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.”
Let be a category.
If is a category, then a diagram in of shape is simply a functor .
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 the “category of diagrams in of shape ”.
If is a quiver, then a diagram in of shape is a functor , or equivalently a graph morphism .
Here denotes the free category on a quiver and 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 and its edges with morphisms in , thereby forming a graph morphism .
For either sort of diagram, may be called the shape, scheme, or index category or graph.
Note that given a diagram , the image of the shape is not necessarily a subcategory of , even if is itself a category. This is because the functor could identify objects of , thereby producing new potential composites which do not exist in . (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.)
If is a category, then a diagram is commutative if it factors through a preorder. Equivalently, a diagram of shape commutes iff any two morphisms in that are assigned to any pair of parallel morphisms in (i.e., with same source and target in ) are equal.
If is a quiver, as is more common when we speak about “commutative” diagrams, then a diagram of shape commutes if the functor factors through a preorder. Equivalently, this means that given any two parallel paths of arbitrary finite length (including zero) in , their images in have equal composites.
The shape of the empty diagram is the initial category with no object and no morphism.
Every category admits a unique diagram whose shape is the empty (initial) category, which is called the empty diagram in .
The shape of the terminal diagram is the terminal category consisting of a single object and a single morphism (the identity morphism on that object).
A diagram of the shape in is the choice of any one morphism in .
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 is the quiver with one object and one endo-edge , then a diagram of shape in consists of a single endomorphism in . Since and the zero-length path are parallel in , 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 : this is a choice of four (not necessarily distinct!) objects in C, together with a choice of (not necessarily distinct) four morphisms , and , in , such that the composite morphism equals the composite .
One typically “draws the diagram” as
in and says that the diagram commutes if the above equality of composite morphisms holds.
Notice that the original poset had, necessarily, a morphism and could have equivalently been depicted as
in which case we could more explicitly draw its image in 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 , one given by the composite and the other by the composite . But the poset in the previous category is the poset reflection of this , 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 with two objects and two morphisms from one to the other.
Notice that if we required to be a poset this would necessarily make these two morphisms equal, and hence reduce this example to the one where . 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;
This is a non-finite commuting diagram.