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Does anyone know if there is a “combinatorial” proof of the right properness of (the standard Quillen model structure on) simplicial sets (i.e. not invoking geometric realization), and if so where it can be found?
It suffices to prove that $\mathrm{Ex}^\infty$ is a fibrant replacement. That can be done purely combinatorially. There is also a direct proof of right properness. Both are found in this paper.
It’s worth pointing out that there is another proof that $\Ex^\infty$ is a fibrant replacement in Simplicial Sets From Categories by Latch, Thomason, Wilson.
Thanks! Why does it follow from $Ex^\infty$ being a fibrant replacement? The appendix of Moss’s paper uses a complicated argument that I don’t yet follow, but you seem to be saying it’s a formal consequence of some property of $Ex^\infty$.
Moss also says that “right-properness follows formally from the excellent properties of the $Ex^\infty$ functor” and cites Goerss-Jardine, but GJ was one of the first places I looked and I didn’t find any reference to right properness; where in it should I be looking?
Goerss and Jardine prove properness using geometric realisation as Corollary 8.6 in Chapter II, but Remark 8.7 explains that you can use $Ex^\infty$ instead. The point is that we have a fibrant replacement functor that preserves fibrations and pullbacks, so we inherit right properness from the category of fibrant objects.
There is also extensive material on properness in Cisinski’s Les préfaisceaux comme modèles des types d’homotopie. Proposition 2.1.5 (page 98 in the pdf linked to) establishes properness for the model structure on simplicial sets, by appeal ultimately to Théorème 1.5.4. There is certainly no appeal to geometric realisation to $\mathsf{Top}$ in this proof, but I’m not sure exactly what you would regard as a combinatorial proof, so am not sure if what Cisinski does is the kind of argument you’re looking for!
The $\mathsf{Ex}^{\infty}$ functor is incidentally also not used in Cisinski’s argument. It is used shortly afterwards (Théorème 2.1.42 is the culmination) to characterise the fibrations in the model structure of Proposition 2.1.5 as Kan fibrations, but not to establish properness itself.
Thanks! I have added this to the page model structure on simplicial sets.
I feel there is something slightly disingenuous in writing $Top$ for a unspecified convenient subcategory of topological spaces. (We need to apply some kind of k-ification just to get the right binary products!) But maybe it doesn’t really matter if you immediately apply the singular simplicial set functor to get back to $sSet$ (or, more relevantly, $Kan$).
That’s a fair point; I’ve added a comment.
The argument in the appendix to Moss’s paper seems pretty nifty. If I understand it correctly, it actually proves that the pullback of a right anodyne map along a left fibration is also right anodyne, which is something I’ve been looking for.
A few observations regarding Zhen Lin’s point. One way to use the actual category $\mathsf{Top}$, rather than compactly generated weak Hausdorff spaces or something like that, is, it seems to me, to use not the usual geometric realisation functor from $\mathsf{Set}^{\Delta^{op}}$ to $\mathsf{Top}$, but a different one constructed as follows.
1) Let $i$ be the functor from $\square_{\mathsf{diag}}$, the category of cubes with diagonals, to $\mathsf{Set}^{\Delta^{op}}$, the category of simplicial sets, which is uniquely determined by sending the 1-truncation of $\square_{\mathsf{diag}}$ to the usual simplicial interval. In the usual way, the functor $i$ gives rise to a functor $i_{!}$ from $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ to $\mathsf{Set}^{\Delta^{op}}$, and to a functor $i^{*}$ from $\mathsf{Set}^{\Delta^{op}}$ to $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ which is right adjoint to it.
Incidentally, if I am not mistaken, $i_{!}$ also admits a left adjoint, arising from a functor from $\Delta$ to $\square_{\mathsf{diag}}$ which sends $n$ to $I^{n}$, and where diagonals are used to define this functor on arrows. Indeed, this functor from $\Delta$ to $\square_{\mathsf{diag}}$ induces a functor from $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ to $\mathsf{Set}^{\Delta^{op}}$ which I believe is $i_{!}$, which in any case admits a right adjoint given by right Kan extension, which would be $i^{*}$, and a left adjoint given by left Kan extension, as claimed.
2) Let $\left| - \right|$ be the obvious geometric realisation functor from $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ to $\mathsf{Top}$, determined by sending the 1-truncation of $\square_{\mathsf{diag}}$ to the usual topological interval.
The geometric realisation functor from $\mathsf{Set}^{\Delta^{op}}$ to $\mathsf{Top}$ that I propose to use is given by $\left| - \right| \circ i^{*}$. We can make the following observations.
1) This functor clearly preserves products. Indeed, $i^{*}$ is a right adjoint, and $\left| - \right|$ does so by construction.
2) Let $j$ be the functor from the usual category $\square$ of cubes to $\square_{\mathsf{diag}}$ which is the identity on 1-truncations. This functor induces a functor $j^{*}$ from $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ to $\mathsf{Set}^{\square^{op}}$, which admits a left adjoint $j_{!}$ and a right adjoint $j_{*}$, given by left and right Kan extension respectively.
The functor $\left| - \right| \circ i^{*}$ factors through $j^{*}$, that it is to say, it is isomorphic to the functor $\left| - \right| \circ j^{*} \circ i^{*}$, where $\left| - \right|$ is now the geometric realisation functor from $\mathsf{Set}^{\square^{op}}$ to $\mathsf{Top}$ which sends the 1-truncation of $\square$ to the usual topological interval. But $j^{*} \circ i^{*}$ is isomorphic to the functor involved in Proposition 8.4.28 of Cisinski’s Les préfaisceaux comme modèles des types d’homotopie, and hence $j^{*} \circ i^{*}$ preserves and reflects weak equivalences.
