Commutative diagrams
### Context

#### Category theory

**category theory**

## Concepts

## Universal constructions

## Theorems

## Extensions

## Applications

# Commutative diagrams

## Idea

In category theory, a *commutative diagram* is a free diagram in which all parallel morphisms obtained by composing morphisms in the diagram agree.

For example that a square diagram of the form

$\array{& X & \overset{f}\rightarrow & Z & \\
g & \downarrow &&\downarrow & g'\\
&Y & \underset{f'}\rightarrow& W & \\
}$

commutes is to say that $g' \circ f = f' \circ g$ (see also at *commuting square*).

## Definitions

### Slick definition

For our purposes, a **free diagram** $D$ in a category $C$ consists of a free category $J$ and a functor to $C$ from the free category on $J$:

$J \overset{D}\to C ,\; J\;\text{a free category}.$

Then this diagram $D$ **commutes** if this functor $D$ factors (up to natural isomorphism) through a poset $P$:

$J \to P \to C \;\cong\; J \to C ,\; P\;\text{a poset} ;$

or equivalently (treating $C$ as a strict category) if the functor factors up to equality through a proset $Q$:

$J \to Q \to C \;\cong\; J \to C ,\; Q\;\text{a proset} .$

In the above, we are identifying posets, and prosets with certain categories in the usual ways.

### Elementary definition

Recall that a **free category** $J$ consists of a set $V$ of *vertices*, a set $E$ of *edges*, and two functions $s,t\colon E \to V$. Given a category $C$, a **free diagram** $D$ of shape $J$ in a category $C$ is consists of a map from $V$ to the objects of $C$ and a map from $E$ to the morphisms of $C$, both denoted $F$, such that $F(s(e)) = S(F(e))$ and $F(t(e)) = T(F(e))$ for each edge $e$, where $S,T$ are the source and target maps in $C$.

Recall that a **path** $p$ in $J$ consists of a list $(v_0,v_1,\ldots,v_n)$ of vertices and a list $(e_1,\ldots,e_n)$ of edges such that $s(e_i) = v_{i-1}$ and $t(e_i) = v_{i}$ for each $i$, where $n$ is any natural number (possibly zero). We say that $v_0$ is the **source** of the path and that $v_n$ is its **target**. Given a path $p$ and a diagram $D$, the **composite** of $p$ under $D$ is the composite $F(e_1);\ldots;F(e_n)\colon F(v_0) \to F(v_n)$ in $C$. (Note that the paths of length zero are composed to the identity $\id_{F(v_0)}\colon F(v_0) \to F(v_0)$.)

A diagram $D$ **commutes** if, given any two vertices $x,y$ in $J$ and any two paths $p,p'$ with source $x$ and target $y$, the composites of $p$ and $p'$ under $D$ are equal in $C$.

## References