nLab
cubical set

Contents

Definition

Notation

We make use of the notation of category of cubes.

Definition

The category of cubical sets is the free co-completion of \square, the category of cubes.

Remark

The free co-completion of a small category can be constructed as the category of presheaves presheaves of sets on this category. Thus we can also think of the category of cubical sets as the category of presheaves of sets on \square.

Notation

We denote the category of cubical sets by Set op\mathsf{Set}^{\square^{op}}.

Definition

A cubical set is an object of Set op\mathsf{Set}^{\square^{op}}.

Remark

When we think of the category of cubical sets as the category of presheaves of sets on \square, we consequently think of a cubical set as a presheaf of sets on \square

Definition

A morphism of cubical sets is an arrow of Set op\mathsf{Set}^{\square^{op}}.

Monoidal structure

The strict monoidal structure of \square gives rise to a (non-strict) monoidal structure on Set op\mathsf{Set}^{\square^{op}}, by Day convolution. The unit of the monoidal structure is 0\square^{0}, in the notation of Notation 8. Whenever we use the symbol \otimes when working with cubical sets or morphisms of cubical sets, we shalll always refer to the functor - \otimes - of this monoidal structure.

Notation

Free standing nn-cube, and an nn-cube of a cubical set

Notation

Let y:Set opy : \square \rightarrow \mathsf{Set}^{\square^{op}} denote the Yoneda embedding functor. Let n0n \geq 0 be an integer. We denote the cubical set y(I n)y(I^{n}) by n\square^{n}.

Terminology

We refer to n\square^{n} as the free-standing nn-cube.

Terminology

Let XX be a cubical set. Let n0n \geq 0 be an integer. By an nn-cube of XX, we shall mean a morphism of cubical sets nX\square^{n} \rightarrow X.

Notation

Let ff be a 1-cube of XX. We shall often depict ff as f:x 0x 1f : x_{0} \rightarrow x_{1} or as follows.

x 0 f x 1 \array{x_{0} & \overset{f}{\rightarrow} & x_{1}}

In this case, x 0x_{0} is to be understood to be the 00-cube fy(i 0)f \circ y(i_{0}) of XX, and x 1x_{1} is to be understood to be the 11-cube fy(i 1)f \circ y(i_{1}) of XX.

Notation

Let σ\sigma be a 2-cube of XX. We shall often depict σ\sigma as follows.

x 0 f 0 x 1 f 2 σ f 1 x 2 f 3 x 3 \array{ x_{0} & \overset{f_{0}}{\rightarrow} & x_{1} \\ f_{2} \downarrow & \sigma & \downarrow f_{1} \\ x_{2} & \underset{f_{3}}{\rightarrow} & x_{3} }

In this case, f 0f_{0} is to be understood to be the 11-cube σy(i 0I 1)\sigma \circ y(i_{0} \otimes I^{1}) of XX, f 1f_{1} is to be understood to be the 11-cube σy(I 1i 1)\sigma \circ y(I^{1} \otimes i_{1}) of XX, f 2f_{2} is to be understood to be the 11-cube σy(I 1i 0)\sigma \circ y(I^{1} \otimes i_{0}) of XX, and f 3f_{3} is to be understood to be the 1-cube σy(i 1I 1)\sigma \circ y(i_{1} \otimes I^{1}) of XX.

It can be checked that this notation is consistent with Notation 11.

Boundary of the free standing nn-cube, and of an nn-cube of a cubical set.

Notation

Let n1n \geq 1 be an integer. We denote by :Set opSet op\partial : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Set}^{\square^{op}} the functor given by defined by sk n1tr n1sk_{n-1} \circ tr_{n-1}, where tr n1tr_{n-1} is the (n1)(n-1)- truncation functor for cubical sets, and sk n1sk_{n-1} is the (n1)(n-1)- skeleton functor for cubical sets.

Terminology

Let n0n \geq 0 be an integer. We refer to n\partial \square^{n} as the boundary of n\square^{n}.

Notation

We also denote by 0\partial \square^{0} (recalling that Set op\mathsf{Set}^{\square^{op}} is, by construction, co-complete) the initial object of Set op\mathsf{Set}^{\square^{op}}.

Notation

Let n1n \geq 1 be an integer. We denote by i n: n ni_{n} : \partial \square^{n} \rightarrow \square^{n} the morphism of cubical sets corresponding, under the adjunction between sk n1sk_{n-1} and tr n1tr_{n-1} described at cubical truncation, skeleton, and co-skeleton, to the identity arrow tr n( n)tr n( n)tr_{n}(\square^{n}) \rightarrow tr_{n}(\square^{n}) in Set n1 op\mathsf{Set}^{\square_{n-1}^{op}}.