It is obvious that $\left| - \right|$ from $\mathsf{Set}^{\square^{op}}$ to $\mathsf{Top}$ preserves and reflects weak equivalences. Thus we have that $\left| - \right| \circ i^{*}$ preserves and reflects weak equivalences.
As an alternative, I think that it is possible to show that $i^{*}$ preserves and reflects weak equivalences. This is discussed towards the end of this post.
3) By a similar argument to that of 2), $\left| - \right| \circ i^{*}$ preserves fibrations: we factor through $\mathsf{Set}^{\square^{op}}$ in the same way; use Cisinski’s work (the Quillen equivalence between the model structure on $\mathsf{Set}^{\square^{op}}$ and the model structure on $\mathsf{Set}^{\Delta^{op}}$) to see that $j^{*} \circ i^{*}$ preserves fibrations; and then use the obvious fact that $\left| - \right|$ preserves Kan fibrations of cubical sets.
Again, as an alternative, I think it possible to show that $i^{*}$ itself preserves fibrations. This is a consequence of the Quillen equivalence that I will discuss later in this post.
To be continued…
…Continuing from the previous post…
4) The functor $\left| - \right|$ preserves pullbacks. To see this, we use the fact that $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ is the classifying topos for strictly bipointed objects in a topos, as proved by Steve Awodey in these notes. Bas Spitters has also worked on this kind of thing. I believe this characterisation of $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ as a classifying topos is a folklore fact, that has been known for some time, but I do not know of a proof in the literature before Awodey’s work. We proceed as follows.
i) The functor $U \circ \left| - \right|$ from $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ to $\mathsf{Set}$, where $U$ is the forgetful functor from $\mathsf{Top}$ to $\mathsf{Set}$, is determined by the universal property of $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ as a classifying topos by the strictly bipointed object of $\mathsf{Top}$ given by the usual interval. Thus $U \circ \left| - \right|$ preserves all finite limits.
ii) It is now once again useful to factor $\left| - \right|$ through the model structure on $\mathsf{Set}^{\square^{op}}$. Indeed, we deduce from i) that $U \circ \left| - \right| \circ j^{*}$ preserves all finite limits, where $\left| - \right|$ is now the geometric realisation functor on $\mathsf{Set}^{\square^{op}}$.
iii) Suppose that we have an equaliser diagram in $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$. The leftmost map in this equaliser diagram, let us denote it by $e$, is a monomorphism, as with any equaliser diagram. Since $j^{*}$ is a right adjoint, it preserves monomorphisms (indeed it preserves the entire equaliser diagram). Thus $j^{*}(e)$ is a monomorphism.
iv) The model structure on $\mathsf{Set}^{\square^{op}}$ is cofibrantly generated, the generating cofibrations being boundary inclusions. Since $\left| - \right|$ preserves colimits, it sends a boundary inclusion to its corresponding topological boundary inclusion, that is to say to a monomorphism.
v) It follows from iii) and iv) that $\left| j^{*}(e) \right|$ is a monomorphism in $\mathsf{Top}$.
vi) It follows from i) and v) that $\left| - \right|$, where this is now the geometric realisation functor on $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$, preserves equalisers.
vi) Since $i^{*}$ is a right adjoint, it preserves equalisers.
vii) It follows from v) and vi) that $\left| - \right| \circ i^{*}$ preserves equalisers.
viii) It follows from 1) and vii) that $\left| - \right| \circ i^{*}$ preserves pullbacks.
We now have established all the properties required of $\left| - \right| \circ i^{*}$, as listed in the section ’Properness’ of model structure on simplicial sets, to be able to deduce properness of the model structure on $\mathsf{Set}^{\Delta^{op}}$.
Whilst I was thinking about the above, the following idea for a proof of a Quillen equivalence between a model structure on and the usual model structure on simplicial sets came to me. I thought I’d share it.
I’ll assume that there is a model structure on $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ with the following properties:
i) every monomorphism is a cofibration;
ii) the weak equivalences are exactly those which are weak equivalences in $\mathsf{Top}$ after geometric realisation;
That there is a model structure with monomorphisms (exactly) as the cofibrations follows immediately from Cisinski’s work, and has been observed in a slightly different way by Bas Spitters, in work that I think is not publically available yet. To establish ii) is harder if one uses Cisinski’s techniques to put the model structure on $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$, but I expect it to hold.
To be more precise, I claim that $i_{!}$ and its left adjoint, let me denote it $i^{?}$, define a Quillen equivalence between the standard model structure on $\mathsf{Set}^{\Delta^{op}}$ and this assumed model structure on $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$. Here is my suggestion for a proof.
To be continued…
…Continuing from the previous post…
1) The functor $i^{?} \circ i_{!}$ is naturally isomorphic to the identity functor, and hence the co-unit $\epsilon_{X} : i^{?} \circ i_{!}(X) \rightarrow X$ of the adjunction between $i^{?}$ and $i_{!}$ is a weak equivalence for any cubical set with diagonals $X$.
2) To show that the unit map $\eta_{X} : X \rightarrow i_{!} \circ i^{?}(X)$, for any simplicial set $X$, is a weak equivalence, we argue as follows.
i) By a standard trick, using 2-out-of-3, the fact that the co-unit is a weak equivalence from i), and one of the two triangles coming from the fact that we have an adjunction, we have that $i^{?}\left( \epsilon_{X}\right)$ is a weak equivalence of cubical sets with diagonals, and hence that $\left| i^{?}\left( \epsilon_{X} \right) \right|$ is a weak equivalence (indeed a homotopy equivalence) in $\mathsf{Top}$.
ii) It is straightforward to see from the definition of $i^{?}$ that the diagram indexed by $\Delta$ which defines $\left| i^{?}(X) \right|$, for any simplicial set $X$, is isomorphic to the diagram defining $\left| X \right|$. Indeed, a topological $n$-cube is isomorphic to the topological $n$-simplex living inside it to which the object $n$ of $\Delta$ is sent under $\left| - \right| \circ i^{?}$. Hence the colimits of the two diagrams are isomorphic, that is to say, $\left| i^{?}(X) \right|$ is isomorphic to $\left| X \right|$.
iii) It follows from ii) that $i^{?}$ preserves and reflects weak equivalences.
iv) We deduce from i) and iii) that $\left| \epsilon_{X} \right|$ is a weak equivalence for any simplicial set $X$, as required.
