nLab commutative diagram

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 graph $J$ and a functor to $C$ from the free category on $J$:

$$F(J) \overset{D}\to C$$

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

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

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

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

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

Elementary definition

Recall that a graph $J$ 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 $C$ comprises a map from $V$ to the objects of $C$ and a map from $E$ to the morphisms of $C$, both denoted $D$, such that $D(s(e)) = s(D(e))$ and $D(t(e)) = t(D(e))$ for each edge $e$, where $s,t$ also denote the source and target maps in $C$.

Recall that a path $p$ in $J$ comprises 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 $F(e_1);\ldots;D(e_n)\colon D(v_0) \to D(v_n)$ in $C$. (Note that the paths of length zero are composed to the identity $\id_{D(v_0)}\colon D(v_0) \to D(v_0)$.)

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 $C$.

Edges cases

Cycles and loops

Cycles (including loops) are generally avoided in commutative diagrams in category theory texts. However, they make sense under the definition above. For instance, commutativity of the following diagram implies that $h g f = 1$, $g f h = 1$, and $f h g = 1$. Similarly, every loop in a commutative diagram is required to be an identity morphism.

Forks

Diagrams like the following are often used to denote forks in category theory. However, note that for this diagram to commute, we require the stronger condition that $g = h$, not just that $g f = h f$.

If this is undesirable, a weaker notion of commutativity may be asked for, in which we only ask for commutativity of parallel paths where the source is a source in the sense that it only has outgoing edges, and the target is a sink in that it only has incoming edges. This convention may sometimes be implicitly encountered in category theory texts.

Related entries

• diagram

• internal diagram

• free diagram

References

• Wikipedia
• Mathworld