nLab
diagram

Idea

Informally, a diagram in a category $C$ consists of some objects of $C$ connected by some morphisms of $C$. Frequently when doing category theory, we “draw diagrams” such as

$$\begin{array}{ccc}A& \stackrel{f}{\to }& B\\ {}^h\downarrow & & {\downarrow }^k\\ C& \underset{g}{\to }& D\end{array}$$\array{A & \overset{f}{\to} & B\\ ^h\downarrow && \downarrow^k\\ C& \underset{g}{\to} & D}

by drawing some objects (or dots labeled by objects) connected by arrows labeled by morphisms.

There are two natural ways to give the notion of “diagram” a formal definition. One is to say that a diagram is a functor, usually one whose domain is a (very) small category. This level of generality is sometimes convenient.

On the other hand, a more direct representation of what we draw on the page, when we “draw a diagram,” only involves labeling the vertices and edges of a directed graph (or quiver) by objects and morphisms of the category. This sort of diagram can be identified with a functor whose domain is a free category, and this is the most common context when we talk about diagrams “commuting.”

Definitions

Let $C$ be a category.

Diagrams shaped like categories

Definition

If $J$ is a category, then a diagram in $C$ of shape $J$ is simply a functor $D:J\to C$.

This terminology is often used when speaking about limits and colimits; that is, we speak about “the limit or colimit of a diagram.” Similarly, it is common to call the functor category $C^J$ the “category of diagrams in $C$ of shape $J$”.

Diagrams shaped like graphs

Definition

If $J$ is a quiver, then a diagram in $C$ of shape $J$ is a functor $D:F(J)\to C$, or equivalently a graph morphism $\overline{D}:J\to U(C)$.

Here $F:\mathrm{Quiv}\to \mathrm{Cat}$ denotes the free category on a quiver and $U:\mathrm{Cat}\to \mathrm{Quiv}$ the underlying quiver of a category, which form a pair of adjoint functors. These are the sorts of diagrams which we “draw on a page” — we draw a quiver, and then label its vertices with objects of $C$ and its edges with morphisms in $C$, thereby forming a graph morphism $J\to U(C)$.

Remarks

• For either sort of diagram, $J$ may be called the shape, scheme, or index category or graph.

• Note that given a diagram $F:J\to C$, the image of the shape $J$ is not necessarily a subcategory of $C$, even if $J$ is itself a category. This is because the functor $F$ could identify objects of $J$, thereby producing new potential composites which do not exist in $J$. (Sometimes one talks about the “image” of a functor as a subcategory, but this really means the subcategory generated by the image in the literal objects-and-morphisms sense.)

Commutative diagrams

If $J$ is a category, then a diagram $J\to C$ is commutative if it factors through a preorder. Equivalently, a diagram of shape $J$ commutes iff any two morphisms in $C$ that are assigned to any pair of parallel morphisms in $J$ (i.e., with same source and target in $J$) are equal.

If $J$ is a quiver, as is more common when we speak about “commutative” diagrams, then a diagram of shape $J$ commutes if the functor $F(J)\to C$ factors through a preorder. Equivalently, this means that given any two parallel paths of arbitrary finite length (including zero) in $J$, their images in $C$ have equal composites.

Examples

• The shape of the empty diagram is the initial category with no object and no morphism.

  Every category $C$ admits a unique diagram whose shape is the empty (initial) category, which is called the empty diagram in $C$.

• The shape of the terminal diagram is the terminal category $J=\{*\}$ consisting of a single object and a single morphism (the identity morphism on that object).

  Specifying a diagram in $C$ whose shape is $\{*\}$ is the same as specifying a single object of $C$, the image of the unique object of $1$. (See global element)

• A diagram of the shape $\{a\to b\}$ in $C$ is the choice of any one morphism $F_{ab}:X_a\to X_b$ in $C$.

  Notice that strictly speaking this counts as a commuting diagram , but is a degenerate case of a commuting diagram, since there is only a single morphism involved, which is necessarily equal to itself.

• If $J$ is the quiver with one object $a$ and one endo-edge $a\to a$, then a diagram of shape $J$ in $C$ consists of a single endomorphism in $C$. Since $a\to a$ and the zero-length path are parallel in $J$, such a diagram only commutes if the endomorphism is an identity. Note, in particular, that a single endomorphism can be considered as a diagram with more than one shape (this one and the previous one), and that whether this diagram “commutes” depends on the chosen shape.

• A diagram of shape the poset indicated by

  $$\left\{\begin{array}{ccc}a& \to & b\\ \downarrow & & \downarrow \\ b\prime & \to & c\end{array}\right\}$$\left\{ \array{ a &\to& b \\ \downarrow && \downarrow \\ b' &\to& c } \right\}

  is a commuting square in $C$: this is a choice of four (not necessarily distinct!) objects $X_a,X_b,X_{b\prime },X_c$ in C, together with a choice of (not necessarily distinct) four morphisms $F_{ab}:X_a\to X_b$, $F_{bc}:X_b\to X_c$ and $F_{ab\prime }:X_a\to X_{b\prime }$, $F_{b\prime c}:X_{b\prime }\to X_c$ in $C$, such that the composite morphism $F_{bc}\circ F_{ab}$ equals the composite $F_{b\prime c}\circ F_{ab\prime }$.

