I'm trying to get my head around what a cograph of an nfunctor is. We (some nLab people) are discussing it here. As a start, I'd be happy to understand what the cograph of a 0functor, i.e. function between sets, is.
Thanks!
I'm trying to get my head around what a cograph of an nfunctor is. We (some nLab people) are discussing it here. As a start, I'd be happy to understand what the cograph of a 0functor, i.e. function between sets, is. Thanks! 


In formal terms, the graph of a map $f: X \to Y$ is just the map $$(1, f): X \to X \times Y.$$ This makes sense in any category with products, denoted $\times$. Of course, one has a certain mental picture, at least in the category of sets: I'm thinking of an $X$axis, and a $Y$axis, and the graph as a curve in the absolutely ordinary, schoolmathematics way. I'm also visualizing the way in which each point on the $X$axis has assigned to it a point on the curve, namely, the one directly above it. This assignment is the map $(1, f)$ itself. The image of this map is the curve. In formal terms, the cograph of a map $f: X \to Y$ is just the map $$[f, 1]: X + Y \to Y.$$ This makes sense in any category with sums (coproducts), denoted $+$. (Usually one would write a column vector instead of $[f, 1]$, but I don't want to figure out how to typeset that, and I hope you know what I mean.) This corresponds to a different mental picture of a map (again sticking to maps of sets): it's the one where you draw the set $X$ on the left, as a bunch of dots in a circle, the set $Y$ on the right, as another bunch of dots in a circle, and arrows going from the various points of $X$ to their images in $Y$. This is discussed in Lawvere and Rosebrugh's book Sets for Mathematics. If you follow the link and download the sample chapter, you'll find a picture of the kind I mean, and the word "cograph", on page 2. 


I believe the usual notion of cograph for a function f : A → B between sets is the quotient of A ∐ B by the relation f(a) ~ b, together with its inclusions from A and B. From a homotopy theoretic point of view, you could also call it the mapping cylinder of f. As discussed in your link, rather than identify f(a) and b (or equivalently glue in a 1cell between f(a) and b) you could insert a noninvertible 1morphism to obtain a category. So there are constructions which yield each of a 0category and a 1category. However, in the ncategorical (or (∞,n)categorical) case it seems that you always just get a ncategory, never an (n+1)category. I think what's going on is the fact that you're looking at a diagram of sets, or more generally ncategories, indexed by a 1category {• → •}. The result is a max(n,1)category. If you consider a natural transformation between two functors between 1categories, maybe you can define a cograph which is naturally a 2category. ( 

