Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




The cellular simplex is one of the basic geometric shapes for higher structures. Variants of the same `shape archetype' exist in several settings, e.g., that of simplicial sets, the topological /cellular one, and categorical contexts, plus others.


Simplicial simplices

For nn \in \mathbb{N}, the standard simplicial nn-simplex Δ[n]\Delta[n] is the simplicial set which is represented (as a presheaf) by the object [n][n] in the simplex category, so Δ[n]=Δ(,[n])\Delta[n]= \Delta(-,[n]).

Cellular (simplicial) simplex

Likewise, there is a standard toplogical nn-simplex, which is (more or less by definition) the geometric realization of the standard simplicial nn-simplex.

Topological simplex

The topological nn-simplex Δ n\Delta^n is a generalization of the standard filled triangle in the plane, from dimension 2 to arbitrary dimensions. Each Δ n\Delta^n is homeomorphic to the closed nn-ball D nD^n, but its defining embedding into a Cartesian space equips its boundary with its cellular decomposition into faces, generalizing the way that the triangle has three edges (which are 1-simplices) as faces, and three points (which are 0-simplices) as corners.

The topological nn-simplex is naturally defined as a subspace of a Cartesian space given by some relation on its canonical coordinates. There are two standard choices for such coordinate presentation, which of course define homeomorphic nn-simplices:

Each of these has its advantages and disadvantages, depending on application, but of course there is a simple coordinate transformation that exhibits an explicit homeomorphism between the two:

Barycentric coordinates

In the following, for nn \in \mathbb{N} we regard the Cartesian space n\mathbb{R}^n as equipped with the canonical coordinates labeled x 0,x 1,,x n1x_0, x_1, \cdots, x_{n-1}.


For nn \in \mathbb{N}, the topological nn-simplex is, up to homeomorphism, the topological space whose underlying set is the subset

Δ n{x n+1| i=0 nx i=1andi.x i0} n+1 \Delta^n \coloneqq \{ \vec x \in \mathbb{R}^{n+1} | \sum_{i = 0 }^n x_i = 1 \; and \; \forall i . x_i \geq 0 \} \subset \mathbb{R}^{n+1}

of the Cartesian space n+1\mathbb{R}^{n+1}, and whose topology is the subspace topology induces from the canonical topology in n+1\mathbb{R}^{n+1}.


For nn \in \mathbb{N}, n1\n \geq 1 and 0kn0 \leq k \leq n, the kkth (n1)(n-1)-face (inclusion) of the topological nn-simplex is the subspace inclusion

δ k:Δ n1Δ n \delta_k : \Delta^{n-1} \hookrightarrow \Delta^n

induced under the barycentric coordinates of def. 1, by the inclusion

n n+1 \mathbb{R}^n \hookrightarrow \mathbb{R}^{n+1}

which omits the kkth coordinate

(x 0,,x n1)(x 0,,x k1,0,x k,,x n1). (x_0, \cdots , x_{n-1}) \mapsto (x_0, \cdots, x_{k-1} , 0 , x_{k}, \cdots, x_{n-1}) \,.

The inclusion

δ 0:Δ 0Δ 1 \delta_0 : \Delta^0 \to \Delta^1

is the inclusion

{1}[0,1] \{1\} \hookrightarrow [0,1]

of the “right” end of the standard interval. The other inclusion

δ 1:Δ 0Δ 1 \delta_1 : \Delta^0 \to \Delta^1

is that of the “left” end {0}[0,1]\{0\} \hookrightarrow [0,1].


For nn \in \mathbb{N} and 0k<n0 \leq k \lt n the kkth degenerate nn-simplex (projection) is the surjective map

σ k:Δ nΔ n1 \sigma_k : \Delta^{n} \to \Delta^{n-1}

induced under the barycentric coordinates of def. 1 under the surjection

n+1 n \mathbb{R}^{n+1} \to \mathbb{R}^n

which sends

(x 0,,x n)(x 0,,x k+x k+1,,x n). (x_0, \cdots, x_n) \mapsto (x_0, \cdots, x_{k} + x_{k+1}, \cdots, x_n) \,.

The collection of face inclusions, def. 2 and degenracy projections, def. 3 satisfy the (dual) simplicial identities. Equivalently, they constitute the components of a functor

Δ :ΔTop \Delta^\bullet : \Delta \to Top

from the simplex category Δ\Delta to the category Top of topological spaces. This is, up to isomorphism, the canonical cosimplicial object in TopTop.

