nLab
diagonal functor

Idea

The diagonal functor is a categorification of the diagonal function.

Definitions

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

More generally, let $J$ and $C$ be arbitrary categories. The $J$-ary diagonal functor of $C$ is the functor ${\Delta }_J:C\to C^J$ sending each object $c$ to the constant functor $\Delta c$ (the functor having value $c$ for each object of $J$ and value $1_c$ for each arrow of $J$), and each arrow $f:c\to c\prime$ of $C$ to the the natural transformation $\Delta f:\Delta c\stackrel{.}{\to }\Delta c\prime$ which has the same value $f$ at each object $j$ of $J$.

Properties

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

Proposition

Let $P$ and $C$ be arbitrary categories. Then ${\Delta }_P:C\to C^P$ preserves all limits that exist in $C$.

Proof

First, recall that limits in functor categories are calculated pointwise. In some detail, if for an object $p\in \mathrm{obj}(P)$ we write $E_p:X^P\to X$ for the ”evaluate at $p$” functor (with $E_p(H:P\to X)=H(p)$ and $E_p(\sigma :H\stackrel{.}{\to }H\prime )={\sigma }_p:H(p)\to H\prime (p)$), then we have the following fact (Theorem V.3.1 on p. 115 of Categories Work): If $S:J\to X^P$ is such that for each object $p$ of $P$, $E_{p}S:J\to X$ has a limiting cone ${\tau }_p:L(p)\stackrel{.}{\to }{E}_{p}S$, then there exists a unique functor $L$ with object function $p\mapsto L(p)$ such that $\tilde{\tau }=\{{\tilde{\tau }}_{j,p}\}$ with ${\tilde{\tau }}_{j,p}:={\tau }_{p,j}$ is a cone $\tilde{\tau }:{\Delta }_J(L)\stackrel{.}{\to }S$; moreover, this $\tilde{\tau }$ is a limiting cone from $L\in \mathrm{obj}(X^P)$ to $S:J\to X^P$.

Back to the proof of the proposition, let $F:J\to C$ be a functor with a limiting cone $\nu :{\Delta }_J(\ell )\stackrel{.}{\to }F$. We would like to show that ${\Delta }_P\nu :{\Delta }_P\circ ({\Delta }_J(\ell ))\stackrel{.}{\to }{\Delta }_P\circ F$ is a limiting cone. Noting that ${\Delta }_P\circ ({\Delta }_J(\ell ))={\Delta }_J({\Delta }_P(\ell ))$ (where the first ${\Delta }_J$ is $C\to C^J$ and the second is $C^P\to (C^P{)}^J$), the last cone may be written as ${\Delta }_P\nu :{\Delta }_J({\Delta }_P(\ell ))\stackrel{.}{\to }{\Delta }_P\circ F$.

First, we note that for each object $p$ of $P$, $E_p\circ ({\Delta }_P\circ F)$ is just $F$, and therefore has the limiting cone $\nu :\ell \stackrel{.}{\to }F$ by assumption. Hence, it is clear that ${\Delta }_P\circ F$ has a limit, but we must verify that ${\Delta }_P\nu$ is a limiting cone.

One functor $P\to X$ with object function $p\mapsto \ell$ is just ${\Delta }_P(\ell )$. For this functor, we have our cone ${\Delta }_P\nu :{\Delta }_J({\Delta }_P(\ell ))\stackrel{.}{\to }{\Delta }_P\circ F$. Since for all $j$ and $p$ we have $({\Delta }_P\nu {)}_{j,p}={\nu }_j=j\text{th component of the limiting cone of}{E}_p\circ ({\Delta }_P\circ F)$, we are done by the theorem on pointwise limits.

Related concepts

• diagonal morphism