Let $\Delta$ be the category of finite ordered sets $[n]$ with morphisms $[n] \to [m]$ being the order-preserving ones. A simplicial set is a contravariant functor $X : \Delta \to \text{Set}$; concretely, call $X^k = X([k])$ to be the set of $k$-simplices in $X$, and let $D_i : [n] \to [n+1]$ be defined by "skipping $i$" and $S_i : [n+1] \to [n]$ by "repeating $i$".

Then these give rise to maps $d_i : X^{n+1} \to X^n$ (to be thought as assigning an $n+1$-simplex it's $i$-th face) and $s_i : X^n \to X^{n+1}$ (to be thought as assigning an $n$-simplex the $i$-th degenerate face that collapses to it)

These relations are straightforward to check: $d_i d_j = d_{j - 1} d_i$ if $i < j$, $d_i s_j = s_{j - 1} d_i$ if $i < j$, $d_i s_i = d_{i+1} s_i = \text{id}$, $d_i s_j = s_j d_{i-1}$ if $i > j+1$ and $s_i s_j = s_{j+1} s_i$ if $i \leq j$ (feed both sides $[n]$ and keep track of the $i$-th and $j$-th slots)

This gives the purely combinatorial interpretation: a simplicial set is a sequences of sets $X^0, X^1, \cdots$ (the $k$-skeleta, consisting of all $k$-simplices along with the degenerate ones), with face maps $d_i : X^{n+1} \to X^n$ and degeneracy maps $s_i : X^n \to X^{n+1}$ (note carefully the direction of the arrows) for all $0 \leq i \leq n$ which satisfies the five axioms above

A morphism of simplicial sets $X, Y : \Delta \to \text{Set}$ is a natural transformation $X \Rightarrow Y$

Denote $\Delta^n$ to be the standard $n$-simplex thought as a simplicial set, and for any simplicial set $X$ define it's geometric realization $|X| = \coprod_{k \geq 1} X^k \times |\Delta^k| /\sim$ where $(x, D_i(p)) \sim (d_i(x), p)$ for $x \in X^{n+1}$ and $p \in |\Delta^n|$, and $(x, S_i(p)) \sim (s_i(x), p)$ for $x \in X^{n-1}$ and $p \in |\Delta^n|$.

What the first relation tells is to glue the point $p$ in the $i$-th face $d_i(x)$ of some $(n+1)$-simplex $x$ to the point $D_i(p)$ in the $(n+1)$-simplex $d_i(x)$, so paste the faces of the $(n+1)$-simplex, floating in space as individual entities, back to the $(n+1)$-simplex, robbing them off their individuality.

Similarly, the second relation tells us to glue the point $p$ in the $i$-th $n$-simplex $s_i(x)$ that degenerates to $x$ to the point $S_i(p)$ in the $i$-th $(n-1)$-simplex $x$, so you're again robbing the degenerate simplices off their individuality

Giving each $X^k$, $k \geq 0$ the discrete topology, we can give $|X|$ the quotient topology from the disjoint union, giving $|\Delta^k|$ the standard topology as a simplex. Given any geometric simplicial complex, one can take the corresponding simplicial set (which algebraically "fattens" it, by adding all the degenerate simplices individually), and taking it's realizing gives it back again.

So $X \mapsto |X|$ "tucks away the algebraic fat"

Let $\text{SSet}$ be the category of simplicial sets; define $\mathscr{S} : \text{Top} \to \text{SSet}$ to be the functor $\mathscr{S}(X) = \text{Hom}_{\text{Top}}(\Delta^n, X)$ - this is a simplicial set with $d_i$ being restriction to a face and $s_i$ being composing with a degenerate simplex. This is basically just the singular chain complex, simplicialsetified

$|.| : \text{SSet} \to \text{Top}$ is the geometric realization functor described above. Then in fact these are adjoint functors

Namely, define $\text{Hom}_{\text{SSet}}(X, \mathscr{S}(Y)) \to \text{Hom}_{\text{Top}}(|X|, Y)$ as follows: given a natural transformation $F : X \Rightarrow \mathscr{S}(Y)$ define $f : \{x\} \times \Delta^n \to Y$ for any $x \in X([n])$ by $f = F([n])$ (acting on $\{x\} \times \Delta^n$ instead of $\Delta^n$). These descent on the quotient to a map $f : |X| \to Y$.