Cartesian coordinates


The standard topological nn-simplex is, up to homeomorphism, the subset

Δ n{x n|0x 1x n1} n \Delta^n \coloneqq \{ \vec x \in \mathbb{R}^n | 0 \leq x_1 \leq \cdots \leq x_n \leq 1 \} \hookrightarrow \mathbb{R}^n

equipped with the subspace topology of the standard topology on the Cartesian space n\mathbb{R}^n.


This definition identifies the topological nn-simplex with the space of interval maps (preserving top and bottom) {0<1<<n+1}I\{0 \lt 1 \lt \ldots \lt n+1\} \to I into the topological interval. This point of view takes advantage of the duality between the simplex category Δ\Delta and the category \nabla of finite intervals with distinct top and bottom. Indeed, it follows from the duality that we obtain a functor

Δ opInt(,I)Top.\Delta \simeq \nabla^{op} \stackrel{Int(-, I)}{\to} Top.
  • For n=0n = 0 this is the point, Δ 0=*\Delta^0 = *.

  • For n=1n = 1 this is the standard interval object Δ 1=[0,1]\Delta^1 = [0,1].

  • For n=2n = 2 this is a triangle sitting in the plane like this:

    {(x 0,x 1)|0x 0x 11}={ (1,1) (12,12) (12,1) (0,0) (0,12) (0,1)} \left\{ (x_0,x_1) | 0 \leq x_0 \leq x_1 \leq 1 \right\} = \left\{ \array{ && && (1,1) \\ && & \nearrow & \downarrow \\ && (\tfrac{1}{2}, \tfrac{1}{2}) && (\tfrac{1}{2},1) \\ & \nearrow & && \downarrow \\ (0,0) &\stackrel{}{\to}& (0,\tfrac{1}{2}) & \to & (0,1) } \right\}

Transformation between Barycentric and Cartesian coordinates

For nn \in \mathbb{N}, write now explicitly

Δ bar n n+1 \Delta^n_{bar} \hookrightarrow \mathbb{R}^{n+1}

for the topological nn-simplex in barycentric coordinate presentation, def. 1, and

Δ cart n n \Delta^n_{cart} \hookrightarrow \mathbb{R}^{n}

for the topological nn-simplex in Cartesian coordinate presentation, def. 4.


S n: n+1 n S_n : \mathbb{R}^{n+1} \to \mathbb{R}^n

for the continuous function given in the standard coordinates by

(x 0,,x n)(x 0,x 0+x 1,, i=0 kx i,, i=0 nx i). (x_0, \cdots, x_{n}) \mapsto (x_0, x_0 + x_1, \cdots, \sum_{i = 0}^k x_i, \cdots, \sum_{i = 0}^n x_i) \,.

By restriction, this induces a continuous function on the topological nn-simplices

Δ bar n n+1 S n| Δ bar n p n Δ cart n n. \array{ \Delta^n_{bar} &\hookrightarrow& \mathbb{R}^{n+1} \\ \downarrow^{\mathrlap{S_n|_{\Delta^n_{bar}}}} && \downarrow^{p_n} \\ \Delta^n_{cart} &\hookrightarrow& \mathbb{R}^n } \,.

For every nn \in \mathbb{N} the function S nS_n is a homeomorphism and respects the face and degenracy maps.

Equivalently, S S_\bullet is a natural isomorphism of functors Δ nTop\Delta^n \to Top, hence an isomorphism of cosimplicial objects

S :Δ bar Δ cart . S_\bullet : \Delta^\bullet_{bar} \stackrel{\simeq}{\to} \Delta^\bullet_{cart} \,.

Singular simplex


For XX \in Top and nn \in \mathbb{N}, a singular nn-simplex in XX is a continuous map

σ:Δ nX. \sigma : \Delta^n \to X \,.


(SingX) nHom Top(Δ n,X) (Sing X)_n \coloneqq Hom_{Top}(\Delta^n , X)

for the set of singular nn-simplices of XX.

As nn varies, this forms the singular simplicial complex of XX.


Relation to globes

The orientals related simplices to globes.

Revised on March 15, 2016 06:13:53 by Urs Schreiber (