A canonical and categorical construction for geometric realization

Question

There is a very intimate connection between categories, simplicial sets, and topological spaces. On one hand, simplicial sets are the presheaf category on the category $\Delta$ and $\Delta$ is a canonically defined "invariant" of the theory of categories. (e.g. the machinery of Mark Weber "spits out" $\Delta$ when you "plug in" the free category monad: http://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html)

However, $\Delta$ is also linked with topological spaces. The key to this link is the functor $\Delta \to Top$ which assigns the category $[n]$ the standard n-simplex $\Delta^n$. It is this functor which produces the adjunction between the geometric realization functor and the singular nerve functor which allow you to transfer the model structure on $Top$ to $Set^{\Delta^{op}}$ so that this adjunction becomes a Quillen equivalence.

My question is the following:

Is there a deep categorical justification for the functor $\Delta \to Top$ being defined exactly how it is? If we didn't know about the standard n-simplices, how could we "cook up" such a functor? I would like a construction of this functor which is truly canonical.

The closest to an answer I've found is Drinfeld's paper http://arxiv.org/abs/math/0304064. However, this doesn't quite "nail it home" to me. First of all, the definition is just made, but not motivated. The definition shouldn't be a "guess that works", but something canonical. Moreover, if you unwind it enough, it is secretly using the fact that finite subsets of the interval with cardinality $n$ correspond to points in (the interior of) the $(n+1)$-simplex. Plus, there's some funny business going on for geometric realization of non-finite simplicial sets. (Don't get me wrong- I think it's a great paper. It just doesn't totally answer my question).

EDIT: A possible lead:

$Set^{\Delta^{op}}$ is the classifying topos for interval objects and the standard geometric realization functor $Set^{\Delta^{op}} \to Top$ is uniquely determined by its sending the generic interval to $[0,1]$. This reduces the question to "why is [0.1] the canonical interval?". Is it perhaps the unique interval object whose induced functor $Set^{\Delta^{op}} \to Top$ is both left-exact and conservative?

EDIT: I've proposed a partial answer to this below, along the lines of the above lead. I would love any feedback that anyone has on this.

Answer 1

As to "why is the unit interval the canonical interval?", there is an interesting universal property of the unit interval given in some observations of Freyd posted at the categories list, characterizing $[0, 1]$ as a terminal coalgebra of a suitable endofunctor on the category of posets with distinct top and bottom elements.

There are various ways of putting it, but for the purposes of this thread, I'll put it this way. Recall that the category of simplicial sets is the classifying topos for the (geometric) theory of intervals, where an interval is a totally ordered set (toset) with distinct top and bottom. (This really comes down to the observation that any interval in this sense is a filtered colimit of finite intervals -- the finitely presentable intervals -- which make up the category $\Delta^{op}$.) Now there is a join $X \vee Y$ on intervals $X$, $Y$ which identifies the top of $X$ with the bottom of $Y$, where the bottom of $X \vee Y$ is identified with the bottom of $X$ and the top of $X \vee Y$ with the top of $Y$. This gives a monoidal product $\vee$ on the category of intervals, hence we have an endofunctor $F(X) = X \vee X$. A coalgebra for the endofunctor $F$ is, by definition, an interval $X$ equipped with an interval map $X \to F(X)$. There is an evident category of coalgebras.

In particular, the unit interval $[0, 1]$ becomes a coalgebra if we identify $[0, 1] \vee [0, 1]$ with $[0, 2]$ and consider the multiplication-by-2 map $[0, 1] \to [0, 2]$ as giving the coalgebra structure.

Theorem: The interval $[0, 1]$ is terminal in the category of coalgebras.

