Contents

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.

This terminology is often used when speaking about limits and colimits; that is, we speak about “the limit or colimit of a diagram.”

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

### Diagrams shaped like categories

We discuss here diagrams of the “shape of a small catgeory”, as well as the concept of cones/cocones over these and limiting/colimiting (co-)cones. There is a quick abstract functorial definition

and there is a more long-winded but more explicit definition in terms of components

#### Functorial definition

We state the concise functorial definition of diagrams of the shape of categories.

###### Definition

(functorial definition)

Let $\mathcal{C}$ be a category and let $\mathcal{I}$ small category,

Then

1. a diagram $X$ of shape $\mathcal{I}$ in $\mathcal{C}$ is a functor of the form

$X \;\colon\; \mathcal{I} \longrightarrow \mathcal{C} \,,$
2. the category of $\mathcal{I}$-shaped diagrams in $\mathcal{C}$ is the functor category $Funct(\mathcal{I}, \mathcal{C})$;

3. a diagram $X \colon \mathcal{I} \to \mathcal{C}$ is constant if it is a constant functor

$const_{\tilde X} \;\colon\; \mathcal{I} \overset{\exists!}{\longrightarrow} \ast \overset{\tilde X}{\longrightarrow} \mathcal{C}$

in which case it is given by the data of a single object $\tilde X$;

4. a cone $C$ over a diagram $X \colon \mathcal{I} \to \mathcal{C}$ with tip an object $\tilde X \in \mathcal{C}$ is a natural transformation from the constant diagram $const_{\tilde X} \colon \mathcal{I} \to \ast \to \mathcal{C}$ to $X$:

$C \;\colon\; const_{\tilde X} \Rightarrow X$
5. a cocone $C$ under a diagram $X \colon \mathcal{I} \to \mathcal{C}$ is a natural transformation to a constant diagram $const_{\tilde X} \colon \mathcal{I} \to \ast \to \mathcal{C}$ from $X$:

$C \;\colon\; X \Rightarrow const_{\tilde X}$
6. the limiting cone (or limit, for short) over a diagram $X$ is, if it exists, the terminal object in the category of cones over $X$, which means that it is a cone $C_{lim}$ with tip denoted $\underset{\longleftarrow}{\lim}_i X_i$ such that for every other cone $C$ with tip $\tilde X$ there is a unique natural transformation $\phi \colon const_{\tilde X} \Rightarrow const_{\underset{\longleftarrow}{\lim}_i X_i}$ such that

$C = C_{lim} \circ \phi$
7. the colimiting cone (or colimit, for short) under a diagram $X$ is, if it exists, the initial object in the category of cocones under $X$, which means that it is a co-cone $C_{lim}$ with tip denoted $\underset{\longrightarrow}{\lim}_i X_i$ such that for every other cocone $C$ with tip $\tilde X$ there is a unique natural transformation $\phi \colon const_{\underset{\longrightarrow}{\lim}_i X_i} \Rightarrow const_{\tilde X}$ such that

$C = \phi \circ C_{lim} \,.$

#### Component definition

We state an explicit component-based definition of diagrams of the shape of categories.

###### Definition

(diagram in a category)

A diagram $X_\bullet$ in a category is

1. a set $\{ X_i \}_{i \in I}$ of objects in the category;

2. for every pair $(i,j) \in I \times I$ of labels of objects a set $\{ X_i \overset{ f_\alpha }{\longrightarrow} X_j\}_{\alpha \in I_{i,j}}$ of morphisms between these objects;

3. for every label $i \in I$ a choice of element $\epsilon_i \in I_{i,i}$;

4. for each triple $i,j,k \in I$ a function

$comp_{i,j,k} \;\colon\; I_{i,j} \times I_{j,k} \longrightarrow I_{i,k}$

such that

1. the pairing $comp$ is associative and unital with the $f_{\epsilon_i}$-s the neutral elements;

2. for every $i \in I$ then $f_{\epsilon_i} = id_{X_i}$ is the identity morphism on the $i$-th obect;

3. for every composable pair of morphisms

$X_i \overset{f_{\alpha} }{\longrightarrow} X_j \overset{ f_{\beta} }{\longrightarrow} X_k$

then the composite of these two morphisms equals the morphism of the diagram that is labeled by the value of $comp_{i,j,k}$ on their labels:

$f_{\beta} \circ f_\alpha \,=\, f_{comp_{i,j,k}( \alpha, \beta )} \,.$

The last condition we depict as follows:

