Zoran Skoda


In an arbitrary (strict) 2-category, given a parallel pair of 1-cells F,G:𝒜F,G : \mathcal{A}\to\mathcal{B} their pseudocoequalizer is a triple (𝒞,p,σ)(\mathcal{C},p,\sigma) where p:𝒞p : \mathcal{B}\to\mathcal{C} is a functor, σ:pFpG\sigma : p\circ F\implies p\circ G is an invertible 2-cell such that for any other 1-cell r:𝒟r : \mathcal{B}\to\mathcal{D} and an invertible 2-cell τ:rFrG\tau : r\circ F\implies r\circ G there is a unique 1-cell u:𝒞𝒟u : \mathcal{C}\to\mathcal{D} such that r=upr = u\circ p and τ=uσ\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 𝒞 0\mathcal{C}_0 whose vertices (objects) are the objects of \mathcal{B} and which has the set of arrows MorSS 1\Mor\mathcal{B}\coprod S\coprod S^{-1} where

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

(ii) formal arrows F(a)s aG(a)F(a) \stackrel{s_a}\longrightarrow G(a), for all aOb𝒜a\in\Ob\mathcal{A} form a set SS

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

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

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

Given r:𝒟r : \mathcal{B}\to\mathcal{D} and invertible τ:rFrG\tau :r F\implies r G as above, one defines first the functor u 0:𝒞 0𝒟u_0 : \mathcal{C}_0\to\mathcal{D} by u 0(b)=r(b)u_0(b) = r(b) for all objects bb in \mathcal{B}, u 0(f)=r(f)u_0(f) = r(f) for all fMorf\in\Mor\mathcal{B}, u 0(s a)=τ au_0(s_a) = \tau_a and u 0(s a 1)=τ a 1u_0(s_a^{-1}) = \tau_a^{-1} for all aOb𝒜a\in \Ob\mathcal{A}. This functor trivially extends to a functor u f:𝒞 f𝒟u_f : \mathcal{C}_f\to\mathcal{D} on the free category 𝒞 f\mathcal{C}_f. Finally one checks that the extension u fu_f factors down to a (unique) functor u:𝒞𝒟u : \mathcal{C}\to\mathcal{D}. Then r=upr = u p and τ=us\tau = u s.

More properties in Cat

In cat\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:𝒟r,r' : \mathcal{B}\to\mathcal{D}, a 2-cell γ:rr\gamma : r\implies r' and invertible 2-cells τ:rFrG\tau: rF\implies rG, τ:rFrG\tau': r'F\implies r'G such that the diagram

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

Created on February 21, 2011 at 14:52:33. See the history of this page for a list of all contributions to it.