I've always felt that in proving that co/ends are co/limits, Mac Lane's CWM makes use of a category apparently coming out of nowhere.


Let $C$ be a category; I define the *subdivision graph* of $C$ to be the digraph having a vertex $c^\S$ for each object $c\in C$, and a vertex $f^\S$ for each morphism $f : c\to c'$ in $C$, and edges all the arrows $s(f)^\S \to f$ and $t(f)^\S \to f^\S$, as $f$ runs over morphisms of $C$.


Formally adding identities, and taking the trivial function as composition law (so that $u\circ v$ is only defined when at least one arrow is an identity) turns the subdivision graph of $C$ into a category, the *subdivision category of $C$*.


Now, you can say many things about this definition, but certainly not that it is easy to see where it comes from: Mac Lane himself stresses how outside of page 220 (1st edition), where it is used to show that every functor $F : C°\times C\to D$ induces a functor $F^\S : C^\S \to D$, and the end of $F$ is the limit of $F^\S$, there will be no other mention of $C^\S$. So



> 
> where does this definition come from? What is the intuition behind it? How did Mac Lane (or whoever else mentioned it first) come up with it?
> 
> 
> 



Intuition behind Mac Lane's "subdivision category"
Sorry, the following is a bit half-assed, but too long for a comment:


I believe that the nerve of the subdivision category is the edgewise subdivision of the nerve. The edgewise subdivision is the left Kan extension of the functor $[n] \mapsto Nerve([n] + [n]^{op})$, $\Delta \to sSet$.


This should be related to Lurie's quasicategorical treatement of the twisted arrow category. He realizes the twisted arrow category as the right adjoint to the edgewise subdivision (in *Higher Algebra*, I think).


The same story should play out 1-categorically, and the theorem given in Mac Lane should be related to the way Lurie uses the twisted arrow category to treat coends (actually I forget what exactly Lurie does -- but surely he at least treats examples like composition of bimodules).


