diagonal functor

The diagonal functor


The diagonal functor is a categorification of the diagonal function.


Let CC be a category. The (binary) diagonal functor of CC is the functor Δ:CC×C\Delta\colon C \to C \times C given by Δ(x)=(x,x)\Delta(x) = (x,x), regardless of whether xx is an object or an arrow of CC.

More generally, let JJ and CC be arbitrary categories. The JJ-ary diagonal functor of CC is the functor Δ J:CC J\Delta_J\colon C\to C^J sending each object cc to the constant functor Δc\Delta c (the functor having value cc for each object of JJ and value 1 c1_c for each arrow of JJ), and each arrow f:ccf\colon c\to c' of CC to the the natural transformation Δf:Δc.Δc\Delta f\colon \Delta c \stackrel{.}{\to} \Delta c' which has the same value ff at each object jj of JJ.


Since CC is JJ-cocomplete (JJ-complete) iff Δ\Delta has a left (right) adjoint, the general adjoint functor theorem may be used in some cases to prove cocompleteness (completeness). For this to work, Δ\Delta must at least preserve small limits (colimits).


Let PP and CC be arbitrary categories. Then Δ P:CC P\Delta_P\colon C\to C^P preserves all limits that exist in CC.


First, recall that limits in functor categories are calculated pointwise. In some detail, if for an object pobj(P)p\in \mathrm{obj}(P) we write E p:X PXE_p:X^P\to X for the ‘’evaluate at pp’‘ functor (with E p(H:PX)=H(p)E_p(H\colon P\to X)=H(p) and E p(σ:H.H)=σ p:H(p)H(p)E_p(\sigma\colon H\stackrel{.}{\to} H')=\sigma_p\colon H(p)\to H'(p)), then we have the following fact (Theorem V.3.1 on p. 115 of Categories Work): If S:JX PS\colon J\to X^P is such that for each object pp of PP, E pS:JXE_p S\colon J\to X has a limiting cone τ p:L(p).E pS\tau_p\colon L(p)\stackrel{.}{\to} E_p S, then there exists a unique functor LL with object function pL(p)p\mapsto L(p) such that τ˜={τ˜ j,p}\tilde{\tau}=\{\tilde{\tau}_{j,p}\} with τ˜ j,p:=τ p,j\tilde{\tau}_{j,p}:=\tau_{p,j} is a cone τ˜:Δ J(L).S\tilde{\tau}\colon \Delta_J(L)\stackrel{.}{\to} S; moreover, this τ˜\tilde{\tau} is a limiting cone from Lobj(X P)L\in \mathrm{obj}(X^P) to S:JX PS\colon J\to X^P.

Back to the proof of the proposition, let F:JCF\colon J\to C be a functor with a limiting cone ν:Δ J().F\nu\colon \Delta_J(\ell) \stackrel{.}{\to} F. We would like to show that Δ Pν:Δ P(Δ J()).Δ PF\Delta_P\nu\colon \Delta_P\circ \bigl(\Delta_J(\ell)\bigr) \stackrel{.}{\to} \Delta_P\circ F is a limiting cone. Noting that Δ P(Δ J())=Δ J(Δ P())\Delta_P\circ \bigl(\Delta_J(\ell)\bigr)=\Delta_J(\Delta_P(\ell)) (where the first Δ J\Delta_J is CC JC\to C^J and the second is C P(C P) JC^P\to (C^P)^J), the last cone may be written as Δ Pν:Δ J(Δ P()).Δ PF\Delta_P\nu\colon \Delta_J(\Delta_P(\ell)) \stackrel{.}{\to} \Delta_P\circ F.

First, we note that for each object pp of PP, E p(Δ PF)E_p\circ(\Delta_P\circ F) is just FF, and therefore has the limiting cone ν:.F\nu\colon\ell \stackrel{.}{\to} F by assumption. Hence, it is clear that Δ PF\Delta_P\circ F has a limit, but we must verify that Δ Pν\Delta_P\nu is a limiting cone.

One functor PXP\to X with object function pp\mapsto \ell is just Δ P()\Delta_P(\ell). For this functor, we have our cone Δ Pν:Δ J(Δ P()).Δ PF\Delta_P\nu\colon \Delta_J(\Delta_P(\ell)) \stackrel{.}{\to} \Delta_P\circ F. Since for all jj and pp we have (Δ Pν) j,p=ν j=jth component of the limiting cone of E p(Δ PF)(\Delta_P\nu)_{j,p}=\nu_j=j\text{th component of the limiting cone of }E_p\circ(\Delta_P\circ F), we are done by the theorem on pointwise limits.

Last revised on December 13, 2011 at 10:11:12. See the history of this page for a list of all contributions to it.