Let's think about this. Given any coalgebra structure $f: X \to X \vee X$, any value $f(x)$ lands either in the "lower" half (the first $X$ in $X \vee X$), the "upper" half (the second $X$ in $X \vee X$), or at the precise spot between them. Thus, you could think of a coalgebra as an automaton where on input $x_0$ there is output of the form $(x_1, h_1)$, where $h_1$ is either upper or lower or between. By iteration, this generates a behavior stream $(x_n, h_n)$. Interpreting upper as 1 and lower as 0, the $h_n$ form a binary expansion to give a number between 0 and 1, and therefore we have an interval map $X \to [0, 1]$ which sends $x_0$ to that number. Of course, should we ever hit $(x_n, between)$, we have a choice to resolve it as either $(bottom_X, upper)$ or $(top_X, lower)$ and continue the stream, but these streams are identified, and this corresponds to the identification of binary expansions

$$.h_1... h_{n-1} 100000... = .h_1... h_{n-1}011111...$$

as real numbers. In this way, we get a unique well-defined interval map $X \to [0, 1]$, so that $[0, 1]$ is the terminal coalgebra.

(Side remark that the coalgebra structure is an isomorphism, as always with terminal coalgebras, and the isomorphism $[0, 1] \vee [0, 1] \to [0, 1]$ is connected with the interpretation of the Thompson group as a group of PL automorphisms $\phi$ of $[0, 1]$ that are monotonic increasing and with discontinuities at dyadic rationals.)

Answer 2

Too long for a comment, too trivial for an answer:

It seems that if $K$ is a finite simplicial complex and $K'$ is the set of simplices of $K$ topologized as a quotient space of $K$ then the quotient map is a weak equivalence. Proof: $K$ is the union of contractible open sets, the open stars of its vertices, with the intersection of any two or more of these being contractible or empty. $K'$ is the union of corresponding contractible open sets, with intersections again being contractible or empty according to the same rule. Now repeatedly use the fact (consequence of Van Kampen and (for singular homology with local coefficients) excision): A continuous map from $X=U_1\cup U_2$ to $Y=V_1\cup V_2$ must be a weak equivalence if it gives weak equivalences $U_1\to V_1$, $U_2\to V_2$, and $U_1\cap U_2\to V_1\cap V_2$.

Replace simplicial complex by simplicial set and you get into a little trouble -- some subdivision is needed.

Answer 3

Here are some crazy ideas. I am posting now just to get the ball rolling, and if anyone (myself included) comes up with anything of substance in this direction you should just edit this answer. I am making it community wiki for ease of editing, and because the ideas are currently just kind of wonky.

I think recovering the standard geometric realization functor from pure abstract nonsense might be hard, just because there would have to be an implicit abstract nonsense construction of the closed interval as a topological space out of just finite linear orders, and I feel I would have seen that somewhere before.

On the other hand, I recently learned that every finite CW complex is weakly equivalent to a finite topological space. For example the circle is weakly equivalent to a topological space with 4 points. In fact "for any finite abstract simplicial complex K, there is a finite topological space $X_K$ and a weak homotopy equivalence $f : |K| \to X_K$ where $|K|$ is the geometric realization of $K$." (according to wikipedia). So maybe we could make this construction functorial from finite simplicial complexes to finite topological spaces. Maybe this functor (which factors through geometric realization, and is "just as good" as far as algebraic topology is concerned) could then be extended to simplicial sets. Since the construction should be more combinatorial, and not involve the reals in any way, I feel like this new functor (if it exists) might be more amenable to an "abstract nonsense" description. As far as algebraic topology is concerned, this new functor might be "just as good" as geometric realization.

I have some ideas for what this functor might look like, but I am still playing around with small examples. Feel free to join in the madness if you like, and add you edits with your name attached.

Answer 4

Well, I've done some reading and (re)discovered an old paper of Peter Johstone which comes pretty close to answering this question using topos theory. It follows the idea I posted as a "possible lead" in my EDIT.

