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check out semicategory
Check also out the theory of plots (partial margmas with more than one object); see also the comment on this math.SE thread:
You can define “isomorphisms” (yes, without identities), and notice that “being an isomorphism” and “admitting an inverse” are different notions in this world, and that they collapse in the categorical world (a category is an associative plot, where the composition is defined and every object has a 1, in the same vein a monoid is an “extremely smooth partial magma”). You can then define isoids, i.e. plots where every arrow is an isomorphism.
We’re even able to define morphisms of plots (p-unctors), natural transformations (trimmings, if I remember well the name Salvatore and I chose), adjoints, limits, and a chain of free-forgetful adjunctions which connects the category (it is a category) of plots to the category of associative plots, semicategories [in this case we’ve even two different adjunctions for two different fully faithful embeddings], and categories. […] Functional analysis and symplectic geometry provide “natural factories” of examples of such structures: one of our two unitization functors applied to the category of symplectic relations gives precisely the Woodward-Wehrheim category.
The morphisms in a category are not so much like the edges in a graph as like the paths in a graph. Edges do not compose, but paths do. Edges are all alike, but paths are not. In particular, some paths are special in a way that identifies them as identity morphisms: the paths of length zero. In particular, an identity morphism is not a loop (and edge from a vertex to itself, which a path of length 1, not 0.) So in identifying the identity morphisms, we do not pick out one loop from all of the loops at a given vertex, label it the identity, and then demand that this labelling be preserved. Instead, we demand that a path of length 0 be mapped to a path of length 0, which is very different.
This is a moral argument, not a logical one. (Urs's and Fosco's answers are also legitimate, after all.) If you define a strict category as a graph with extra structure, then (besides defining composition) you do indeed pick out one loop from all of the loops at a given vertex. But that is missing the point. Turning a graph into a category is not putting some arbitrary structure on the graph; there is a reason for considering this species of structure rather than another (such as the structure of a mere semicategory). And the reason is that we are rethinking the edges of the graph that we started with as paths in some other (perhaps nonexistent) graph. And so each vertex needs some loop to be rethought of as (not a loop at all but) the path of length zero. And this must be preserved because a path length of zero is preserved, not because some arbitrary labelling is arbitrarily preserved.
This moral argument may indeed be just what settles the original question (we’ll see) and I entirely agree with the moral.
But just for completeness maybe one should mention that the distinction between edges in a graph and paths of edges is not so clear-cut. After all, the free-forgetful adjunction between directed graphs and categories, while it sends a graph to its path category, yes, sends a category to the graph whose edges are the morphisms of the category.
(Of course Toby knows this exceedingly well, I am just mentioning it for completeness in the general discussion.)
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