category theory

# The 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 $\Delta :C\to C×C$ given by $\Delta \left(x\right)=\left(x,x\right)$, 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 $p\in \mathrm{obj}\left(P\right)$ we write ${E}_{p}:{X}^{P}\to X$ for the ”evaluate at $p$” functor (with ${E}_{p}\left(H:P\to X\right)=H\left(p\right)$ and ${E}_{p}\left(\sigma :H\stackrel{.}{\to }H\prime \right)={\sigma }_{p}:H\left(p\right)\to H\prime \left(p\right)$), 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 ${\tau }_{p}:L\left(p\right)\stackrel{.}{\to }{E}_{p}S$, then there exists a unique functor $L$ with object function $p↦L\left(p\right)$ such that $\stackrel{˜}{\tau }=\left\{{\stackrel{˜}{\tau }}_{j,p}\right\}$ with ${\stackrel{˜}{\tau }}_{j,p}:={\tau }_{p,j}$ is a cone $\stackrel{˜}{\tau }:{\Delta }_{J}\left(L\right)\stackrel{.}{\to }S$; moreover, this $\stackrel{˜}{\tau }$ is a limiting cone from $L\in \mathrm{obj}\left({X}^{P}\right)$ 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 $\nu :{\Delta }_{J}\left(\ell \right)\stackrel{.}{\to }F$. We would like to show that ${\Delta }_{P}\nu :{\Delta }_{P}\circ \left({\Delta }_{J}\left(\ell \right)\right)\stackrel{.}{\to }{\Delta }_{P}\circ F$ is a limiting cone. Noting that ${\Delta }_{P}\circ \left({\Delta }_{J}\left(\ell \right)\right)={\Delta }_{J}\left({\Delta }_{P}\left(\ell \right)\right)$ (where the first ${\Delta }_{J}$ is $C\to {C}^{J}$ and the second is ${C}^{P}\to \left({C}^{P}{\right)}^{J}$), the last cone may be written as ${\Delta }_{P}\nu :{\Delta }_{J}\left({\Delta }_{P}\left(\ell \right)\right)\stackrel{.}{\to }{\Delta }_{P}\circ F$.

First, we note that for each object $p$ of $P$, ${E}_{p}\circ \left({\Delta }_{P}\circ F\right)$ 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 $p↦\ell$ is just ${\Delta }_{P}\left(\ell \right)$. For this functor, we have our cone ${\Delta }_{P}\nu :{\Delta }_{J}\left({\Delta }_{P}\left(\ell \right)\right)\stackrel{.}{\to }{\Delta }_{P}\circ F$. Since for all $j$ and $p$ we have $\left({\Delta }_{P}\nu {\right)}_{j,p}={\nu }_{j}=j\text{th component of the limiting cone of}{E}_{p}\circ \left({\Delta }_{P}\circ F\right)$, we are done by the theorem on pointwise limits.

Revised on December 13, 2011 10:11:12 by Urs Schreiber (82.169.65.155)