> I've never studied categories, but my reading of the parent comment was that a category was the combination of a graph and a seperate identification of paths within the graph in addition to the bare graph.
Pretty much, yes. In group theory we call this a "presentation": we give the generating elements of the group, followed by the equations to "mod" by. For instance, the group presented by <x, y | xy = yx> is the free commutative group on two elements -- the equation we've modded by forces commutativity, but leaves us otherwise unconstrained. Or the group presented by <x | x^5 = 1> -- that's the cyclic group of order 5.
Categories are novel algebraic structures in some respects, but conceptually relatively standard in others. A naive "presentation" of a category is exactly as you say: you define some fundamental edges and assert any desired equivalences of paths composed from them [0]. But, given the expressive power of categories, sometimes this is still quite explicit and clunky. In particular, you don't usually want to spell out all of the basic arrows you're starting with.
> I guess you could represent that as an extra complex graph, but I'm guessing that would be unwieldly
That being the case, it's often easier to define a model category by stating how it relates to other, already-understood categories. A "category with finite products" is any category that relates in a particular way to the category with two objects and no non-trivial morphisms. Of course, the phrase "a particular way" is load-bearing, but it's formalized on the notion of "functor" and "universal property". It's a much more implicit kind of definition, but in turn it gives you a very powerful tool for speaking about your category.
Pretty much, yes. In group theory we call this a "presentation": we give the generating elements of the group, followed by the equations to "mod" by. For instance, the group presented by <x, y | xy = yx> is the free commutative group on two elements -- the equation we've modded by forces commutativity, but leaves us otherwise unconstrained. Or the group presented by <x | x^5 = 1> -- that's the cyclic group of order 5.
Categories are novel algebraic structures in some respects, but conceptually relatively standard in others. A naive "presentation" of a category is exactly as you say: you define some fundamental edges and assert any desired equivalences of paths composed from them [0]. But, given the expressive power of categories, sometimes this is still quite explicit and clunky. In particular, you don't usually want to spell out all of the basic arrows you're starting with.
> I guess you could represent that as an extra complex graph, but I'm guessing that would be unwieldly
That being the case, it's often easier to define a model category by stating how it relates to other, already-understood categories. A "category with finite products" is any category that relates in a particular way to the category with two objects and no non-trivial morphisms. Of course, the phrase "a particular way" is load-bearing, but it's formalized on the notion of "functor" and "universal property". It's a much more implicit kind of definition, but in turn it gives you a very powerful tool for speaking about your category.
[0] https://ncatlab.org/nlab/show/presentation+of+a+category+by+...