higher category theory - Joins of simplicial sets

It might be helpful to work through some simple examples. You probably know that Δⁿ ★ Δ^k = Δ^n+k+1. This has to do with the ordinal sum: one way of defining joins is as a restriction of the monoidal structure on augmented simplicial sets, which are contravarient functors from the category Δ₊ of all finite ordinals (including the empty ordinal) into sets. The category Δ₊ has a monoidal structure given by ordinary addition with ∅ as the unit, and this induces the aforementioned monoidal structure on augmented simplicial sets. The thing we call n when we are talking about simplicial sets is really the ordinal n+1, so the formula above holds because

(n+1) + (k+1) = (n+k+1)+1.

Of course, this example doesn't illustrate the asymmetry you asked about, but this one will:

∂Δⁿ ★ Δ⁰ = Λⁿ⁺¹[n+1] while Δ⁰ ★ ∂Δⁿ = Λ⁰[n+1].

To work out the details, you'll need to understand how the face maps of S★T are defined, as alluded to above. Here's my notation: (S★T)_n = S_n ∪ T_n ∪ (∪ _{j+k = n+1} S_j × T_k ).

The i-th boundary map d_i : (S★T)_n → (S★T)_n-1 is defined on S_n and T_n using the i-th boundary map on S and T. Given σ∈S_j and τ∈T_k , we have:

d_i (σ, τ) = (d_i σ,τ) if i ≤ j, j ≠ 0.
d_i (σ, τ) = (σ,d_i-j-1 τ) if i > j, k ≠ 0.

If j = 0, d₀(σ, τ) = τ ∈ T_n-1 ⊂ (S★T)_n-1. If k = 0, d_n(σ, τ) = σ ∈S_n-1 ⊂ (S★T)_n-1 .

Try this out for n = 1 or 2 first, to get a feel for things. While these sorts of computations can be quite annoying, I find they do really help me develop my intuition. Best of luck!