Terminology

Let n0n \geq 0 be an integer, and let σ: nX\sigma : \square^{n} \rightarrow X be an nn-cube of a cubical set XX. We refer to the morphism of cubical sets

n i n n σ X \array{\partial \square^{n} & \overset{i_{n}}{\rightarrow} & \square^{n} & \overset{\sigma}{\rightarrow} & X }

as the boundary of σ\sigma.

Notation

Let σ\sigma be a 2-cube of XX as follows.

x 0 f 0 x 1 f 2 σ f 1 x 2 f 3 x 3 \array{ x_{0} & \overset{f_{0}}{\to} & x_{1} \\ f_{2} \downarrow & \sigma & \downarrow f_{1} \\ x_{2} & \underset{f_{3}}{\to} & x_{3} }

We shall often depict the boundary of σ\sigma as follows.

x 0 f 0 x 1 f 2 f 1 x 2 f 3 x 3 \array{ x_{0} & \overset{f_{0}}{\to} & x_{1} \\ f_{2} \downarrow & & \downarrow f_{1} \\ x_{2} & \underset{f_{3}}{\to} & x_{3} }

Horns of the free-standing nn-cube

Notation

We define inductively, for any integer n1n \geq 1, any integer 1in1 \leq i \leq n, and any integer 0ϵ10 \leq \epsilon \leq 1, a cubical set n,i,ϵ\sqcap^{n,i,\epsilon} and a morphism of cubical sets i i,ϵ: n,i,ϵ ni_{i, \epsilon} : \sqcap^{n,i,\epsilon} \rightarrow \square^{n}.

When n=1n=1, we define both 1,1,0\sqcap^{1,1,0} and 1,1,1\sqcap^{1,1,1} to be 0\square^{0}. We define i 1,0: 0 1i_{1,0} : \square^{0} \rightarrow \square^{1} to be y(i 0)y(i_{0}), and define i 1,1: 0 1i_{1,1} : \square^{0} \rightarrow \square^{1} to be y(i 1)y(i_{1}).

Suppose that, for some integer n1n \geq 1, we have defined n,i,ϵ\sqcap^{n,i,\epsilon} and a morphism of cubical sets i i,ϵ: n,i,ϵ ni_{i, \epsilon} : \sqcap^{n,i,\epsilon} \rightarrow \square^{n} for all integers 1in1 \leq i \leq n, and all integers 0ϵ10 \leq \epsilon \leq 1. For 1in1 \leq i \leq n, we define (recalling that Set op\mathsf{Set}^{\square^{op}} is co-complete by construction) n+1,i,ϵ\sqcap^{n+1,i, \epsilon} to be a cubical set fitting into a co-cartesian square in Set op\mathsf{Set}^{\square^{op}} as follows.

n,i,ϵ n,i,ϵ ( n,i,ϵy(i 0))( n,i,ϵy(i 1)) n,i,ϵ 1 i i,ϵi i,ϵ r 0 n n r 1 n+1,i,ϵ \array{ \sqcap^{n,i,\epsilon} \sqcup \sqcap^{n,i,\epsilon} & \overset{\big( \sqcap^{n,i,\epsilon} \otimes y(i_{0}) \big) \sqcup \big( \sqcap^{n,i,\epsilon} \otimes y(i_{1}) \big)}{\rightarrow} & \sqcap^{n,i,\epsilon} \otimes \square^{1} \\ \mathllap{i_{i,\epsilon} \sqcup i_{i,\epsilon}} \downarrow & & \downarrow \mathrlap{r_{0}} \\ \square^{n} \sqcup \square^{n} & \underset{r_{1 }}{\rightarrow} & \sqcap^{n+1,i,\epsilon} }

We denote by i i,ϵ: n+1,i,ϵ n+1i_{i,\epsilon} : \sqcap^{n+1,i,\epsilon} \rightarrow \square^{n+1} the canonical arrow determined, by means of the universal property of n+1,i,ϵ\sqcap^{n+1,i,\epsilon}, by the following commutative square in Set op\mathsf{Set}^{\square^{op}}.

