Cograph

Cograph of a function

The cograph of a function $f:A\to B$ is a quotient of the coproduct $A+B$ by the equivalence relation generated from $\iota_{1}(a)\sim\iota_{2}(b):=(f(a)=b)$.

We can also describe the cograph as a category, using the following (strict) pullback:

In displayed land, this would be the pullback of pointed sets along the functor that picks out the domain and codomain of $f$, along with the map between them. Therefore, the resulting category would have objects $A\rightarrowtriangle[0]$, $B\rightarrowtriangle[1]$, and a proofs that $f(a)=b$ over $[0]\to[1]$.

Cograph of a functor

We can also form the cograph of a functor $F:\mathcal{C}\to\mathcal{D}$; this yields a fibration over the interval category, where $\mathcal{C}_{0}\rightarrowtriangle[0]$, $\mathcal{D}_{0}\rightarrowtriangle[1]$, and $\mathcal{D}(F(c),d)\rightarrowtriangle([0]\to[1])$

Cograph of a profunctor

More generally, the cograph of a profunctor $P:\mathcal{D}\nrightarrow\mathcal{C}$ is a fibration over the interval category, where $\mathcal{C}_{0}\rightarrowtriangle[0]$, $\mathcal{D}_{0}\rightarrowtriangle[1]$, and $P(c,d)\rightarrowtriangle([0]\to[1])$

Cograph of a 2-sided fibration

References

• https://ncatlab.org/nlab/show/graph+of+a+function#cograph
• https://ncatlab.org/nlab/show/cograph+of+a+functor