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Software: 0.03G. 1: 5: . 5. Get free recovery tools: 1: 0: . 07.jpg UPDATED · 9Jul2020
PATCHED Recuva 4 Pro features a couple of options that might appeal to you. D-e-x-t-e-r-a-t-e-d.pdf UPDATED · 13Jul2020 Recover.My.Files.Portable.v3.98.5282.rar (1; 2 of 2).Q:

Confusion about finite categorical groups

In this document, the author seems to talk about what (I assume to be a group) $C$ being a finite category rather than what it is a finite category in the first place (I have no experience in category theory).
It says that the set of objects $Ob(C)$ of $C$ is finite iff the group $\pi_0(C)$ generated by the elements $[x,y]$ of the set $Hom_{C}(x,y)$ of morphisms from $x$ to $y$ in $C$ is finite.
It seems that what makes $Hom(x,y)$ finite is that there are no elements $r\in Hom(x,y)$ which cannot be expressed as a finite product of other elements in $Hom(x,y)$.
Also, an element $r\in Hom(x,y)$ can always be written as a product $r=r_1\dots r_n$ where $r_i \in Hom(x,y)$.
Is this correct?
I would be grateful if anybody can explain.

A:

What you say is correct. It is a standard idea in universal algebra to consider a group $G$ to be a $\mathbb{Z}$-graded group (where there is an internal operation in $G$ of degree $0$ and $1$), and the simplest example of such a thing is a finite group. Of course, then a morphism $f: G \to G’$ has degree $1$ iff $f$ is an isomorphism. A little extra work is needed to show that such morphisms are in fact closed under composition.
More generally, one thinks of a $\mathbb{Z}$-
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