category of cubes





We denote by 1\square_{\leq 1} the category defined uniquely (up to isomorphism) by the following.

1) There are exactly two objects, which we shall denote by I 0I^{0} and I 1I^{1}.

2) There are exactly two arrows i 0,i 1:I 0I 1i_{0}, i_{1} : I^{0} \rightarrow I^{1}.

3) There is exactly one arrow p:I 1I 0p : I^{1} \rightarrow I^{0}.

4) There are no non-identity arrows I 0I 0I^{0} \rightarrow I^{0}.

5) There are exactly two non-identity arrows I 1I 1I^{1} \rightarrow I^{1}, which are i 0pi_{0} \circ p and i 1pi_{1} \circ p.


In particular, because of 4) in Notation , the diagram

I 0 i 0 I 1 id p I 0 \array{ I^{0} & \overset{i_{0}}{\to} & I^{1} \\ & \underset{id}{\searrow} & \downarrow p \\ & & I^{0} }

commutes in 1\square_{\leq 1}, and the diagram

I 0 i 1 I 1 id p I 0 \array{ I^{0} & \overset{i_{1}}{\to} & I^{1} \\ & \underset{id}{\searrow} & \downarrow p \\ & & I^{0} }

commutes in 1\square_{\leq 1}.


The category 1\square_{\leq 1} is isomorphic to the category Δ 1\Delta_{\leq 1}, i.e. it may also be described as

  • The full subcategory of the simplex category Δ\Delta on the objects [0][0] and [1][1].

  • (A skeleton of) the category of linearly ordered sets of cardinality 1 or 2.

  • The indexing category for reflexive equalizers.


The category 1\square_{\leq 1} can also be constructed by beginning with the free category on the directed graph defined uniquely by the fact that 1), 2), and 3) in Notation hold, and by the fact that there are no other non-identity arrows. One then takes a quotient of this free category which forces the diagrams in Remark to commute.

This quotient can be expressed as a colimit in the category of small categories, or, which ultimately amounts to the same, by means of the equivalence relation \sim on the arrows of the free category generated by requiring that pi 0idp \circ i_{0} \sim id and pi 1idp \circ i_{1} \sim id, and by requiring that g 1g 0f 1f 0g_{1} \circ g_{0} \sim f_{1} \circ f_{0} if g 1f 1g_{1} \sim f_{1} and g 0f 0g_{0} \sim f_{0}.


The category of cubes is the free strict monoidal category? on 1\square_{\leq 1} whose unit object is I 0I^{0}.


We denote the category of cubes by \square.


We refer to \square as the category of cubes.


It is not the case that \square is the free strict monoidal category on 1\square_{\leq 1}. Rather, \square is the free strict monoidal category with specified unit on 1\square_{\leq 1}, where the unit is specified to be I 0I^{0}.



Let n0n \geq 0 be an integer. We often denote the object I 1I 1 n\underbrace{I^{1} \otimes \cdots \otimes I^{1}}_{n} of \square by I nI^{n}.


There are several useful variations of \square, to be described on other pages in the future.

Expository material

For expository and other material, see category of cubes - exposition.

Last revised on February 3, 2019 at 10:06:33. See the history of this page for a list of all contributions to it.