n,i,ϵ n,i,ϵ ( n,i,ϵy(i 0))( n,i,ϵy(i 1)) n,i,ϵ 1 i i,ϵi i,ϵ i i,ϵ 1 n n ( ny(i 0))( ny(i 1)) n+1 \array{ \sqcap^{n,i,\epsilon} \sqcup \sqcap^{n,i,\epsilon} & \overset{\big( \sqcap^{n,i,\epsilon} \otimes y(i_{0}) \big) \sqcup \big( \sqcap^{n,i,\epsilon} \otimes y(i_{1}) \big)}{\rightarrow} & \sqcap^{n,i,\epsilon} \otimes \square^{1} \\ \mathllap{i_{i,\epsilon} \sqcup i_{i,\epsilon}} \downarrow & & \downarrow \mathrlap{i_{i,\epsilon} \otimes \square^{1}} \\ \square^{n} \sqcup \square^{n} & \underset{\big( \square^{n} \otimes y(i_{0}) \big) \sqcup \big( \square^{n} \otimes y(i_{1}) \big)}{\rightarrow} & \square^{n+1} }

We define n+1,n+1,ϵ\sqcap^{n+1, n+1, \epsilon} to be a cubical set fitting into a co-cartesian square in Set op\mathsf{Set}^{\square^{op}} as follows.

n,n,ϵ n,n,ϵ (y(i 0) n,n,ϵ)(y(i 1) n,n,ϵ) 1 n,n,ϵ i n,ϵi n,ϵ r 0 n n r 1 n+1,n+1,ϵ \array{ \sqcap^{n,n,\epsilon} \sqcup \sqcap^{n,n,\epsilon} & \overset{\big( y(i_{0}) \otimes \sqcap^{n,n,\epsilon} \big) \sqcup \big( y(i_{1}) \otimes \sqcap^{n,n,\epsilon} \big)}{\rightarrow} & \square^{1} \otimes \sqcap^{n,n,\epsilon} \\ \mathllap{i_{n,\epsilon} \sqcup i_{n,\epsilon}} \downarrow & & \downarrow \mathrlap{r_{0}} \\ \square^{n} \sqcup \square^{n} & \underset{r_{1}}{\rightarrow} & \sqcap^{n+1,n+1,\epsilon} }

We denote by i n+1,ϵ: n+1,n+1,ϵ n+1i_{n+1,\epsilon} : \sqcap^{n+1,n+1,\epsilon} \rightarrow \square^{n+1} the canonical arrow determined, by means of the universal property of n+1,n+1,ϵ\sqcap^{n+1,n+1,\epsilon}, by the following commutative square in Set op\mathsf{Set}^{\square^{op}}.

n,n,ϵ n,n,ϵ (y(i 0) n,n,ϵ)(y(i 1) n,n,ϵ) 1 n,n,ϵ i n,ϵi n,ϵ 1i n,ϵ n n (y(i 0) n)(y(i 1) n) n+1 \array{ \sqcap^{n,n,\epsilon} \sqcup \sqcap^{n,n,\epsilon} & \overset{\big( y(i_{0}) \otimes \sqcap^{n,n,\epsilon} \big) \sqcup \big( y(i_{1}) \otimes \sqcap^{n,n,\epsilon} \big)}{\rightarrow} & \square^{1} \otimes \sqcap^{n,n,\epsilon} \\ \mathllap{i_{n,\epsilon} \sqcup i_{n,\epsilon}} \downarrow & & \downarrow \mathrlap{\square^{1} \otimes i_{n,\epsilon}} \\ \square^{n} \sqcup \square^{n} & \underset{\big( y(i_{0}) \otimes \square^{n} \big) \sqcup \big( y(i_{1}) \otimes \square^{n} \big)}{\rightarrow} & \square^{n+1} }
Terminology

We refer to n,i,ϵ\sqcap^{n, i, \epsilon} together with the morphism i i,ϵi_{i,\epsilon} as a horn of n\square^{n}.

Morphism from n\square^{n} to 0\square^{0}

Notation

We denote by p: n 0p : \square^{n} \rightarrow \square^{0} the arrow y(ppp n)y(\underbrace{p \otimes p \otimes \cdots p}_{n}) of Set op\mathsf{Set}^{\square^{op}}, making use of the fact that 0 0 0 n\underbrace{\square^{0} \otimes \square^{0} \otimes \cdots \square^{0}}_{n} is 0\square^{0}, since 0\square^{0} is the unit of the monoidal structure on Set op\mathsf{Set}^{\square^{op}}.

Expository material

For expository and other material, see cubical set - exposition.

Revised on April 17, 2016 16:10:40 by Richard Williamson (62.16.228.203)