nLab
commutative diagram

Idea

A commutative diagram is a diagram in which composition is path-independent?.

Definitions

Slick definition

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

$$J\stackrel{D}{\to }C,J\text{a quiver}.$$J \overset{D}\to C ,\; J\;\text{a quiver}.

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

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

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

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

Elementary definition

Recall that a quiver $J$ consists of a set $V$ of vertices, a set $E$ of edges, and two functions $s,t:E\to V$. Given a category $C$, a 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 $(v_0,v_1,\dots ,v_n)$ of vertices and a list $(e_1,\dots ,e_n)$ 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 $F(e_1);\dots ;F(e_n):F(v_0)\to F(v_n)$ in $C$.

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

Links

• Wikipedia
• Mathworld
• An Experiment