join of simplicial sets



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



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Homotopy groups

Basic facts




The join STS \star T of two simplicial sets SS and TT is a new simplicial set that may geometrically be thought of as a cone over TT with tip of shape SS. Topologically, it can also be thought of as the union of line segments connecting SS to TT if both are placed in general position.

If the simplicial sets in question are quasi-categories the notion of join on them produces the notion of join of quasi-categories that underlies many constructions in higher category theory such as the definition of limit in a quasi-category.

The join of simplicial sets extends that historically given for simplicial complexes, cf. for instance the description and discussion in Spanier's classical text (page 109 and then pages 114 -116).

The adaptation of this to simplicial sets reveals a neat link with some categorical structure in the category, Δ a\Delta_a, of finite ordinals (including the empty one).

Motivating examples

When S=Δ 0S = \Delta^0 is the point, then the join STS \star T is a genuine cone over TT. Or if S=2S = 2 is the discrete two-point space, the join is the suspension of TT.

For example, consider the two cones over [2][2], the standard 2-simplex. The first picture represents [0][2][0]\star [2], while the second represents [2][0][2]\star [0].

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If you take two non-coplanar line segments in 3\mathbb{R}^3 (such as ABA B and CDC D in the picture below), then join every point in one to every point in the other, you get a 3-simplex (the tetrahedron in the picture). You can think of this as being the union of all the cones on the first segment, with cone points on the second one. We have that the join Δ[1]Δ[1]\Delta[1]\star \Delta[1] is Δ[3]\Delta[3].

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We first define the join of simplicial sets as the restriction to simplicial sets of the extension of the ordinal sum operation on the augmented simplex category Δ a\Delta_a to augmented simplicial sets.

Then we give the more explicit definition in terms of concrete formulas. We first refer to the description of ordinal sum, and then how it induces structure on the category of augmented simplicial sets.

By Day convolution

Via the general process of Day convolution, the ordinal sum monoidal structure on Δ a\Delta_a is lifted to a monoidal structure on presheaves on Δ a\Delta_a, i.e. to the the category asSet or sSet +sSet_+ of augmented simplicial sets. This is given by a coend formula:


The join of simplicial set is equivalently expressed as

:sSet +×sSet +sSet + \star : sSet_+ \times sSet_+ \to sSet_+
(SS)():= [i],[j]Δ a(S i×S j)×Hom Δ a([i][j],). (S \star S')(-) := \int^{[i],[j] \in \Delta_a} (S_i \times S'_j) \times Hom_{\Delta_a}([i] \boxplus [j],-) \,.

This is an abuse of notation because Hom Δ a(,[i][j])Hom_{\Delta_a}(-,[i] \boxplus [j]) is a functor, while (S i×S j)(S_i \times S'_j) is a set. To be precise, the second ×\times should be replaced with \cdot, which denotes the indexed copower.

Note that the join of simplicial sets STS \star T is cocontinuous in each of its separate arguments SS, TT (this is true generally of Day convolution products).


This join tensor product forms part of a closed monoidal structure on the category of augmented simplicial sets, asSet :=Sets Δ a op := Sets^{\Delta_a^{op}}. The internal hom is given by

[X,Y] n=asS(X;Dec n+1Y),[X, Y ]_n =asS(X; Dec^{n+1}Y )\,,

where DecDec is the total décalage? functor (see also at décalage).


For SS a simplicial set, let S^\hat S denote the augmented simplicial set which equals SS in all degrees except in degree -1, where it is the point, (S^) 1=pt({\hat S})_{-1} = pt. This is the trivial augmentation of SS.


The join of two ordinary simplicial sets S 1S_1 and S 2S_2 is the join of their trivial augmentation :

S 1S 2:=S^ 1S^ 2. S_1 \star S_2 := {\hat S_1} \star {\hat S_2} \,.

Note that the join operation is not commutative. This comes from the fact that the monoidal structure given by ordinal sum on the augmented simplex category is not symmetric monoidal. See the discussion at augmented simplex category for more details.

