Simplicial Sets II

Returning to simplicial sets and all that, so where we last left off we’d gotten through the definition of Delta sets and how they’re really just functors from $\Delta^{op} \to Set$ and thus maps between Delta sets are just going to be natural transformations between functors, which is great because the more things we can just describe with categorical concepts we already have the better.

The author (of this paper, just as a reminder) introduces a notion of degeneracy maps that are, essentially, the inverse of the face maps earlier. They repeat vertices in order to bump up the dimension of a simplex to the next level. As an example, we can bump $[0,1]$ to $[0,0,1]$ in order to treat it as a 2-simplex instead of a 1-simplex. This is an example of a degenerate 2-simplex because it isn’t really a 2-simplex in a sense, but we can effectively treat it was one anyway. What’s the point of degeneracy? Why do we need it? Well, in some sense we need it because we need to be able to describe maps that collapse vertices but with maps-as-natural-transformations as described above we really can’t because, in the case of our example in the previous post, we’d need to have a morphism that took us from $[[0,1],[1,2],[0,2],[0,3]]$ to $[[0,1],[1,2],[0,2]]$ that commuted with the face morphisms except, well, we can’t do that in a way that will respect the fact that we’re identifying 0 and 3 since we’d have to map $[0,3]$ onto $[0,1]$ or $[0,2]$ and either of those does something very different than just “shrinking” the line the way we want it. What we want to do is send $[0,3]$ to the degenerate simplex $[0,0]$.

So now we can say that a simplicial set is a family of sets just like the Delta sets, but now has both the family of $d_i$ face maps and the $s_i$ degeneracy maps, where $s_i$ duplicates the $i$-th entry in the simplex. In our example above this means that $s_0 [0] = [0,0]$, $s_0 [1] = [1,1]$, etc. This means that $[0]$ also “contains” $[0,0]$,$[0,0,0]$, etc. at every dimension.

Just as there were algebraic laws for the face maps, there are also algebraic laws for degeneracy maps and the interaction of degeneracy and face maps: $s_i \circ s_j = s_{j+1} \circ s_i$, $d_i \circ s_j = s_{j-1} \circ d_i$, $d_j \circ s_j = d_{j+1} \circ s_j = id$, $d_i \circ s_j = s_j \circ d_{i-1}$.

Following the previous construction of Delta sets, we can have a new category which I’ll also call $\Delta$ (because there really should have been a prime on the previous post) which has the same objects, but now has all all order preserving morphisms rather just the strictly order preserving ones. Just as the strictly order preserving maps were generated by the $Di$s, we can keep the $latex Di$s and introduce another set of morphisms $S_i : \Delta^(n+1) \to \Delta^n$ which map both $i$ and $i+1$ to $i$. Since we already know we’re going to take the opposite category, just as the $D_i$ corresponded to the face maps that select out faces from the simplex, the $S_i$ when reversed are going to be “inflating” faces into a degenerate simplex one level up.

This is one place where my knowledge is faltering a bit, because I know there’s something special about categories where all arrows can broken into a composition of a mono and an epi, which is definitely the case here. I know it’s a property that Set and I believe some (all??) topoi have.

That aside, we come to the topic of realization, which is how one goes the abstractness of a set-valued contravariant functor over $\Delta$ and actually obtains a topological space. This is where I start to get shaky because I need to review a lot of my topology. The construction starts by taking all the $X^n$ for the simplicial sets and then giving them the discrete topology, where every subset is an open set (true story, I’ve gotten the discrete and indiscrete topologies consistently mixed up since I was a kid 16 years ago first learning this stuff) and then for each of the $X^n$ we take its product with $\Delta^n$ and then take the quotient by the relation $(x,D_i(p)) ~ (d_i(x), p)$ and $(x,S_i(p)) ~ (s_i(x),p)$. We then take the (infinite) direct sum of all of these spaces and that’s the topological space.

It was clear as mud when I looked at this definition for the first time, but reading the explanation and doing some sketches I think I get the point: the elements of $\Delta$ are the real “volumes” that we are gluing together on the right points, lines, and faces with the above equivalence relation. What’s I still can’t really visualize is how this operation is a functor? Is it because it’s essentially just built from operations we know are functorial? Despite my inability to see this, the author goes on to explain that this functor has an adjoint in the singular set functor that turns topological spaces into simplicial sets.

The next section on products isn’t terribly remarkable except that the most naive way of doing products of simplicial sets happens to work beautifully.

The last thing I was going to cover today was simplicial objects, but it turns out those are just contravariant functors from $\Delta \to C$ where C is any category. So I’m guessing, without really being sure, that simplicial categories (functors from $\Delta \to Cat$) might somehow be relevant in defining higher-dimensional categories?

In any case, next time I’ll finish with this paper and go through the sections on Kan complexes, simplicial homotopy, and homotopy groups of Kan complexes