3) Exactly the same kind of argument as one we saw earlier demonstrates that $i^{?}$ preserves cofibrations. Indeed, the model structure on $\mathsf{Set}^{\Delta^{op}}$ is cofibrantly generated, the generating cofibrations being boundary inclusions. Since $i^{?}$ preserves colimits, it sends a boundary inclusion to a boundary inclusion, that is to say, to a monomorphism. It follows that $i^{?}$ preserves all monomorphisms, that is to say, all cofibrations.
4) It follows from 2) iii) and 3) that $i^{?}$ preserves trivial cofibrations.
5) By 3) and 4), we have a Quillen adjunction. In addition, it follows from 2) iii), 1), and the 2-out-of-3 property, that, for any simplicial set $X$ and any cubical set with diagonals $Y$, a morphism $X \rightarrow i^{*}(Y)$ is a weak equivalence of simplicial sets if and only if the morphism $i^{?}(X) \rightarrow Y$ corresponding to it under adjunction is a weak equivalence of cubical sets. Thus we have a Quillen equivalence, as we wished to show.
Once we have that $(i^{?}, i_{!})$ defines a Quillen equivalence, it is easy to deduce that $(i_{!}, i^{*})$ is also a Quillen equivalence, if we know that the cofibrations of the model structure on cubical sets are exactly monomorphism. We do this in the following way.
1) Since $i_{!}$ is a right adjoint, it preserves monomorphisms, that is to say, sends cofibrations to cofibrations.
2) Since $i^{?} \circ i_{!}$ is isomorphic to the identity functor, it follows from 2) iii) above that $i_{!}$ both preserves and reflects weak equivalences.
3) It follows from 1) and 2) that $i_{!}$ preserves trivial cofibrations. Thus we have a Quillen adjunction. Moreover, using the version of the homotopy category whose objects are both fibrant and cofibrant, the left derived functor of $i_{!}$ according to the Quillen adjunction between $i_{!}$ and $i^{*}$ is in fact the same as the right derived functor of $i_{!}$ according to the Quillen adjunction between $i^{?}$ and $i_{!}$, and we know that this right derived functor of $i_{!}$ defines an equivalence of homotopy categories. We conclude that $(i_{!}, i^{*})$ defines a Quillen equivalence.
It would be interesting to see how far these kind of ideas can be taken. They suggest, to me at least, that the model structure on $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ is a rather fundamental one.
The really crucial use of it that is made of $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ with regard to properness is the fact that it has a nice description as a classifying topos, which allows us to show that pullbacks are preserved by geometric realisation. This is false for ordinary cubical sets. I don’t know whether it is true for cubical sets with connections, but in any case the use of $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ is more conceptual. I would very much like a direct argument which shows that geometric realisation out of $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ preserves pullbacks, without using the characterisation as a classifying topos. Perhaps such an argument can be extracted from Awodey’s proof characterising $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ as a classifying topos, I have not tried this.
As a final remark, Cisinski’s methods, cofibrant generation, and so on, are all not valid constructively. It would be very nice to have a constructively valid theory of test categories.
Regarding Moss’s paper…I guess one person’s nifty is another person’s horrific! Clearly there is a great deal of cleverness in the paper, I’m not putting down the achievements of it at all. But these kind of intricate (and completely non-constructive, so we don’t really gain anything from that point of view by using them to avoid the theory of minimal fibrations with its reliance on the axiom of choice) combinatorial arguments are not something that I would wish to rely on in my own work, so I look at a paper like this as kind of provisional: there may be some really nice things in there, but I would have to find a more conceptual way to understand/establish them before I would ever use them in my own work!
I don’t follow your proof that geometric realisation preserves finite limits.
By definition, $\square_{\mathsf{diag}}$ is the free cartesian category on its $1$-truncation. The usual interval in $\mathsf{Top}$ defines a functor from the 1-truncation of $\square_{\mathsf{diag}}$ to $\mathsf{Top}$. This functor determines a product preserving functor from $\square_{\mathsf{diag}}$ to $\mathsf{Top}$ by means of the universal property of the former. By the universal property of $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ as a presheaf category, this functor from $\square_{\mathsf{diag}}$ to $\mathsf{Top}$ determines a colimit preserving functor from $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ to $\mathsf{Top}$, which is the geometric realisation functor I am working with. Since the functor from $\square_{\mathsf{diag}}$ to $\mathsf{Top}$ preserves products, and since the product functor on $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ preserves colimits in each variable (it is indeed the monoidal structure obtained by Day convolution from that on $\square_{\mathsf{diag}}$), we have that the geometric realisation functor preserves products, as claimed.
That the geometric realisation functor preserves products for these canonical reasons is precisely the reason I am using it!
Regarding your second point: you are correct that I need to show a little more, thank you very much for catching this. But in fact the argument that I gave does show that we have a subspace inclusion, if we start from the fact that $\left| - \right|$ takes a boundary inclusion to a subspace inclusion, rather than just observing that it is taken to a monomorphism, as I did originally. We then appeal to the fact that the colimits used to build up all monomorphisms from the generating cofibrations produce subspace inclusions, not just monomorphisms. If you would like a reference, it seems that you can follow the links in the proof of Lemma 1 here.
You are assuming that binary products in $Top$ preserve colimits in each variable. As far as I know, that is not true and is one of the reasons why we use a convenient category of topological spaces.
I haven’t been reading Richard’s posts, but I think Zhen is right that binary products in Top don’t preserve colimits in each variable. If they did, the smash product in $Top$ would be associative, but there is a counterexample to that in the introduction to May-Sigurdsson’s Parametrized homotopy theory.
I haven’t been following Richard’s posts either, but you guys (Zhen Lin and Mike) are sounding awfully modest here (-:.
For categories $E$ topological over $Set$, a functor like $X \times -: E \to E$ preserves colimits iff it has a right adjoint; meanwhile we know $Top$ is not cartesian closed and the issue of exponentiability in $Top$ has been analyzed in great detail.
True, there is an adjoint functor theorem.
As Todd mentioned, it is certainly not the case that the functor $X \times -$ from $\mathsf{Top}$ to itself preserves colimits for an arbitrary topological space $X$. As an example, in case anybody is wondering, the space $\mathbb{Q} \times \left( \mathbb{Q} / \sim \right)$ is not isomorphic to $(\mathbb{Q} \times \mathbb{Q}) / \approx$, where $\sim$ is the equivalence relation generated by identifying all integers with $0$, and $\approx$ is the equivalence relation identifying $(q_{0},q_{1})$ with $(q_{0}', q_{1}')$ if $q_{1} \sim q_{1}'$. Here $\mathbb{Q}$ is the set of rationals with the subspace topology coming from the standard topology on the reals. Of course, these quotients can expressed as colimits, for instance as coequalisers.
For the purposes of the question at hand, we are interested in whether the geometric realisation of a cubical set (or whatever) is exponentiable. Unfortunately, this is not the case either in general. An example is given by glueing infinitely many copies of the interval together at $0$. This is apparently sometimes referred to as a ’hedgehog space’, for example here. It is straightforward to see that this space is not locally compact. It is Hausdorff, though, so, by the characterisation of exponentiability in $\mathsf{Top}$ at exponential law for spaces, there is no hope for showing that it is exponentiable.
In some ways this example is rather frustrating, because:
1) It is straightforward, as far as I see, to prove that the geometric realisation of a cubical set (or whatever) is locally compact, and hence exponentiable, as long as it is ’locally finite’, in the sense that no single $n$-cube is glued to infinitely many other $m$-cubes. In fact, the ’hedgehog’ may be the only obstruction: it seems to me to be plausible that the geometric realisation of a cubical/simplicial set will be exponentiable if and only if it does not contain a hedgehog!
2) I’m not sure that these ’non-locally finite’ cubical (or whatever) sets are good for much from the point of view of homotopy theory.
I wonder if there is some nice way to exclude ’hedgehogs’? One possibility that occurs to me is to begin with presheaves of finite sets rather than presheaves of arbitrary sets, and then try to make use of ind-categories somehow. Though I’ve not thought about it in depth, it seems that it may be the case that directed colimits of locally compact topological spaces are preserved by products in $\mathsf{Top}$ in each variable.
Anyhow, you are completely correct, Zhen Lin, that I was relying on the fact that the product in $\mathsf{Top}$ with a geometric realisation of a cubical set preserves products. The reason for this oversight is no doubt that in all cases in practise that I have ever needed (eg, products with $n$-cubes when working with homotopies, products with horns, etc), I have a locally compact, and thus exponentiable, space on one side, so that colimits are indeed preserved on the other side. I have become so used to this that I forgot not to take it for granted here!
However, the essence of the argument that I gave is the fact that the realisation functor I defined preserves products of representables for conceptual reasons. Thus, if one works with $k$-spaces or whatever, rather than $\mathsf{Top}$, one has a completely canonical argument for showing that products of simplicial sets are taken to products under it, rather than having to give an explicit, and rather involved and messy, argument to show that products of representables are taken to products, as is the case with the usual realisation functor. And if one is working with simplicial homotopies, simplicial horns, or whatever, then one can use $\mathsf{Top}$ to carry out the same canonical argument.
So although I have to modify my initial suggestion that $\mathsf{Top}$ can be used (unless the ideas with ind-categories or whatever can be got to work!), I still think that we gain something. (Perhaps I should also say that most of the ideas in the arguments I gave are independent of the question of whether $\mathsf{Top}$ can be used or not, and might be of interest to somebody in themselves).
Incidentally, I should perhaps emphasise that I see no reason for the realisation functor I described to admit a right adjoint, and I am not proposing that it can replace the usual realisation functor for all purposes. But I think an idea with a lot of promise to it is to take the model structure on $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ as primary, and try to re-organise the story of the relationship between $\mathsf{Top}$ and $\mathsf{Set}^{\Delta^{op}}$ in such a way as to make use of $\mathsf{Set}^{\square_{\mathsf{diag}}^{op}}$ as an intermediary.
I think the usual realization is left exact for a very conceptual reason: it’s the inverse image part of the geometric morphism $Top \to Set^{\Delta^{op}}$ that classifies the topological interval $[0,1]$. It only gets a bit messy because $Top$ is not a topos, but that’s the essence of the point.
The parts of geometric realization that I had a hand in writing were written in order to try to bring out the conceptual point that Mike mentioned (that the topos $Set^{\Delta^{op}}$ classifies intervals). The idea is to see the left exactness $Set^{\Delta^{op}} \to Space$, where “$Space$” is some convenient category of topological spaces, by exploiting the left exactness of its underlying set functor $Set^{\Delta^{op}} \to Set$, which is the underlying conceptual part exploiting the fact that $Set$ is a topos.
Part of the left exactness of $Set^{\Delta^{op}} \to Space$ is not sensitive to whether we use a convenient category $Space$ or just $Top$. In particular, the arguments at geometric realization show that equalizers are preserved regardless of which is used. The only “messiness” there is to check that realization takes monomorphisms to (closed) subspace inclusions; since equalizers in $Top$ are equalizers in $Set$ equipped with the subspace topology, that’s all that is needed.
The “messiness” of proving that realization preserves products can be confined to a very small part: that in simplicial sets $Set^{\Delta^{op}}$, a product of representables $\Delta(-, [m]) \times \Delta(-, [n])$ is a quotient of a finite coproduct of representables. (If someone can improve on my proof of Lemma 2, please let me know!) Once you accept this, then you know that
${|\Delta(-, [m]) \times \Delta(-, [n])|}$ is compact,
${|\Delta(-, [m])|} \times {|\Delta(-, [n])|}$ is Hausdorff,
the canonical map ${|\Delta(-, [m]) \times \Delta(-, [n])|} \to {|\Delta(-, [m])|} \times {|\Delta(-, [n])|}$ is a bijection at the underlying set level (that’s the conceptual part),
and so ${|\Delta(-, [m]) \times \Delta(-, [n])|} \to {|\Delta(-, [m])|} \times {|\Delta(-, [n])|}$, being a continuous bijection from a compact space to a Hausdorff space, is a homeomorphism.
Then the remainder of the proof of product-preservation proceeds by the usual yoga of coends and Day convolution and whatnot – that part is completely conceptual, and is where the cartesian closedness of $Space$ really enters.
Regarding #22: I certainly agree that this is a conceptual reason for geometric realisation out of simplicial sets into a topos to preserve products. But $\mathsf{Top}$ is not a topos, and I wouldn’t agree at all that pretending that it is a topos is a conceptual reason for why the geometric realisation out of simplicial sets into it preserves products.
I agree it holds for conceptual reasons that composing the geometric realisation functor with the forgetful functor to sets gives a functor which preserves finite limits. But we still have to go from this back to $\mathsf{Top}$, and it is this part that I do not regard as conceptual.
I do think that the approach that Todd describes in #23 is about as optimal as possible in this regard. Surely there is a simpler and more conceptual proof of Lemma 2, which is intuitively obvious, though? We could observe that:
i) any levelwise finite $n$-truncated simplicial set is a finite colimit of representables (one representable for every $m$-simplex, for $0 \leq m \leq n$);
ii) the product of an $m$-truncated and an $n$-truncated simplicial set is $m+n$-truncated;
iii) a pair of representable simplicial sets are certainly levelwise finite and $n$-truncated.
Just now I can’t think of a simple proof of ii) that improves much on the proof of Lemma 2, though.
Here is a more conceptual argument. The finite co-completion of $\Delta$ has products. It is tautologically the case that any object of this category is a finite colimit of representables, thus in particular it is the case for a product of representables. Passing from the finite co-completion of $\Delta$ to its full co-completion, namely to $\mathsf{Set}^{\Delta^{op}}$, preserves finite colimits and products.
I’m not sure if the use of the finite co-completion of $\Delta$ really allows this argument to qualify as simple, though! I think that the argument goes through if one works with presheaves of finite sets instead of the finite co-completion, and if so, maybe that argument gives the best balance between simplicity and conceptuality.
Anyhow, even with a conceptual proof of Lemma 2, I wouldn’t describe the overall proof that Todd describes as conceptual, because we rely on certain rather specific facts about topological spaces that we are not fruitfully going to be able to abstract to many interesting categories. In other words, I think it is about as conceptual as possible, but that the result that the product of the (ordinary) geometric realisations to $\mathsf{Top}$ of representable simplicial sets just does not admit an entirely conceptual proof.
Now, one might say: well, just use a topos instead of $\mathsf{Top}$, such as the topological topos of Johnstone. To this I would say: the proof of the pudding is in the eating! That is to say: maybe this can work, but there are things that need to be pursued. Firstly: does the topological topos admit a proper model structure that is Quillen equivalent to the one on $\mathsf{Set}^{\Delta^{op}}$?
I actually think that this may be fairly straightforward to show, if one takes the notion of weak equivalence of an arrow in the topological topos to be one which is a weak equivalence in $\mathsf{Set}^{\Delta^{op}}$ after applying the singular complex functor $S$. Because I think the unit of the adjunction $X \rightarrow S \left| X \right|$ is exactly the same as the usual one where we use $\mathsf{Top}$, and we know that this is a weak equivalence. It follows formally that the co-unit is also a weak equivalence. And because one seems to be able to work with CW-complexes and so on in the topological topos much as one does in $\mathsf{Top}$, I would not be at all surprised if one of the usual proofs of the existence of the Serre model structure went through. I would not be surprised either if the topological topos admits a Hurewicz-type model structure.
All this said, though, and even if the approach with the topological topos can work, there are still things that I prefer about the approach I suggested. These principally come down to simplicity: there is some non-trivial work involved in showing that simplicial sets are a classifying topos for intervals; the notion of a classifying topos itself is much more involved than having a free cartesian category; and it is nice to be able to know that geometric realisation into categories which are not topoi, such as $\mathsf{Cat}$ to give a simple example, preserve products for conceptual reasons, and in such a way that one has to check very little.
On a different note, I did in fact use the ’classifying topos’ approach to show that the realisation functor from cubical sets with diagonals to sets preserves equalisers. So I do not regard this particular step in my argument to be as conceptual as I would like. But, as I suggested, I strongly suspect that there is a direct proof that this realisation functor preserves equalisers/pullbacks. It may even be a consequence of the universal property of a free cartesian category: the nLab page free cartesian category suggests that this is at least the case when we have a discrete category.
As an aside, it is possible to show that the model structure on cubical sets with diagonals is Quillen equivalent to the one on unadorned cubical sets. Bas Spitters has a proof of this which I think is not available publically yet. I explained a proof to Bas over email a couple of weeks ago, using techniques which are similar to those I used above to show that the model structure on cubical sets with diagonals is Quillen equivalent to the one on simplicial sets.
If one combines the Quillen equivalence between simplicial sets and cubical sets with diagonals with one between the latter and unadorned cubical sets, we obtain a Quillen equivalence between simplicial sets and cubical sets. This is known by work of Cisinski, but the proof that I am suggesting is simpler (at least in my opinion!).
$Top$ is very close to being a quasitopos, and many quasitoposes embed in toposes (and one can even argue directly about geometric morphisms between quasitopoi). So it seems perfectly conceptual to me.
I think that what you are giving are reasons to believe that the result holds for $\mathsf{Top}$, or that suggest that the result holds for $\mathsf{Top}$. What I am speaking of is a conceptual proof that it holds for $\mathsf{Top}$ itself, that it is to say a conceptual explanation for why it actually is true for $\mathsf{Top}$.
Todd, regarding your Lemma 2. If all that you need to conclude is that $|\Delta[m] \times \Delta[n]|$ is compact, then it suffices to observe that $\Delta[m] \times \Delta[n]$ has finitely many non-degenerate simplices. That’s clear since non-degenerate $k$-simplices in the nerve of a poset $P$ are exactly injective order preserving maps $[k] \to P$.
Karol: well, you’re right; thanks! And the same observation is already there in the proof, but you just condensed it for me. So maybe I’ll replace the proof with what you just told me.
Perhaps I was weighed down by the tradition of analyzing the simplicial structure of prisms, which seems to be part and parcel of just about every other proof I’ve seen that realization is left exact. Is prisms the word I want? I mean products of simplices.
Okay. I would tend to interprete the phrase “conceptual reason”, as used in your #24, to refer to “a reason to believe”.
(And, of course, it doesn’t actually hold for $Top$ itself; one has to modify it.)
@Todd - I’ve been asked twice for a proof recently that fat geometric realisation preserves pullbacks. Would your argument adapt to this case? I would guess that instead of $|\Delta[m] \times \Delta[n]|$ being compact, we would show that $||\Delta[m] \times \Delta[n]|| \to ||\Delta[0]||$ is proper, and so on.
David, I don’t know – I don’t have the requisite experience with fat realization. Maybe it’s time I sat down with it.
My guess is that it should be not too hard to show that the fibres of the map in #32 are compact, so we’d need to show that it was closed.
The only reference I’m aware of is a claim without proof in an unpublished preprint of Henriques and Gepner.
Regarding #31…true, but, as in #21:
1) changing $\mathsf{Top}$ to the category of $k$-spaces, or whatever, doesn’t change the point I’m making regarding having a conceptual proof;
2) it does hold most of the time for $\mathsf{Top}$ itself, in particular in most cases that occur in practise.
Regarding #32: I think that it is straightforward to see that it is the case for simplicial sets rather than simplicial topological spaces. Indeed, consider the functor $\Delta_{\mathsf{semi}} \rightarrow \Delta$. This functor gives rise to a functor $i^{*} : \mathsf{Set}^{\Delta^{op}} \rightarrow \mathsf{Set}^{\Delta_{\mathsf{semi}}^{op}}$, a functor $i_{*} : \mathsf{Set}^{\Delta_{\mathsf{semi}}^{op}} \rightarrow \mathsf{Set}^{\Delta^{op}}$ which is right adjoint to it, and a functor $i_{!} : \mathsf{Set}^{\Delta_{\mathsf{semi}}^{op}} \rightarrow \mathsf{Set}^{\Delta^{op}}$ which is left adjoint to $i^{*}$. It seems to me that the geometric realisation functor you refer to (I can’t stand the name of it!) is $\left| - \right| \circ i_{*} \circ i^{*}$, where $\left| - \right|$ is the usual geometric realisation functor from $\mathsf{Set}^{\Delta^{op}}$ to $\mathsf{Top}$ (or whatever variant of this category you choose). Since both $i^{*}$ and $i_{*}$ are right adjoints, they both preserve pullbacks, so the question reduces to showing that the usual geometric realisation functor $\left| - \right|$ preserves pullbacks.
I have not thought about the case of simplicial topological spaces.
Thanks, Richard. Yes, it’s the case of simplicial spaces that most of interest. But your argument for simplicial sets is very nice!
Without having checked anything or thought about it in any detail, I would guess that, using enriched category theory, one could give essentially the same argument in the case of simplicial topological spaces. In other words, I would guess that one can reduce in the same way to showing that the ordinary geometric realisation functor from simplicial topological spaces to $\mathsf{Top}$ preserves pullbacks. It is particularly this that I have not thought about.
Looking at the proof that products are preserved, if we assume we are in a category of spaces where colimits and pullbacks commute (i.e. coequalisers and pullbacks commute), then I think once we know that the map $||\Delta[m] \times \Delta[n]|| \to ||\Delta[m]|| \times_{||\Delta[0]||}||\Delta[n]||$ is an isomorphism we can just proceed as written, mutatis mutandis.
Yes, I guess that, viewing $\Delta$ as a discrete $\mathsf{Top}$-enriched category, the category of simplicial topological spaces is the free $\mathsf{Top}$-enriched co-completion of $\Delta$, as long as we have a cartesian closed version of $\mathsf{Top}$. And, if so, then I think that indeed the same proof as for simplicial sets goes through for products.
For equalisers, if one can reduce to representables (as apparently is possible for compactly generated weak Hausdorff spaces), then the same argument as for simplicial sets seems to go through.
In short, I think I agree! And I think that, using $\mathsf{Top}$-enriched left and right Kan extensions, the argument that I gave to show that the other geometric realisation functor preserves pullbacks will go through, as I suggested in #38.
A point perhaps worth mentioning is that, in a situation where one can reduce the case of equalisers to representables, one can use the same kind of argument as in #23, which I think is easier than the one showing that we have a subspace inclusion by appeal to (part of) cofibration generation. In this case, the fact that we have a finite colimit of representables is more or less immediate.
Richard, if you see a way clear to simplifying the equalizer part of the argument here, I cordially invite you to do so. I’m very much interested in optimizing the argument, as so many texts seem to make the overall argument more difficult than I think it really is.
The argument I had in mind is as follows. Suppose that we have a pair of maps $f, g : X \rightarrow Y$ in $\mathsf{Set}^{\Delta^{op}}$, where $X$ is representable. Let $E$ be an equaliser of this pair of maps, and let $e : E \rightarrow X$ be the canonical map. Since $X$ is $n$-truncated and levelwise finite, so is $E$. Thus $E$ is clearly a finite colimit of representables. Hence $\left| E \right|$ is compact. Let $E'$ be the equaliser of $\left| f \right|$ and $\left| g \right|$ in (an appropriate variant of) $\mathsf{Top}$. Let $\mathsf{can} : \left| E \right| \rightarrow E'$ be the canonical map. We know, because $U \circ \left| - \right|$ preserves finite limits by the classifying topos argument, that $U(\mathsf{can})$ is an isomorphism of sets (where $U$ is the forgetful functor from $\mathsf{Top}$ to $\mathsf{Set}$). Moreover, $E'$ is Hausdorff, because it is a subspace of $\left| X \right|$, which is certainly Hausdorff. In summary, we have that $\mathsf{can}$ is a continuous bijection from a compact space to a Hausdorff space. Thus it is a homeomorphism, as required.
If equalisers preserve colimits in our variant of $\mathsf{Top}$, then we can reduce preservation of arbitrary equalisers $f, g : X \rightarrow Y$ to preservation of equalisers where $X$ is representable, to which the argument I have just given applies.
I would say that this argument is complementary to the one currently on the nLab: it is, I would say, a little simpler, but relies on stronger assumptions about $\mathsf{Top}$.
I agree with your concluding remarks and I think that the overall approach you have come up with is nice, and as about as optimal as it can be, even if some particular bits can be carried out in different ways. I guess it is correct to attribute the approach to you? The use of the characterisation of $\mathsf{Set}^{\Delta^{op}}$ is of course an old idea, due to Joyal I believe, but I mean the way to get from this to a result in (a variant of) $\mathsf{Top}$.
For David and anybody else interested: I realised that my argument is #36 is not quite correct. The thickened (I will use this in preference to ’fat’) geometric realisation is given by $\left| - \right| \circ i_{!} \circ i^{*}$, not $\left| - \right| \circ i_{*} \circ i^{*}$ as I wrote (too quickly!).
As penance, I will explain how to give a proof of (part of) Proposition 9 on the nLab page, namely that the thickened geometric realisation preserves products if we map to $\mathsf{Top} / \left|| \ast \right||$.
What I will actually do is show that if we view $i_{!} \circ i^{*}$ as a functor $\mathsf{Set}^{\Delta^{op}} \rightarrow \mathsf{Set}^{\Delta^{op}} / i_{!}i^{*}( \Delta^{0})$, then it preserves binary products. The part of Proposition 9 that concerns finite products is an immediate corollary. The argument I will give is purely abstract: it works for any adjunction for which $i^{*}$ is conservative, and this conservativity holds for any ’forgetful functor’ between presheaf categories of the kind we have here.
To be continued…
Continuing from the previous post…
Let $P$ be the product of $i_{!}i^{*}(X)$ and $i_{!}i^{*}(Y)$ in $\mathsf{Set}^{\Delta^{op}} / i_{!}i^{*}( \Delta^{0})$, that is to say, the pullback of the morphisms $i_{!}i^{*}(e_{X}) : i_{!}i^{*}(X) \rightarrow i_{!}i^{*}(\Delta^{0})$ and $i_{!}i^{*}(e_{X}) : i_{!}i^{*}(X) \rightarrow i_{!}i^{*}(\Delta^{0})$, where $e_{X}$ and $e_{Y}$ are the canonical morphisms coming from the fact that $\Delta^{0}$ is a final object of $\mathsf{Set}^{\Delta^{op}}$. Let $q_{X} : P \rightarrow i_{!}i^{*}(X)$ and $q_{Y} : P \rightarrow i_{!}i^{*}(Y)$ be the structural morphisms exhibiting $P$ as a pullback.
There is a canonical morphism $p : i_{!}i^{*}(X \times Y) \rightarrow P$ coming from the morphisms $i_{!}i^{*}(p_{X}) : i_{!}i^{*}(X \times Y) \rightarrow i_{!}i^{*}(X)$ and $i_{!}i^{*}(p_{Y}) : i_{!}i^{*}(X \times Y) \rightarrow i_{!}i^{*}(Y)$ , where $p_{X}$ and $p_{Y}$ are the structural morphisms exhibiting $X \times Y$ as product.