Concrete formulas

The join of two non-augmented simplicial sets is given by the formula

(SS) n:=S nS n( i+j=n1S i×S j). (S \star S')_n := S_n \cup S'_n \cup (\cup_{i+j = n-1} S_i \times S'_j) \,.

The ii-th boundary map

d i:(ST) n(ST) n1 d_i : (S \star T)_n \to (S \star T)_{n-1}

is defined on S nS_n and T nT_n using the iith boundary map on SS and TT.

Given σS j\sigma \in S_j and τT k\tau \in T_k , we have:

d i(σ,τ)={(d iσ,τ) ifij,j0 (σ,d ij1) ifi>j,k0 d_i (\sigma, \tau) = \left\{ \array{ (d_i \sigma, \tau) & if\; i \leq j , j \neq 0 \\ (\sigma, d_{i-j-1}) & if\; i \gt j, k \neq 0 } \right.

If j=0j = 0 then

d 0(σ,τ)=τT n1(ST) n1. d_0(\sigma, \tau) = \tau \in T_{n-1} \subset (S \star T)_{n-1} \,.

If k=0k = 0 then

d n(σ,τ)=σS n1(ST) n1. d_n(\sigma, \tau) = \sigma \in S_{n-1} \subset (S \star T)_{n-1} \,.

Join of quasi-categories

If the simplicial sets in question are quasi-categories, their join computes the corresponding join of quasi-categories, effectively an over quasi-category construction.

In this sense the join can then also be computed – up to equivalence of quasi-categories – as the homotopy pushout of the two projections out of S×SS \times S'.

In this form, the join is used in definition, p. 42 of HTT


Recall that the join of simplicial sets STS \star T is a cocontinuous functor in each of its separate arguments SS, TT (this is true generally of Day convolution products).

This observation can help simplify calculations. For example, simplicial joins preserve unions in the first argument SS, and inasmuch as horns are unions of face simplices, this allows to compute joins of horns with simplices.

Joins with the point: cones

For {v}=Δ[0]\{v\} = \Delta[0] the point, a join with the point is called a cone with cone vertex vv: for SsSetS \in sSet we say

  • S :={v}SS^\triangleleft := \{v\} \star S is the cone over SS;

  • S :=S{v}S^\triangleright := S \star \{v\} is the co-cone under SS;

Universal images of cones and cocones over a fixed base SS in a quasi-category CC are limits and colimits in that quasi-category.

For instance the cone over the interval Δ[1]\Delta[1] is the 2-simplex

{v}Δ[1]=( v 0 1)Δ[2]. \{v\} \star \Delta[1] = \left( \array{ && v \\ & \swarrow &\swArrow& \searrow \\ 0 &&\to&& 1 } \right) \simeq \Delta[2] \,.

More generally, the cone over the nn-simplex is the (n+1)(n+1)-simplex

Δ[n] Δ[n+1]. \Delta[n]^{\triangleleft} \simeq \Delta[n+1] \,.

Cones of 2-horns are simplicial 2-squares Δ[1]×Δ[1]\simeq \Delta[1] \times \Delta[1]:

Δ[1]×Δ[1]{v}Λ 2[2]=(v 1 0 2) \Delta[1] \times \Delta[1] \simeq \{v\} \star \Lambda_2[2] = \left( \array{ v &\to& 1 \\ \downarrow &{}_{\swArrow}\searrow^{\swArrow}& \downarrow \\ 0 &\to& 2 } \right)


Δ[1]×Δ[1]Λ 0[2]{v}=(0 1 2 v). \Delta[1] \times \Delta[1] \simeq \Lambda_0[2] \star \{v\} = \left( \array{ 0&\to& 1 \\ \downarrow &{}_{\swArrow}\searrow^{\swArrow}& \downarrow \\ 2 &\to& v } \right) \,.

Joins of simplices

Effectively by the definition from ordinal sum, we have that the join of two simplices is another simplex:

Δ[k]Δ[l]=Δ[k+l+1]. \Delta[k] \star \Delta[l] = \Delta[k + l + 1] \,.

