## Background

Let $X$ and $S$ be simplicial sets, i.e. presheaves on $\Delta$, the so-called *topologist's simplex category*, which is the category of finite nonempty ordinals with morphisms given by order preserving maps.

How can we derive the structure of the face and degeneracy maps of the join from either of the two equivalent formulas for it below:

The *Day Convolution*, which extends the monoidal product to the presheaf category:

$$(X\star S)_{n}:=\int^{[c],[c^\prime] \in \Delta_a}X_{c}\times S_{c^\prime}\times Hom_{\Delta_a}([n],[c]\boxplus[c^\prime])$$

Where $\Delta_a$ is the augmented simplex category, and $\boxplus$ denotes the ordinal sum. The augmented simplex category is the category of all finite ordinals (note that this includes the empty ordinal, written $[-1]:=\emptyset$.

The join formula (for $J$ a finite nonempty linearly ordered set):

$$(X\star S)(J)=\coprod_{I\cup I=J}X(I) \times S(I')$$ Where $\forall (i \in I \text{ and } i' \in I'),$ $i < i'$, which implies that $I$ and $I'$ are disjoint.

Then we would like to derive the following formulas for the face maps (and implicitly the degeneracy maps):

The $i$-th face 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 face map on $S$ and $T$. Given $\sigma \in S_j\text{ and }\tau\in T_k$ , we have:

$$d_i (\sigma, \tau) = (d_i \sigma,\tau)\text{ if } i \leq j, j ≠ 0.$$ $$d_i (\sigma, \tau) = (\sigma,d_{i-j-1} \tau) \text{ if } i > j, k ≠ 0.$$ $$d_0(\sigma, \tau) = \tau \in T_{n-1} \subseteq (S\star T)_{n-1} \text{ if } j = 0$$ $$d_n(\sigma, \tau) = \sigma \in S_{n-1} \subset (S\star T)_{n-1}\text{ if } k = 0$$

We note that the special cases at the bottom come directly from the inclusion of augmentation in the formula for the join.

Edit: Another note here: I got these formulas from a different source, so the indexing may be off by a factor of -1.

## Question

How can we derive the concrete formulas for the face and degeneracy maps from the definition of the join (I don't want a geometric explanation. There should be a precise algebraic or combinatorial reason why this is the case.)?

Less importantly, how can we show that the two definitions of the join are in fact equivalent?

**Edit:**

Ideally, an answer would show how to induce one of the maps by a universal property.

Note also that in the second formula, we allow $I$ or $I'$ to be empty, and we extend the definition of a simplicial set to an augmented simplicial set such that $X([-1])=*$, i.e. the set with one element.

A further note about the first formula for the join: $\boxplus$ denotes the ordinal sum. That is, $[n]\boxplus [m]\cong [n+m+1]$. However, it is important to notice that there is no natural isomorphism $[n]\boxplus [m]\to [m]\boxplus [n]$. That is, there is no way to construct this morphism in a way that is natural in both coordinates of the bifunctor. This is important to note, because without it, it's not clear that the ordinal sum is asymmetrical.