Zoran Skoda pseudocoequalizer

Definition

In an arbitrary (strict) 2-category, given a parallel pair of 1-cells $F,G : \mathcal{A}\to\mathcal{B}$ their pseudocoequalizer is a triple $(\mathcal{C},p,\sigma)$ where $p : \mathcal{B}\to\mathcal{C}$ is a functor, $\sigma : p\circ F\implies p\circ G$ is an invertible 2-cell such that for any other 1-cell $r : \mathcal{B}\to\mathcal{D}$ and an invertible 2-cell $\tau : r\circ F\implies r\circ G$ there is a unique 1-cell $u : \mathcal{C}\to\mathcal{D}$ such that $r = u\circ p$ and $\tau = u\sigma$.

Construction in Cat

When the 2-category is Cat, the pseudocoequalizers exist for all parallel pairs of functors. The following explicit construction does the job.

First form a graph $\mathcal{C}_0$ whose vertices (objects) are the objects of $\mathcal{B}$ and which has the set of arrows $\Mor\mathcal{B}\coprod S\coprod S^{-1}$ where

(i) morphisms from $\mathcal{B}$ form a copy of the set $\Mor\mathcal{B}$

(ii) formal arrows $F(a) \stackrel{s_a}\longrightarrow G(a)$, for all $a\in\Ob\mathcal{A}$ form a set $S$

(iii) formal arrows $G(a) \stackrel{s_a^{-1}}\longrightarrow F(a)$ for all $a\in\Ob\mathcal{A}$ form a set $S^{-1}$

Then form a free category $\mathcal{C}_f$ on this graph with composition $\circ_f$. The category $\mathcal{C}$ is the quotient of $\mathcal{C}_f$ by the smallest equivalence relation $\sim$ on the set of morphisms containing the relations from $\mathcal{B}$ (in other words $h\circ_f h' \sim h\circ_\mathcal{B} h'$ for all pairs $h,h'$ of morphisms in $\mathcal{B}$), the relations $s_a\circ s_a^{-1} \sim \id_{G(a)}$, $s_a^{-1}\circ s_a \sim \id_{F(a)}$ for all $a \in \Ob\mathcal{A}$ and $G(f)\circ s_a \sim s_{a'}\circ F(f)$ for all morphism $f : a\to a'$ in $\mathcal{A}$.

Let $[f]$ be the class in $\mathcal{C}$ of the morphism $f\in\Mor\mathcal{B}$. The tautological map $p : f\mapsto [f]$ is in fact a functor (which is identity on objects). The maps $s_a$ are in fact components of a natural isomorphism of functors $s : p F\implies p G$. The triple $(\mathcal{C},p,s)$ is in fact a pseudocoequalizer of the parallel pair $F,G$.

Given $r : \mathcal{B}\to\mathcal{D}$ and invertible $\tau :r F\implies r G$ as above, one defines first the functor $u_0 : \mathcal{C}_0\to\mathcal{D}$ by $u_0(b) = r(b)$ for all objects $b$ in $\mathcal{B}$, $u_0(f) = r(f)$ for all $f\in\Mor\mathcal{B}$, $u_0(s_a) = \tau_a$ and $u_0(s_a^{-1}) = \tau_a^{-1}$ for all $a\in \Ob\mathcal{A}$. This functor trivially extends to a functor $u_f : \mathcal{C}_f\to\mathcal{D}$ on the free category $\mathcal{C}_f$. Finally one checks that the extension $u_f$ factors down to a (unique) functor $u : \mathcal{C}\to\mathcal{D}$. Then $r = u p$ and $\tau = u s$.

More properties in Cat

In $\cat$, the pseudocoequalizer has an additional property which does not hold for pseudocoequalizers in an arbitrary 2-category. Namely, suppose we are given 1-cells $r,r' : \mathcal{B}\to\mathcal{D}$, a 2-cell $\gamma : r\implies r'$ and invertible 2-cells $\tau: rF\implies rG$, $\tau': r'F\implies r'G$ such that the diagram

commutes. Let $u,u' : \mathcal{C}\to\mathcal{D}$ be the functors induced by universality of pseudocoequalizers satisfying $u p=r$ and $u'p =r'$. Then there is a unique 2-cell $\delta : u\implies u'$ such that $\delta\circ p = \gamma$. Its components are in fact identical to the components of $\gamma$, i.e. for all $b \in\Ob\mathcal{B}$, one has $\delta_b = \gamma_b$. The fact that these components form a natural transformation $\delta : u\implies u'$, includes both the naturality of $\gamma$ and the identity above for each object $a\in\mathcal{A}$, and for each morphism $g:b\to b'$ in $\mathcal{B}$ the following diagrams commute: