# A canonical and categorical construction for geometric realization

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.

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It would be very surprising to me if a nice categorical construction could start off with finite linear orders, increasing functions, and somehow spit out the topology of the real numbers. – Steven Gubkin Jun 16 '10 at 15:01
I wouldn't be surprised, but, perhaps I have too much faith in category theory. As a side note, really you want the topology of the compact unit interval to be spat out, not the reals. – David Carchedi Jun 17 '10 at 9:45
As a cute "observation", we can use the topology on the unit interval as a "seed" to produce all the standard n-simplices. The topologically-enriched free commutative monoid on the unit interval is the disjoint union all the standard n-simplices. If this somehow fit in, it would be nice. – David Carchedi Jun 17 '10 at 9:48
@Harry, the singular complex functor USES the definition of the standard n-simplices. But, having the definition of each standard n-simplex (and the maps between them) gives us the functor $\Delta \to Top$ which if we left-Kan extend we get geometric realization. So, this doesn't gain us anything. – David Carchedi Jun 18 '10 at 1:25
I'm getting a sense, especially from remarks about recovering the unit interval from nonsense, that an unasked question is "why work in $Top$?" --- btw, the unit interval seems to be a terminal coalgebra for a reasonable functor, as observed by Peter Freyd. – some guy on the street Aug 25 '10 at 21:51

## 6 Answers

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.)

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Yes, I mentioned that... (dang! comments have no anchors!) – some guy on the street Sep 2 '10 at 22:47
Yes, I see now that you refer to this result and give a link. Why didn't you write it as an answer? IMO it comes close to answering David's plea for a canonical categorical justification. For example, I mentioned to David in email that the terminal coalgebra for this particular endofunctor could be seen as a universal solution to the problem of constructing an interval which is invariant with respect to <i>subdivision</i> (which may help justify why this particular endofunctor). – Todd Trimble Sep 3 '10 at 2:11
oh, why not is probably that I'm just so thoroughly used to homotopy being about the interval; so that when asked "whence this geometric realization functor" what occurs to me isn't "because the interval is terminal in ...", but rather that the cone monad --- using the interval --- gives a natural topological simplex category. And that's the answer I did give. – some guy on the street Sep 6 '10 at 23:43
Well, your first comment on my answer gave me the impression that you were annoyed that I hadn't acknowledged your earlier mention (and if you were, then please accept my apologies). I think the point however is that any interval (toset with distinct top and bottom) induces a left exact geometric realization, so the question still remains: why choose this one? Is there some sort of abstract conceptual reason? The same question applies to the cone monad: for any interval there is an associated cone monad, so what's the reason for choosing [0, 1] as the interval? – Todd Trimble Sep 7 '10 at 2:28
This is really pretty awesome. – Steven Gubkin Nov 5 '10 at 22:24

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.

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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.

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I suspect the finite space you want to use as a replacement for $\Delta^n$ is the one with elements the simplices of $\Delta^n$, and topology given by the preorder of face inclusions i.e. so the faces are the closed subsets. – Oscar Randal-Williams Jun 16 '10 at 18:17
Yes, that is the first idea I had, and what I am playing around with. The problem that I am having is figuring out how to glue these guys appropriately: How can I recover (a finite substitute for) a circle, or a torus? – Steven Gubkin Jun 16 '10 at 18:24
Since there aren't any references for finite spaces on Wikipedia, I'll point out that Peter May's website contains some nice notes on the topic, and there are also recent papers by Minian and Barmak on the arXiv. The original source for the result (stated in the question) about weak homotopy equivalence is McCord's paper (Duke J., 1966). Richard Stong also wrote some early papers on the topic. – Dan Ramras Jun 17 '10 at 0:28
This does seem very interesting, but, I have some concerns. If we could definte a "natural" functor $\Phi$ from $\Delta$ to $FinTop$ (finite topological spaces), just because for each $[n]$ $\Phi([n])$ is weakly equivalent to $\Delta^{n}$, it doesn't seem obvious to me that its left Kan-extension (denote it also by $\Phi$) would have $\Phi(X)$ w.e. to $|X|$ for each simplicial set $X$. In a more fundamental way, I am against replacing $|X|$ by something which is homotopy equivalent and saying it's "good enough" for algebraic topology. (will continue in another comment). – David Carchedi Jun 17 '10 at 9:40
@David: Your concerns are valid; e.g. the standard simplicial circle (having only two nondegenerate simplices) has realization, using the finite spaces mentioned by Oscar Randall-Williams, as the Sierpinski two-point space, which is contractible. – Tyler Lawson Jun 17 '10 at 13:52

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".

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Have you looked at Peter Freyd's "Algebraic Real Analysis"? I have barely cracked it open, but it seems like if you are looking for a categorical analysis of the unit interval, that is the place to go. – Steven Gubkin Jun 19 '10 at 12:37
I've taken a brief look, but as of yet, have not penetrated too deeply. I'll let you know if I find anything of interest. Thanks for the reference! – David Carchedi Jun 20 '10 at 13:57
I don't think "I is an interval object in the topos $\mathcal E$" is equivalent to "Hom$(E,I)$ is an interval object in $Set$ for all $E\in\mathcal E$." Consider, for example, the case of $\mathcal E=Set$, $I=[0,1]$, and $E=2$. I don't see a linear ordering on Hom$(2,I)$ induced by the linear ordering on $I$. (This doesn't affect the rest of your answer.) Generally, this sort of "definition via Yoneda embedding" works fine for concepts defined by universal Horn sentences, but not for notions like "linear order" or "field". For these, one needs the internal logic of the topos. – Andreas Blass Aug 25 '10 at 20:24

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.

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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.

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The multiplication may as well be [[x, s], t] |-> [x, st], or even [[x, s], t] |-> [x, min(s, t)], and the unit x |-> [x, 1]. Under the min formulation, you can similarly form a monad C_I for any interval I, where C_I X is the pushout of the inclusion X -> X x I (at the endpoint "top" of I) along X -> 1. You can define a notion of I-contractibility as involving a retraction h: C_I X -> X of the unit (or even demand h to be an algebra structure). You can go on to define an I-analogue of the topological simplex category. So what is so canonical about the usual choice I = [0, 1]? – Todd Trimble Sep 7 '10 at 2:49
As it happens, I've thought about this kind of thing a fair amount. If you'd like to discuss this further, you can write me at the address topological.musings@gmail.com. Regards, Todd. – Todd Trimble Sep 7 '10 at 2:53
ahah! ... well spotted, sir. – some guy on the street Sep 7 '10 at 13:17