  One typically “draws the diagram” as

  $$\begin{array}{ccc}{X}_a& \stackrel{F_{ab}}{\to }& X_b\\ {}^{F_{ab\prime }}\downarrow & & {\downarrow }^{F_{bc}}\\ X_{b\prime }& \stackrel{F_{b\prime c}}{\to }& X_c\end{array}$$\array{ X_a &\stackrel{F_{a b}}{\to}& X_b \\ {}^{\mathllap{F_{a b'}}}\downarrow && \downarrow^{\mathrlap{F_{b c}}} \\ X_{b'} &\stackrel{F_{b' c}}{\to}& X_{c} }

  in $C$ and says that the diagram commutes if the above equality of composite morphisms holds.

  Notice that the original poset had, necessarily, a morphism $a\to c$ and could have equivalently been depicted as

  $$\left\{\begin{array}{ccc}a& \to & b\\ \downarrow & \searrow & \downarrow \\ b\prime & \to & c\end{array}\right\}$$\left\{ \array{ a &\to& b \\ \downarrow &\searrow& \downarrow \\ b' &\to& c } \right\}

  in which case we could more explicitly draw its image in $C$ as

  $$\begin{array}{ccc}{X}_a& \stackrel{F_{ab}}{\to }& X_b\\ {}^{F_{ab\prime }}\downarrow & {\searrow }^{\stackrel{F_{bc}\circ F_{ab}}{=F_{b\prime c}\circ F_{ab\prime }}}& {\downarrow }^{F_{bc}}\\ X_{b\prime }& \stackrel{F_{b\prime c}}{\to }& X_c\end{array}$$\array{ X_a &\stackrel{F_{a b}}{\to}& X_b \\ {}^{\mathllap{F_{a b'}}}\downarrow &\searrow^{\stackrel{F_{b c}\circ F_{a b}}{= F_{b' c}\circ F_{a b'}}}& \downarrow^{\mathrlap{F_{b c}}} \\ X_{b'} &\stackrel{F_{b' c}}{\to}& X_{c} }

• By contrast, a diagram whose shape is the quiver

  $$\left\{\begin{array}{ccc}a& \to & b\\ \downarrow & & \downarrow \\ b\prime & \to & c\end{array}\right\}$$\left\{ \array{ a &\to& b \\ \downarrow && \downarrow \\ b' &\to& c } \right\}

  is a not-necessarily-commuting square. The free category on this quiver differs from the poset in the previous example by having two morphisms $a\to c$, one given by the composite $a\to b\to c$ and the other by the composite $a\to b\prime \to c$. But the poset in the previous category is the poset reflection of this $F(J)$, so a diagram of this shape commutes, in the sense defined above, iff it is a commuting square in the usual sense.

• A pair of objects is a diagram whose shape is a discrete category with two objects.

• A pair of parallel morphisms is a diagram whose shape is a category $J=\{a\overrightarrow{\to }b\}$ with two objects and two morphisms from one to the other.

  Notice that if we required $\{a\overrightarrow{\to }b\}$ to be a poset this would necessarily make these two morphisms equal, and hence reduce this example to the one where $J=\{a\to b\}$. In other words, a diagram of this shape only commutes if the two morphisms are equal.

• A span is a diagram whose shape is a category with just three objects and single morphisms from one of the objects to the other two;

  $$J=\left\{\begin{array}{ccc}& & a\\ & \swarrow & & \searrow \\ b& & & & c\end{array}\right\}$$J = \left\{ \array{ && a \\ & \swarrow && \searrow \\ b &&&& c } \right\}

  dually, a cospan is a diagram whose shape is opposite to the shape of a span.

  $$J=\left\{\begin{array}{ccccc}b& & & & c\\ & \searrow & & \swarrow \\ & & a\end{array}\right\}$$J = \left\{ \array{ b &&&& c \\ & \searrow && \swarrow \\ && a } \right\}

• A transfinite composition diagram is one of the shape the poset indicated by

  $$J=\left\{\begin{array}{ccccc}{a}_0& \to & a_1& \to & \cdots \\ & \searrow & \downarrow & \swarrow & \cdots \\ & & b\end{array}\right\},$$J = \left\{ \array{ a_0 &\to& a_1 &\to& \cdots \\ & \searrow & \downarrow & \swarrow & \cdots \\ && b } \right\} \,,

  where the indices may range over the natural numbers or even some more general ordinal number.

  This is a non-finite commuting diagram.