In particular the cone over the nn-simplex is the (n+1)(n+1)-simplex

Δ[0]Δ[n]=Δ[n+1] \Delta[0] \star \Delta[n] = \Delta[n+1]

and hence

Δ[n]=Δ[0]Δ[0]. \Delta[n] = \Delta[0] \star \cdots \star \Delta[0] \,.

Notice that while thus Δ[n+1]Δ[0]Δ[n]Δ[n]Δ[0]\Delta[n+1] \simeq \Delta[0]\star\Delta[n] \simeq \Delta[n] \star \Delta[0] the identifications of the cone point of course differ in both cases. The asymmetry is seen for instance by restricting attention to the cone over the boundary of the nn-simplex, where we have

Δ[n]Δ[0]=Λ n+1[n+1] \partial \Delta[n] \star \Delta[0] = \Lambda_{n+1}[n+1]


Δ[0]Δ[n]=Λ 0[n+1]. \Delta[0] \star \partial \Delta[n] = \Lambda_0[n+1] \,.

Simplicial nn-sphere

Let Δ[1]=Δ[0]Δ[0]\partial \Delta[1] = \Delta[0] \coprod \Delta[0] the simplicial 0-sphere: just the disjoint union of the point. Then the nn-fold join of Δ[1]\partial \Delta[1] with itself is a simplicial model for the nn-sphere

S 0:=Δ[0] \mathbf{S}^0 := \partial \Delta[0]
S n:=S 0S n1 \mathbf{S}^n := \mathbf{S}^0 \star \mathbf{S}^{n-1}

for nn \in \mathbb{N}, n>0n \gt 0. The geometric realization of S n\mathbf{S}^n is equivalent to the topological nn-sphere.

See Ehlers/Porter p. 8.


Compatibility with quasi-categories


If S,SS, S' \in sSet are both quasi-categories, then so is their join SSS \star S'.

This is due to Andre Joyal. A proof appears as HTT, prop.

Compatibility with homotopy coherent nerve

There is also a join operations on categories and sSet-categories:


Let C,DsSetCatC,D \in sSet Cat. Then define CDC \star D to be the sSetsSet-category given by

Obj(CD)=Obj(C)Obj(D) Obj(C \star D) = Obj(C) \coprod Obj(D)
CD(x,y)={C(x,y) forx,yC D(x,y) forx,yD forxD,yC * forxC,yD C \star D(x,y) = \left\{ \array{ C(x,y) & for\; x,y \in C \\ D(x,y) & for\; x,y \in D \\ \emptyset & for \;x \in D, y \in C \\ * & for\; x \in C , y \in D } \right.

with the obvious composition operations.


τ hc:sSetsSetCat \tau_{hc} : sSet \to sSet Cat

for the left adjoint of the homotopy coherent nerve functor (denoted \mathfrak{C} in HTT. )


For S,SS, S' two simplicial sets we have that

  • the two inclusions τ hc(S),τ hc(S)τ hc(SS)\tau_{hc}(S), \tau_{hc}(S') \to \tau_{hc}(S\star S') are full and faithful.

  • τ hc(SS)\tau_{hc}(S \star S') is in general not isomorphic to τ hc(S)τ hc(S)\tau_{hc}(S) \star \tau_{hc}(S');

  • the canonical morphism

    τ hc(SS)τ hc(S)τ hc(S) \tau_{hc}(S \star S') \to \tau_{hc}(S) \star \tau_{hc}(S')

    is an equivalence in the model structure on sSet-categories.

This is HTT, corollary


The join operation was studied by P. J. Ehlers, in his thesis

  • Algebraic Homotopy in Simplicially Enriched Groupoids, 1993, University of Wales Bangor, available here, (see also the reference below and the Menagerie notes available from Tim Porter‘s homepage.),

but was there ascribed to Jack Duskin and Don van Osdol in some unpublished notes. The main ideas were derived there from earlier work of Bill Lawvere.

A useful published reference is

A useful discussion emphasizing the Day convolution operation is also in section 3.1 of

Discussion in homotopy type theory (with application to n-image factorization) is in

Last revised on October 19, 2018 at 09:07:50. See the history of this page for a list of all contributions to it.