The join $S \star T$ of two simplicial sets $S$ and $T$ is a new 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.
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, $\Delta_a$, of finite ordinals (including the empty one).
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 $[2]\star [0]$.
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 $\Delta[3]$.
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 ordinal sum, and then how it induces structure on the category of augmented simplicial sets.
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:
The join of simplicial set is equivalently expressed as
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 $S$, $T$ (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 $:= Sets^{\Delta_a^{op}}$. The internal hom is given by
where $Dec$ is the total décalage? functor (see also at décalage).
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 $S$.
The join of two ordinary simplicial sets $S_1$ and $S_2$ is the join of their trivial augmentation :
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.
The join of two non-augmented simplicial sets is given by the formula
The $i$-th boundary map
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:
If $j = 0$ then
If $k = 0$ then
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 $S \times S'$.
In this form, the join is used in definition 1.2.8.1, p. 42 of HTT
Recall that the join of simplicial sets $S \star T$ is a cocontinuous functor in each of its separate arguments $S$, $T$ (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 to compute joins of horns with simplices.
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
$S^\triangleleft := \{v\} \star S$ is the cone over $S$;
$S^\triangleright := S \star \{v\}$ 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
More generally, the cone over the $n$-simplex is the $(n+1)$-simplex
Cones of 2-horns are simplicial 2-squares $\simeq \Delta[1] \times \Delta[1]$:
and
Effectively by the definition from ordinal sum, we have that the join of two simplices is another simplex:
In particular the cone over the $n$-simplex is the $(n+1)$-simplex
and hence
Notice that while thus $\Delta[n+1] \simeq \Delta[0]\star\Delta[n] \simeq \Delta[n] \star \Delta[0]$ 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
and
Let $\partial \Delta[1] = \Delta[0] \coprod \Delta[0]$ 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
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.
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.
There is also a join operations on categories and sSet-categories:
Let $C,D \in sSet Cat$. Then define $C \star D$ to be the $sSet$-category given by
with the obvious composition operations.
Write
for the left adjoint of the homotopy coherent nerve functor (denoted $\mathfrak{C}$ in HTT. )
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 $\tau_{hc}(S) \star \tau_{hc}(S')$;
the canonical morphism
is an equivalence in the model structure on sSet-categories.
This is HTT, corollary 4.2.1.4.
The join operation was studied by P. J. Ehlers, in his thesis
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