graph of a functor

In as far as the notion of functor generalizes that of function and that of profunctor generalizes that of relation, the notion of *graph* of a (pro)functor generalizes that of graph of a function.

Just as the graph of a function $f : X \to Y$, or more generally that of a relation $R \subset X \times Y$ for $X,Y \in Set = 0 Cat$ is nothing but the category of elements of the corresponding characteristic function $\chi_R : X \times Y \to (-1)Cat = \{0,1\}$, so the graph of a functor $F\colon C \to D$, or more generally that of a profunctor $\chi : C^{op} \times D \to 0 Cat = Set$, is nothing but its category of elements.

Generally, the graph of a functor $f : C \to D$ between $(n,r)$-categories is the Grothendieck construction of the corresponding correspondence: the fibration classified by the correspondence, i.e. the comma object

$\array{
Graph(f) &\to& {*}
\\
\downarrow && \downarrow
\\
C^{op} \times D &\stackrel{\chi_f}{\to}& (n-1,r-1)Cat
}$

whenever this makes sense. For instance in the context of $(\infty,1)$-category theory the graph may be taken to be the fibration classified by $\chi_f : C \times D^{op} \to (\infty,0)$ as described at universal fibration of (∞,1)-categories.

For $n \leq \infty$ let $(n-1) Cat$ and $n Cat$ be a realization of the notions of $n$-category of $(n-1)$-categories and of the $(n+1)$-category of $n$-categories, respectively, such that standard constructions of category theory work, in particular a version of the Yoneda lemma. See higher category theory.

Then with $C,D \in n Cat$ let $f : C \to D$ be a ($n$-)functor.

By the general logic of distributors this defines an $n$-correspondence

$\chi_f : C^{op} \times D \stackrel{f^{op} \times Id}{\to} D^{op}
\times D \stackrel{Func(-,-)}{\to} (n-1) Cat$

The **graph** of $f$ is the fibration $Graph(f) \to C^{op} \times D$ classified by $\chi_f$.

Mike Shulman: It’s not obvious to me that this is the best thing to call the graph of a functor; there are lots of other graphy things one can construct from a functor that all reduce to the usual notion of the graph of a function. To start with, there is of course also the induced opfibration oven $C\times D^{op}$, would you call that the “opgraph”? But actually, the two-sided fibration $D \leftarrow P \to C$ (an opfibration over $C$ and a fibration over $D$) looks to me more like a graph. And then there is of course the *other* profunctor induced by $f$, which gives a fibration over $C\times D^{op}$, an opfibration over $C^{op}\times D$, and a two-sided fibration from $C$ to $D$.

Urs Schreiber: I would be inclined to loosely say “graph” for all of these and to introduce terminology like “opgraph” when it really matters which specific realization we mean. Because all these seem to be so similar to me that I am not sure if it is worth distinguishing them a lot. For instance, wouldn’t an analogous discussion be possible concerning what we call $F^{op} : C^{op} \to D^{op}$ given a functor $F : C \to D$? I don’t actually know what a standard term is, does one say “opfunctor” for this? But I’d say it doesn’t matter much either way, calling $F^{op}$ just a functor which effectively is the functor $F$ doesn’t do much harm.

Colin Zwanziger: Aren’t we better off defining graph of a function as a span to avoid an arbitrary choice of $\langle 1, f\rangle$ or $\langle f, 1\rangle$ and then treating the two-sided fibration as the graph of a functor?Edit: Actually, we would still have to choose whether we were taking the graph of the representable or corepresentable profunctor induced by the functor, since these yield different spans. But we have that two functors F and G are adjoint iff (Lawvere's definition) the (graph of F)_A and (graph of G)_B agree. One level down we would have two functions f and g are adjoint (=inverse) iff (graph of f)_A and (graph of g)_B agree, but the two notions of graph turn out to be the same at this level.

To reproduce the ordinary notion of graph of a function let $(n,r) = (0,0)$. then $(n,r)$-categories $X,Y$ are just sets and a functor $f : X \to Y$ is just a function between sets. Moreover, the category of $(n-1,r) = (-1,0)$-categories is the set $\{0,1\}$ of truth values, as described at (-1)-category. The profunctor corresponding to $f : X \to Y$ is therefore the characteristic function

$\chi_f : X \times Y \to \{0,1\}$

that maps

$\chi_f(x,y) =
\left\lbrace
\array{
1 & if f(x) = y
\\
0 & otherwise
}
\right.
\,.$

(Notice that in this case $X^{op} = X$.)

The 2-pullback of ${*} = {1} \to \{0,1\}$ along $\chi_f$ is just the ordinary pullback

$\array{
Graph(f) &\to& {*}
\\
\downarrow && \downarrow
\\
X \times Y
&\stackrel{\chi_f}{\to}&
\{0,1\}
}$

which identifies $Graph(f) \hookrightarrow X \times Y$ with the subset of pairs $(x,y)$ for which $f(x) = y$. This is the ordinary notion of graph of a function.

For $f : C \to D$ an ordinary functor, with corresponding profunctor $\chi_f : C^{op} \times D \to Set$, the category $Graph(f)$ is the category of elements $\int \chi_f$ of $\chi_f$.

If we regard $C$ and $D$ as 2-categories under the embedding $1Cat \hookrightarrow 2Cat$ then the profunctor $\chi_f$ corresponding to $f$ is of the form $\chi_f : C^{op} \times D \to Cat$ and in this context $Graph(f) \to C^{op} \times D$ is the Grothendieck construction on $\chi_f$.

**graph of a functor**, cograph of a functor- cograph of a profunctor

Revised on August 17, 2014 18:50:41
by Colin Zwanziger
(174.63.87.107)