The inverse $\text{Hom}_{\text{Top}}(|X|, Y) \to \text{Hom}_{\text{SSet}}(X, \mathscr{S}(Y))$ is sort of obvious; compose a map $f : |X| \to Y$ with a simplex $\Delta^n \to X$ in the geometric realization $|X|$. These are very clearly inverses to each other

Some annoying details: If $X, Y$ are simplicial sets then $X \times Y$ can be made into a simplicial set by doing factorwise face and degeneracy maps, so that the "projections" $X \times Y \implies X$ and $X \times Y \implies Y$ are morphism of simplicial sets

Fact: $|X \times Y| \cong |X| \times |Y|$ in $\text{CGHaus}$

One can define the internal hom in $\text{SSet}$ by defining $\mathbf{Hom}_{\text{SSet}}(X, Y)$ to be the one whose $k$-th skeleton is $\mathbf{Hom}_{\text{SSet}}(X, Y)_k = \text{Hom}_{\text{SSet}}(\Delta^k \times X, Y)$ and defining the face and degeneracy maps using the face and degeneracy maps for $\Delta^k$

Fact: $\text{Hom}(X, \mathbf{Hom}(Y, Z)) \cong \text{Hom}(X \times Y, Z)$ hence $\mathbf{Hom}(X, \mathbf{Hom}(Y, Z)) \cong \mathbf{Hom}(X \times Y, Z)$ as simplicial sets

The key point of internal homs is the following. Given a simplicial set $X$, there is a natural bijection between it's $n$-simplices and $\text{Hom}_{\text{SSet}}(\Delta^n, X)$. The reason is the following $X$ and $\Delta^n = \text{Hom}_{\text{Set}}(-, [n])$ are contravariant functors $\Delta \to \text{Set}$, so by Yoneda lemma $X([n]) \cong \text{Nat}(\text{Hom}_{\text{Set}}(-, [n]), X) = \text{Hom}_{\text{SSet}}(\Delta^n, X)$.

Then $\mathbf{Hom}_{\text{SSet}}(\Delta^0, X)[n] = \text{Hom}_{\text{SSet}}(\Delta^n, X) \cong X([n])$, and the face/degeneracy maps are clearly natural in this correspondence, so $\mathbf{Hom}_{\text{SSet}}(\Delta^0, X) \cong X$ as simplicial sets.

The fact above is fairly straightforward to check by the way, since a natural transformation $X \Rightarrow \mathbf{Hom}(Y, Z)$ assigns over $[n]$ a set-homomorphism $X[n] \to \mathbf{Hom}(Y, Z)[n] = \text{Hom}_{\text{Set}}(Y \times \Delta^n, Z)$, which by adjunction is just a homomorphism $\text{Hom}_{\text{Set}}(X \times Y \times \Delta^n, Z)$

LOL

I got confused on the last message. A natural transformation $X \Rightarrow \mathbf{Hom}(Y, Z)$ gives over each $[n]$ a Set-hom $X[n] \to \mathbf{Hom}(Y, Z)[n] = \text{Hom}_{\text{SSet}}(Y \times \Delta^n, Z)$.

Let's call this $f$. Then for every $n$-simplex $x \in X[n]$ of $X$, $f(x)[n] \in \text{Hom}_{\text{Set}}((Y \times \Delta^n)[n], Z[n])$. $(Y \times \Delta^n)[n] \cong Y[n] \times \Delta^n[n]$. Forget about the second factor, that gives a map $Y[n] \to Z[n]$.

Jointly this defines $(X \times Y)[n] = X[n] \times Y[n] \to Z[n]$, defining an element of $\text{Hom}(X \times Y, Z)$.

This is the bijection $\text{Hom}(X, \mathbf{Hom}(Y, Z)) \cong \text{Hom}(X \times Y, Z)$.

Hi @KarlKronenfeld!

Hey

It's like opening a door and finding a wall behind it

You caught me in one of my less glorious moments, learning simplicial sets

I remember one thing I found confusing is that there are different simplicial sets corresponding to the same space after taking geometric realization. Like it's completely obvious why something like that would happen, but e.g. I remember not realizing that there are some simplicial sets which are bad to work with (eg they're not Kan complexes), even though its geometric realization is a sphere.