$\array{ && X_j \\ & {}^{\mathllap{f_{\alpha}}}\nearrow && \searrow^{\mathrlap{f_{\beta}}} \\ X_i && \underset{ comp_{i,j,k}(\alpha,\beta) }{\longrightarrow} && X_k } \,.$
###### Definition

(cone over a diagram)

Consider a diagram

$X_\bullet \,=\, \left( \left\{ X_i \overset{f_\alpha}{\longrightarrow} X_j \right\}_{i,j \in I, \alpha \in I_{i,j}} \,,\, \mathrm{comp} \right)$

in some category (def. ). Then

1. a cone over this diagram is

1. an object $\tilde X$ in the category;

2. for each $i \in I$ a morphism $\tilde X \overset{p_i}{\longrightarrow} X_i$ in the category

such that

• for all $(i,j) \in I \times I$ and all $\alpha \in I_{i,j}$ then the condition

$f_{\alpha} \circ p_i = p_j$

holds, which we depict as follows:

$\array{ && \tilde X \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p'_j}} \\ X_i && \underset{f_\alpha}{\longrightarrow} && X_j }$
2. a co-cone over this diagram is

1. an object $\tilde X$ in the category;

2. for each $i \in I$ a morphism $q_i \colon X_i \longrightarrow \tilde X$ in the category

such that

• for all $(i,j) \in I \times I$ and all $\alpha \in I_{i,j}$ then the condition

$q_j \circ f_{\alpha} = q_i$

holds, which we depict as follows:

$\array{ X_i && \overset{f_\alpha}{\longrightarrow} && X_j \\ & {}_{\mathllap{q_i}}\searrow && \swarrow_{\mathrlap{q_j}} \\ && \tilde X } \,.$
###### Definition

Consider a diagram

$X_\bullet \,=\, \left( \left\{ X_i \overset{f_\alpha}{\longrightarrow} X_j \right\}_{i,j \in I, \alpha \in I_{i,j}} \,,\, \mathrm{comp} \right)$

in some category (def. ). Then

1. its limiting cone (or just limit for short) is, if it exists, the cone

$\left\{ \array{ && \underset{\longleftarrow}{\lim}_i X_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ X_i && \underset{f_\alpha}{\longrightarrow} && X_j } \right\}$

over this diagram (def. ) which is universal or initial among all possible cones, in that it has the property that for

$\left\{ \array{ && \tilde X \\ & {}^{\mathllap{p'_i}}\swarrow && \searrow^{\mathrlap{p'_j}} \\ X_i && \underset{f_\alpha}{\longrightarrow} && X_j } \right\}$

any other cone, then there is a unique morphism

$\phi \;\colon\; \tilde X \overset{}{\longrightarrow} \underset{\longrightarrow}{\lim}_i X_i$

that factors the given cone through the limiting cone, in that for all $i \in I$ then

$p'_i = p_i \circ \phi$

which we depict as follows:

$\array{ \tilde X \\ {}^{\mathllap{\phi}}\downarrow & \searrow^{\mathrlap{p_i}} \\ \underset{\longrightarrow}{\lim}_i X_i &\underset{p_i}{\longrightarrow}& X_i }$
2. its colimiting cocone (or just colimit for short) is, if it exists, the cocone

$\left\{ \array{ X_i && \underset{f_\alpha}{\longrightarrow} && X_j \\ & {}^{\mathllap{q_i}}\searrow && \swarrow^{\mathrlap{q_j}} \\ \\ && \underset{\longrightarrow}{\lim}_i X_i } \right\}$

under this diagram (def. ) which is universal or terminal among all possible co-cones, in that it has the property that for

$\left\{ \array{ X_i && \underset{f_\alpha}{\longrightarrow} && X_j \\ & {}^{\mathllap{q'_i}}\searrow && \swarrow_{\mathrlap{q'_j}} \\ && \tilde X } \right\}$

any other cocone, then there is a unique morphism

$\phi \;\colon\; \underset{\longrightarrow}{\lim}_i X_i \overset{}{\longrightarrow} \tilde X$

that factors the given co-cone through the co-limiting cocone, in that for all $i \in I$ then

$q'_i = \phi \circ q_i$

which we depict as follows:

$\array{ X_i &\overset{q_i}{\longrightarrow}& \underset{\longrightarrow}{\lim}_i X_i \\ {}^{\mathllap{\phi}}\downarrow & \swarrow^{\mathrlap{q'_i}} \\ \tilde X }$

### Diagrams shaped like directed graphs

###### Definition

(free diagram)

If $J$ is a directed graph with free category $F(J)$, 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

Last revised on October 29, 2018 at 12:41:25. See the history of this page for a list of all contributions to it.