# What is the definition of a commutative diagram?

Reading in ACC (the joy of cats) I was quite surprised to meet a real definition of diagram. This word had been used exactly $27$ times allready (just believe me) and the question 'what is it?' had not reached my thinking whatsoever. Okay, good to have a definition (11.1(1) see under), but it triggered me to expect a definition of commutative diagram as well. Alas, it was not there. In my intuition I think of a diagram having a poset as scheme, but I don't trust that intuition enough to take this for granted. So I am asking you here:

What is the definition of a commutative diagram?

11.1 (1) A diagram in a category $\mathcal{A}$ is a functor $D:\mathcal{I}\rightarrow\mathcal{A}$ with codomain $\mathcal{A}$. The domain, $\mathcal{I}$, is called the scheme of the diagram.

• Your intuition probably works, though one can get commutative diagrams with less restrictions, since you just need the images of all arrows to agree (when they share domain and codomain), rather than all arrows themselves agreeing. Jan 10 '14 at 9:27
• There are some exceptional conventions to learn. For instance one may wonder what it means to commute, when working with a diagram with a pair of arrows between a couple of vertices, see here. Jan 10 '14 at 9:32
• There are no exceptions. The general definition (see my answer) also covers this case. Jan 10 '14 at 9:34
• @MartinBrandenburg When one explicitly writes those pairs like in my link, one ignores the fact that they are non-equal as the answer points out. As I understand it, your answer does not cover this case. Jan 10 '14 at 9:36
• @KarlKronenfeld, while that situation does arise in the literature, it is an abuse of language to call such a thing a commutative diagram. Martin's definition is the standard definition, really. Jan 10 '14 at 9:40

Let $\Gamma=(V,E,s,t)$ be a directed graph ($V$ = vertices, $E$ = edges, $s$ = source, $t$ = target). Let $\mathcal{C}$ be a category. A diagram of shape $\Gamma$ in $\mathcal{C}$ is a family of objects $X(v) \in \mathcal{C}$ for every vertex $v \in V$ and morphisms $X(e)$ in $\mathcal{C}$ for every edge $e \in E$ such that $s(X(e))=X(s(e))$ and $t(X(e))=X(t(e))$ for all $e \in E$. Thus, every edge $e : v \to w$ is mapped to a morphism $X(e) : X(v) \to X(w)$.

For a path $\gamma$ in $\Gamma$ we define a morphism $X(\gamma)$ in $\mathcal{C}$ by induction: If $\gamma$ is the empty path at a vertex $v$, let $X(\gamma):=\mathrm{id}_{X(v)}$. If $\gamma = \beta \circ e$ for a shorter path $\beta$ and an edge $e$, define $X(\gamma) := X(\beta) \circ X(e)$.

The diagram $X$ is called commutative if for all vertices $v,w \in V$ and all two paths $\gamma,\gamma'$ in $\Gamma$ from $v$ to $w$ we have $X(\gamma)=X(\gamma')$.

Convince yourself that this coincides with the usual definition for simple examples such as

$$\Gamma = \begin{array}{c} \bullet & \rightarrow & \bullet \\ \downarrow && \downarrow \\ \bullet & \rightarrow & \bullet \end{array}$$

There is a close connection between diagrams and functors. If $\mathsf{Path}(\Gamma)$ denotes the path category, then functors $\mathsf{Path}(\Gamma) \to \mathcal{C}$ correspond to diagrams of shape $\Gamma$ in $\mathcal{C}$. These in turn correspond to homomorphisms of directed graphs $\Gamma \to U(\mathcal{C})$, where $U(-)$ is the forgetful functor from categories to directed graphs.

If one even regards arbitrary functors (with small domain category) as diagrams, then we may use the same definition of commutativity as above: Every two chains of morphisms between two given objects are mapped to the same morphism.

Sometimes one only demands the commutativity condition for certain paths. For example, when dealing with sheaves or simplicial sets, usually a diagram of the shape $$\begin{array}{c} \bullet & \rightrightarrows & \bullet \\ \downarrow && \downarrow \\ \bullet & \rightrightarrows & \bullet \end{array}$$ is called commutative if the two squares which consist of the two upper resp. lower horizontal morphisms are commutative.

PS: I have just found that these definitions can also be found in Grothendieck's Tohoku paper.

• I haven't absorbed this yet, but how about my mentioned intuition (poset as scheme). Can you say some explicit about that? Is it too restricted? I expect so, because - if not - then your answer would be redundant, wouldn't it? Jan 10 '14 at 9:45
• A directed graph $\Gamma$ as in Martin's anwer generates freely a category, which Maritn writes $Patth(\Gamma)$ and you can quotient it by the relation that identifies two arrows whenever they are parallel (that is, they have the same domain and codomain) to get a new category $\bar\Gamma$. A commutative diagram of shape $\Gamma$ is a functor defined on a category of the form $\bar\Gamma$. For suitably restricted $\Gamma$, $\bar\Gamma$ is a poset. (You need $\Gamma$ to be acyclic and probably more) But you do want, in general, cycles. For example, if $G$ is an group, you can view it as a Jan 10 '14 at 9:49
• (...) category with one object, and a space with action of $G$ is the same as a «diagram of shape $G$» with values in topological spaces: in this situation, the domain category is all loops. You can codify more complicated things in this way: you can have a poset and a group at each vertex acting and so on and so forth. This things do arise in nature. Jan 10 '14 at 9:50
• @MarianoSuárez-Alvarez Thank you. Intuition is essential and cannot be missed. But trustworthy it is not. I have a lot to learn again. By the way what is the percentage of mathematicians that are speaking of commutative diagrams and do not know what it really is? At the moment I am still one of them, but I have hope now that that will change. Jan 10 '14 at 9:52
• @drhab In Wikipedia (en.wikipedia.org/wiki/Commutative_diagram) commutative diagrams are defined exactly as you say with posets categories as schemes/indices. This is basically the same definition given by Martin for commutative diagrams. I do not see it as more restrictive than the one given with paths by Martin. Indeed also nlab (nlab.mathforge.org/nlab/show/commutative+diagram) describes a commutative diagram as a quiver which factors via a poset Jan 10 '14 at 16:11