First some background information:

It was shown by Joyal that simplicial sets is the classifying topos for "interval objects" (this is explained for instance in "Sheaves in Geometry and Logic" by Mac Lane and Moerdijk, in which they refer to interval objects as linear orders). By an interval object in $Set$, one roughly means a linearly ordered set together with a top and bottom element. You can say $I$ in a topos $\mathcal{E}$ is an interval object if and only if $Hom(E,I)$ is an interval object in $Set$ for all $E \in \mathcal{E}$. Since $Set^{\Delta^{op}}$ is the classifying topos for interval objects, for any topos $\mathcal{E}$, there is an equivalence of categories between the category of geometic morphisms $Hom(\mathcal{E},Set^{\Delta^{op}})$ and the category of interval objects in $\mathcal{E}$, $Int(\mathcal{E})$. Notice that the inverse image functor $f^*:Set^{\Delta^{op}} \to \mathcal{E}$ of a geometric morphism $f:\mathcal{E} \to Set^{\Delta^{op}}$ is always left-exact, so, this can be thought of as a "geometric realization functor with values in $\mathcal{E}$".

Now, the classical geometric realization functor $Set^{\Delta^{op}} \to Top$ nearly fits in this framework- it is left-exact and is uniquely determined by the fact that the universal interval object of simplicial sets is mapped to the standard unit interval $[0,1]$. However, $Top$ is not a topos. This is where Peter Johstone's 1977 Paper "On a topological topos" comes in. In this paper he constructs a topos $\mathcal{T}$ which contains sequential topological spaces (and hence e.g. CW-complexes) as a reflective subcategory. (In case you are interested, this topos is the topos of sheaves with respect to the canonical topology on the fullsubcategory of $Top$ consisting of the one-point space and the one-point compactification of $\mathbb{N}$.) Moreover, the inclusion of sequential spaces into $\mathcal{T}$ preserves lots of colimits- e.g. all colimits you'd use to construct CW-complexes. Now, since the standard unit interval $[0,1]$ is an object of $\mathcal{T}$, it corresponds to a unique geometric morphism $r:\mathcal{T} \to Set^{\Delta^{op}}$. Johnstone then proves that if $X$ is a simplicial set, then $r^*(X)$ is exactly $|X|$ (as a sequential space considered as an object of $\mathcal{T}$) AND that if $T \in \mathcal{T}$ is a sequential space, then $r_*(T) \cong Sing(T)$.

This is somewhat satisfying. However, for it to truly be satisfying, we'd have to either make sense out of why $\mathcal{T}$ is a natural choice, or, show that any "suitable choice" of a topos would give the same answer. Moreover, although intuively somehow clear, I would like to make sense out of in what way the "standard unit interval" $[0,1]$ is really a "canonical interval object".

Answer 5

Geometric realization is just $\operatorname{hocolim}\limits_{\Delta^{op}}$ — or more precisely, the explicit construction for this homotopy colimit functor (lifting it from HoTop to Top). An n-simplex arise there as a cofibrant replacement for the map from n points to 1 point. And the convex hull of n points certainly seems to be the most natural contractible set containing n distinct points, although I can't think of a precise statement here.

Answer 6

I'd just like to point out that there is a monad on $Top$, (which in the homotopy category looks rather dull,) assigning to each space $X$ its cone $CX$, the mapping cylinder of $X\to * $. The unit map is the inclusion of $X \to CX$, and the composition $CCX\to CX$ may as well be the map $ [[x,s],t]\mapsto [x,s+t-ts].$ (This ugly formula is just a natural obfuscation of the heuristic description of $CX$ as the union of convex combinations of points $x\in X$ and the new point $*$.) Another way to think of it is that $CX$ is the underlying space of the free contraction of $X$.

The topological (realization of the) simplex category is just the orbit of a one-point space $\star$ under this monad, together with the maps derived from the monad and the maps $C\star\to \star$ and $\star\to *\to C\star$. I think this gets at what Grigory M means above by "most natural" contractible set on $n$ points. Somewhere in this nonsense I should say "Bar construction", but I can't remember precisely where.