  But because that free Stone category might not be profinite, troubles arise.
There is lots of work on this by Jorge Almeida and his former students.  The original goal was to find profinite identities defining semidirect products of pseudovarieties of monoids, but the approach has not been 100% successful and the profinite versus Stone issue is part of the difficulties.
This paper might give some flavor of what people look at.  There are older papers referred to within.
A: I would like to post my answer as well. In the Chapter 3 of my thesis:
"Automorphism-preserving color substitutions on Profinite Graphs" (2022). Electronic Thesis and Dissertation Repository. 8795.
https://ir.lib.uwo.ca/etd/8795 ,
I came up with a certain categorical notion that worked for what I intended to do: prove the common properties of structures such as profinite groups, graphs and rings for one general category. The name: profinite category is already being used for a different approach: see for example the paper by J. Almeida and P. Weil https://www.sciencedirect.com/science/article/pii/S0022404996000837 , or the post by @Benjamin Steinberg. I called my category instead a Profinite structure.
Since profinite objects are essentially completions of finite objects, I decided that I first needed a functor $F$, into the category of sets that let me make a difference between finite and infinite objects. I also wanted these completions to exist, so I required that $F$ transforms a projective limit of finite objects into a projective limit of sets. With these points in mind I defined a "preprofinite" category, which is essentially the category of objects I want to complete: one can think of for example rings or groups. The definition of a preprofinite category is:
A couple $(\mathcal{C},F)$ with $\mathcal{C}$ being a category and $F$ a functor with:

*

*$F$ faithful

*If $(X_i)_{i \in I}$ together with transition maps $\phi_{i,j}$ is a a projective system in $\mathcal{C}$, such that $F(X_i)$ is finite for all indexes $i \in I$, then there the system has a projective limit. Furthermore if $(X,(\pi_i)_{i \in I})$ is a projective limit of such a system, then $(F(X),(F(\pi_i)_{i \in I})$ is a projective limit of the projective system $(F(X_i))_{i \in I}$ with $F(\phi_{i,j})$ being the transition maps.

*For all objects $A,B,C$ in $\mathcal{C}$ and for all morphisms $u$ from $B$ to $A$, $v$ from $B$ to $C$ and $f$ a map from $F(A)$ to $F(C)$, if $F(u)$ is surjective and $f \circ F(u)=F(v)$, then there exists a morphism $w: A \rightarrow C$, such that $f=F(w)$.

Typically one should think of $F$ as a forgetful functor. The last axiom establishes a certain relation between maps and maps that are morphisms and was useful to me to prove the results that I needed.
Now that we defined what I called a preprofinite category, one can simply define a profinite stucture as a completion of finite objects in that category. That is:
If $(\mathcal{C},F)$ is a preprofinite category, define a profinite structure on $(\mathcal{C},F)$ as a category $\mathcal{P}$, whose objects are limits of projective systems of finite objects in $\mathcal{C}$ and whose morphisms are morphisms $u$ in $\mathcal{C}$, such that $F(u)$ is continuous for the induced profinite topologies on the sets.
