nLab commutative square

Commutative squares

Commutative squares

Definition and notation

Let CC be a category. A square of morphisms of CC consists of objects X,Y,Z,WX,Y,Z,W of CC and morphisms f:XZf\colon X \to Z, g:XYg\colon X \to Y, f:YWf'\colon Y \to W, and g:ZWg'\colon Z \to W. This is often pictured as a square

X f Z g g Y f W \array{& X & \overset{f}\rightarrow & Z & \\ g & \downarrow &&\downarrow & g'\\ &Y & \underset{f'}\rightarrow& W & \\ }

The square is commutative if gf=fgg' \circ f = f' \circ g.

The class of commutative squares in CC is written C\square C.


This class has partial compositions 1\circ_1 and 2\circ_2 which are vertical and horizontal:

\array{ \bullet & {\to} & \bullet & \\ \downarrow &&\downarrow \\ \bullet & {\to}& \bullet \\ \downarrow & & \downarrow\\ \bullet & \to & \bullet } \quad \quad \array{\bullet & {\to} & \bullet & \to & \bullet \\ \downarrow &&\downarrow && \downarrow \\ \bullet & {\to}& \bullet & \to & \bullet }

thus forming a (strict) double category, also written C\square C, whose objects are those of CC, whose horizontal and vertical 1-cells are given by morphisms in CC, and whose 2-cells exhibit commutative squares. It contains the vertical category 1C\square_1 C and the horizontal category 2C\square _2 C.

One can also form multiple compositions [a ij][a_{ij}] of arrays (a ij)(a_{ij}), i=1,,m;j=1,,ni=1, \ldots, m; j=1, \ldots , n, of commutative squares provided that in the obvious sense adjacent squares are composible. One checks by induction that:

any composition of commutative squares is commutative.


Let 22 denote the walking arrow: the category with two objects 0,10,1 and one arrow 010 \to 1. This has the structure of cocategory. Then the class of commutative squares in CC can also be described as Cat(2×2,C)Cat(2 \times 2, C).

If DD is a category, then Cat(D, 1C)Cat(D, \square_1 C) can be regarded as the class of natural transformations of functors DCD \to C. Then the category structure 2C\square _2 C induces a category structure on Cat(D, 1C)Cat(D,\square _1 C) giving the functor category CAT(D,C)CAT(D,C): the category of functors and natural transformations. (This account is due to Charles Ehresmann.)

One deduces that if also EE is a category then there is a natural bijection

Cat(E×D,C)(E,CAT(D,C)),Cat(E \times D, C) \cong (E, CAT(D,C)),

which thus states that the category of (small if you like!) categories is cartesian closed.

The commutative squares serve as the morphisms in the arrow category of CC, which is the functor category CAT(2,C)CAT(2,C).

Last revised on April 16, 2021 at 22:56:57. See the history of this page for a list of all contributions to it.