category theory

# Contents

## 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

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

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 $D\colon J \to C$.

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 $D\colon F(J) \to C$, or equivalently a graph morphism $\bar{D}\colon J \to U(C)$.

Here $F\colon Quiv \to Cat$ denotes the free category on a quiver and $U\colon Cat \to Quiv$ 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 $D:J\to C$, the image of the shape $J$ is not necessarily a subcategory of $C$, even if $J$ is itself taken to be a category. This is because the functor $D$ 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.)

• $C$ must be a strict category to make sense of $U(C)$; however, $F(J)$ always makes sense.

## Commutative diagrams

If $J$ is a category, then a diagram $J\to C$ is commutative if it factors through a thin category. 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 thin category. 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 $D_{a b} : X_a \to X_b$ 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

$\left\{ \array{ a &\to& b \\ \downarrow && \downarrow \\ b' &\to& c } \right\}$

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

One typically “draws the diagram” as

$\array{ X_a &\stackrel{D_{a b}}{\to}& X_b \\ {}^{\mathllap{D_{a b'}}}\downarrow && \downarrow^{\mathrlap{D_{b c}}} \\ X_{b'} &\stackrel{D_{b' c}}{\to}& X_{c} }$

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

$\left\{ \array{ a &\to& b \\ \downarrow &\searrow& \downarrow \\ b' &\to& c } \right\}$

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

$\array{ X_a &\stackrel{D_{a b}}{\to}& X_b \\ {}^{\mathllap{D_{a b'}}}\downarrow &\searrow^{\stackrel{D_{b c}\circ D_{a b}}{= D_{b' c}\circ D_{a b'}}}& \downarrow^{\mathrlap{D_{b c}}} \\ X_{b'} &\stackrel{D_{b' c}}{\to}& X_{c} }$
• By contrast, a diagram whose shape is the quiver

$\left\{ \array{ a &\to& b \\ \downarrow && \downarrow \\ b' &\to& c } \right\}$

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 $a \to b'\to c$. 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 $J = \{a \stackrel{\to}{\to} b\}$ with two objects and two morphisms from one to the other.

Notice that if we required $\{a \stackrel{\to}{\to} b\}$ 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;

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

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

$J = \left\{ \array{ b &&&& c \\ & \searrow && \swarrow \\ && a } \right\}$
• A transfinite composition diagram is one of the shape the poset indicated by

$J = \left\{ \array{ a_0 &\to& a_1 &\to& \cdots \\ & \searrow & \downarrow & \swarrow & \cdots \\ && b } \right\} \,,$

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

This is a non-finite commuting diagram.

• tower diagram

Revised on August 17, 2014 20:57:52 by Toby Bartels (75.88.46.170)