nLab join of simplicial sets

Idea

The join $S \star T$ of a pair $S, T$ of simplicial set (Ehlers 1993) is a simplicial set that may geometrically be thought of as a cone over $T$ with tip of shape $S$. Topologically, it can also be thought of as the union of line segments connecting $S$ to $T$ if both are placed in general position, cf. join of topological spaces.

The join operation on simplicial sets generalizes the historically earlier notion of join of simplicial complexes, cf. Spanier 1966. It is naturally discussed in view of augmented simplicial sets [Ehlers & Porter 2000]

If the simplicial sets in question are quasi-categories, then their join produces the corresponding join of quasi-categories that underlies many constructions in higher category theory, such as a definition of limits in quasi-categories.

Motivating examples

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

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

$$[0]$$

If you take two non-coplanar line segments in $\mathbb{R}^3$ (such as $A B$ and $C D$ 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 $\Delta[1]\star \Delta[1]$ is $\Delta[3]$.

$$g$$

Definition

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 $\Delta_a$ to augmented simplicial sets.

Then we give the more explicit definition in terms of concrete formulas. We first refer to the description of the 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 $\Delta_a$ is lifted to a monoidal structure on presheaves on $\Delta_a$, i.e., to the the category asSet or $sSet_+$ of augmented simplicial sets. This is given by a coend formula:

Definition/Proposition

The join of simplicial sets is equivalently expressed as

$$\star : sSet_+ \times sSet_+ \to sSet_+$$

$$(S \star S')(-) \coloneqq \int^{[i],[j] \in \Delta_a} (S_i \times S'_j) \times Hom_{\Delta_a}(-,[i] \boxplus [j]) \,.$$

Remark

This is an abuse of notation because $Hom_{\Delta_a}(-,[i] \boxplus [j])$ is a functor, while $(S_i \times S'_j)$ 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 $S \star T$ is cocontinuous in each of its separate arguments $S$ and $T$ (this is true generally of Day convolution products).

Proposition

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

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

where $Dec$ is the total décalage functor (see also at décalage).

Definition

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

Definition

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

$$S_1 \star S_2 \coloneqq {\hat S_1} \star {\hat S_2} \,.$$

Remark

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

$$(S \star S')_n \coloneqq S_n \cup S'_n \cup (\cup_{i+j = n-1} S_i \times S'_j) \,.$$

The $i$-th boundary map

$$d_i : (S \star T)_n \to (S \star T)_{n-1}$$

is defined on $S_n$ and $T_n$ using the $i$th boundary map on $S$ and $T$.

Given $\sigma \in S_j$ and $\tau \in T_k$ , we have:

$$d_i (\sigma, \tau) = \left\{ \array{ (d_i \sigma, \tau) & if\; i \leq j , j \neq 0 \\ (\sigma, d_{i-j-1} \tau) & if\; i \gt j, k \neq 0 } \right.$$

If $j = 0$, then

$$d_0(\sigma, \tau) = \tau \in T_{n-1} \subset (S \star T)_{n-1} \,.$$

If $k = 0$, then

$$d_n(\sigma, \tau) = \sigma \in S_{n-1} \subset (S \star T)_{n-1} \,.$$

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 also be computed – up to equivalence of quasi-categories – as the homotopy pushout of the two projections out of $S \times S'$.

In this form, the join is used in definition 1.2.8.1, p. 42 of HTT

Examples

Recall that the join of augmented simplicial sets $S \star T$ is a cocontinuous functor in each of its separate arguments $S$ and $T$ (this is true generally of Day convolution products).

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

As an operation on simplicial sets, $S \star T$ merely commutes with connected colimits (since these preserve the property of having trivial augmentation). However, for each $S$, the natural inclusion $S \cong S \star \varnothing \to S \star T$ induces a functor $S \star (-) : sSet \to S/sSet$ that does preserve colimits. Similarly, $(-) \star T : sSet \to T/sSet$ preserves colimits.

Joins with the point: cones

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

• $S^\triangleleft \coloneqq \{v\} \star S$ is the cone over $S$;

• $S^\triangleright \coloneqq S \star \{v\}$ is the co-cone under $S$;

Universal images of cones and cocones over a fixed base $S$ in a quasi-category $C$ are limits and colimits in that quasi-category.

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

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

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

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

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

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

and

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

Joins of simplices

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

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

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

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

and hence

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

Notice that while $\Delta[n+1] \simeq \Delta[0]\star\Delta[n] \simeq \Delta[n] \star \Delta[0]$, 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 $n$-simplex, where we have

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

and

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

Simplicial $n$-sphere

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

$$\mathbf{S}^0 \coloneqq \partial \Delta[1]$$

$$\mathbf{S}^n \coloneqq \mathbf{S}^0 \star \mathbf{S}^{n-1}$$

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

See Ehlers/Porter p. 8.

Properties

Compatibility with quasi-categories

Proposition

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

This is due to Andre Joyal. A proof appears as HTT, prop. 1.2.8.3.

Compatibility with homotopy coherent nerve

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

Definition

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

$$Obj(C \star D) = Obj(C) \coprod Obj(D)$$

$$C \star D(x,y) = \left\{ \array{ C(x,y) & for\; x,y \in C \\ D(x,y) & for\; x,y \in D \\ \emptyset & for \;x \in D, y \in C \\ * & for\; x \in C , y \in D } \right.$$

with the obvious composition operations.

Write

$$\tau_{hc} : sSet \to sSet Cat$$

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

Proposition

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

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

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

• the canonical morphism

  $$\tau_{hc}(S \star S') \to \tau_{hc}(S) \star \tau_{hc}(S')$$

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

This is HTT, corollary 4.2.1.4.

Related concepts

• join of topological spaces

• join of categories, join of quasi-categories

• join of maps

References

The historical predecessor notion of the join of simplicial complexes:

• Edwin Spanier, pp. 109 & 114-116 in: Algebraic topology, McGraw-Hill (1966), Springer (1982) [doi:10.1007/978-1-4684-9322-1]

Early discussion of the join operation on simplicial sets:

• Philip J. Ehlers, Algebraic Homotopy in Simplicially Enriched Groupoids, PhD thesis, University of Wales Bangor (1993) [pdf]

attributed there to unpublished notes of Jack Duskin and Don van Osdol.

Published discussion, relating to augmented simplicial sets:

• Phil Ehlers, Tim Porter: Joins for (Augmented) Simplicial Sets. Journal of Pure and Applied Algebra 145 1 (2000) 37-44 [doi:10.1016/S0022-4049(98)00065-6, arxiv:math/9904039]

Discussion emphasizing relation to Day convolution

• Dominic Verity, section 3.1 of: Weak complicial sets I [arXiv:math/0604414]

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

• Egbert Rijke, The join construction [arXiv:1701.07538]