I have been testing the waters of Category theory and came across the following theorem in Hungerfords Algebra:
Theorem 7.3
If $(P,\{\pi_i\})$ and $(Q,\{\psi_i\})$ are both products of the family $\{A_i:i\in I\}$ of objects in a category $\mathcal{C}$, then $P$ and $Q$ are equivalent.
The proof given in the text basically composes the two following diagrams:
and argues by uniqueness that $f\circ g=\mathrm{id}$.
When I tried to prove it before looking at the proof provided in the book my first thought was to try looking at isomorphisms of directed graphs. Indeed it does seem natural to consider graphs when studying categories. Similar questions have been asked before like: Is Category Theory similar to Graph Theory, but this question didn't really provide insight into whether or not a graph theory approach is useful to the study of categories — while the theories may have similar aesthetic in terms of how we visualize them, can graph theory be used in proving results about categories? However, I do know that graphs themselves are not quite powerful enough to completely determine categories and their morphisms Is every finite category identifiable with a multigraph? (and vice versa?), but none the less my gut tells me there must be some applications of directed graphs to category theory even if in a somewhat limited domain. Any suggested readings would be appreciated!
Best Answer
As far as I know, there aren't a lot of theorems in Category Theory that were proved by considering certain aspects of infinite directed graphs. There are, in fact, several ways Mathematicians have understood Category Theory by appealing to Graph Theoretic notions.
Sketches are often defined by a 4-tuple of graphs (though not in the n-Lab page...). These come up when studying accessible categories/locally presentable categories. See p. 124 of Toposes, Triples, and Theories for a definition in terms of the 4-tuple of graphs. Even categories with set-sized collections of objects and morphisms have a large (at least proper class size) category of Sketches.
Every small (or accessible, if using Choice) category determines a skeleton, which has essentially all of the properties of the Category, and removes extraneous isomorphisms.
Tom Leinster wrote a post on the n-Category Cafe discussing which graphs can be given a Category structure. This may be of interest to you.