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
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.”
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
We state the concise functorial definition of diagrams of the shape of categories.
(functorial definition)
Let $\mathcal{C}$ be a category and let $\mathcal{I}$ small category,
Then
a diagram $X$ of shape $\mathcal{I}$ in $\mathcal{C}$ is a functor of the form
the category of $\mathcal{I}$-shaped diagrams in $\mathcal{C}$ is the functor category $Funct(\mathcal{I}, \mathcal{C})$;
a diagram $X \colon \mathcal{I} \to \mathcal{C}$ is constant if it is a constant functor
in which case it is given by the data of a single object $\tilde X$;
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 $X$:
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 $X$:
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 $\phi \colon const_{\tilde X} \Rightarrow const_{\underset{\longrightarrow}{\lim}_i X_i}$ such that
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
We state an explicit component-based definition of diagrams of the shape of categories.
A diagram $X_\bullet$ in a category is
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;
for every label $i \in I$ a choice of element $\epsilon_i \in I_{i,i}$;
for each triple $i,j,k \in I$ a function
such that
the pairing $comp$ is associative and unital with the $f_{\epsilon_i}$-s the neutral elements;
for every $i \in I$ then $f_{\epsilon_i} = id_{X_i}$ is the identity morphism on the $i$-th obect;
for every composable pair of morphisms
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:
The last condition we depict as follows:
Consider a diagram
in some category (def. 2). Then
a cone over this diagram is
an object $\tilde X$ in the category;
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
holds, which we depict as follows:
a co-cone over this diagram is
an object $\tilde X$ in the category;
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
holds, which we depict as follows:
Consider a diagram
in some category (def. 2). Then
its limiting cone (or just limit for short) is, if it exists, the cone
over this diagram (def. 3) which is universal or initial among all possible cones, in that it has the property that for
any other cone, then there is a unique morphism
that factors the given cone through the limiting cone, in that for all $i \in I$ then
which we depict as follows:
its colimiting cocone (or just colimit for short) is, if it exists, the cocone
under this diagram (def. 3) which is universal or terminal among all possible co-cones, in that it has the property that for
any other cocone, then there is a unique morphism
that factors the given co-cone through the co-limiting cocone, in that for all $i \in I$ then
which we depict as follows:
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 $J\to U(C)$.
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.
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.
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
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
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
in which case we could more explicitly draw its image in $C$ as
By contrast, a diagram whose shape is the quiver
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;
dually, a cospan is a diagram whose shape is opposite to the shape of a span.
A transfinite composition diagram is one of the shape the poset indicated by
where the indices may range over the natural numbers or even some more general ordinal number.
This is a non-finite commuting diagram.