To demonstrate that $p$ is an isomorphism, as we require, it suffices, because $i^{*}$ is conservative, to show that $i^{*}(p)$ is an isomorphism. Because $i^{*}$ preserves limits (it is a right adjoint) , we have that $i^{*}(p_{X})$ and $i^{*}(p_{Y})$ exhibit $i^{*}(X \times Y)$ as the product of $i^{*}(X)$ and $i^{*}(Y)$. The morphisms $i^{*}(\epsilon_{X}) \circ i^{*}(q_{X})$ and $i^{*}(\epsilon_{Y}) \circ i^{*}(q_{Y})$ give rise to a canonical morphism $q : i^{*}(P) \rightarrow i^{*}(X \times Y)$, where $\epsilon : i_{!} i^{*} \rightarrow id$ is the co-unit for the adjunction between $i_{!}$ and $i^{*}$.
We now verify that $i^{*}(p)$ and $i^{*}i_{!}(q) \circ \eta_{i^{*}(P)}$ are inverse to one another, where $\eta : id \rightarrow i^{*}i_{!}$ is the unit of the adjunction between $i^{!}$ and $i^{*}$. To show that $i^{*}(p) \circ i^{*}i_{!}(q) \circ \eta_{i^{*}(P)}$ is the identity, we use the universal property of $i^{*}(P)$ as a pullback (appealing to the fact that $i^{*}$ preserves pullbacks, being a right adjoint). I’m sure that anybody reading could complete this part of the argument without my giving the details, but I’ll do so in case it saves anybody some time.
1) We have that:
$i^{*}(q_{X}) \circ i^{*}(p) \circ i^{*}i_{!}(q) \circ \eta_{i^{*}(P)}$
= $i^{*}\big( q_{X} \circ p \circ i_{!}(q) \big) \circ \eta_{i^{*}(P)}$
= $i^{*}\big(i_{!}i^{*}(p_{X}) \circ i_{!}(q) \big) \circ \eta_{i^{*}(P)}$
= $i^{*}\Big( i_{!}\big( i^{*}(p_{X}) \circ q \big) \Big) \circ \eta_{i^{*}(P)}$
= $i^{*}\Big( i_{!} \big( i^{*}(\epsilon_{X}) \circ i^{*}(q_{X}) \Big) \circ \eta_{i^{*}(P)}$
= $i^{*}i_{!}i^{*}(\epsilon_{X}) \circ i^{*}i_{!}i^{*}(q_{X}) \circ \eta_{i^{*}(P)}$
= $i^{*}i_{!}i^{*}(\epsilon_{X}) \circ \eta_{i^{*}i_{!}i^{*}(P)} \circ i^{*}(q_{X})$
= $id \circ i^{*}(q_{X})$
= $i^{*}(q_{X}).$
The penultimate equality comes from one of the triangles which commute as a consequence of the fact that $i_{!}$ and $i^{*}$ are adjoint.
2) By an entirely analogous argument, we have that $i^{*}(q_{Y}) \circ i^{*}(p) \circ i^{*}i_{!}(q) \circ \eta_{i^{*}(P)} = i^{*}(q_{Y})$.
3) We have that $i^{*}(q_{X}) \circ id = i^{*}(q_{X})$ and that $i^{*}(q_{Y}) \circ id = i^{*}(q_{Y})$.
4) We conclude from 1), 2), 3), and the universal property of $i^{*}(P)$ that $i^{*}(p) \circ i^{*}i_{!}(q) \circ \eta_{i^{*}(P)} = id$, as required.
To show that $i^{*}i_{!}(q) \circ \eta_{i^{*}(P)} \circ i^{*}(p)$ is the identity, we use the universal property of $i^{*}(X \times Y)$ as a product. The details are as follows.
1) We have that:
$i^{*}(p_{X}) \circ q \circ i^{*}(p)$
= $i^{*}(\epsilon_{X}) \circ i^{*}(q_{X}) \circ i^{*}(p)$
= $i^{*}(\epsilon_{X}) \circ i^{*}(q_{X} \circ p)$
= $i^{*}(\epsilon_{X}) \circ i^{*}i_{!}i^{*}(p_{X})$
= $i^{*}\big(\epsilon_{X} \circ i_{!}i^{*}(p_{X}) \big)$
= $i^{*}(p_{X} \circ \epsilon_{X \times Y})$
= $i^{*}(p_{X}) \circ i^{*}(\epsilon_{X \times Y}).$
2) By an entirely analogous argument, we have that $i^{*}(p_{Y}) \circ q \circ i^{*}(p) = i^{*}(p_{Y}) \circ i^{*}(\epsilon_{X \times Y})$.
3) We deduce from 1), 2), and the universal property of $i^{*}(X \times Y)$ as a product, that $q \circ i^{*}(p) = i^{*}(\epsilon_{X \times Y})$.
4) We conclude that
$i^{*}i_{!}(q) \circ \eta_{i^{*}(P)} \circ i^{*}(p)$
= $\eta_{i^{*}(X \times Y)} \circ q \circ i^{*}(p)$
= $\eta_{i^{*}(X \times Y)} \circ i^{*}(\epsilon_{X \times Y})$
= $id$,
as required. The second equality is a consequence of 3), and the final equality again comes from one of the triangles which commute as a consequence of the fact that $i_{!}$ and $i^{*}$ are adjoint.
We are done! We have demonstrated that $i^{*}(p)$ is an isomorphism, from which we conclude that $p$ is an isomorphism.
I would think that exactly the same kind of argument will work to show that $i_{!}i^{*}$ moreover preserves pullbacks, when viewed as a functor into $\mathsf{Set}^{\Delta^{op}} / i_{!}i^{*}(\Delta^{0})$, but have not gone through this.
Richard, thank you! I’m not yet clear on “if equalizers preserve colimits”, though (i.e., how to turn that from hypothesis to fact). Perhaps I need to wake up more.
But if that works out easily, I think what I’d like to do is use that proof you gave in the beginning of #43, but at least archive the proof that’s there now and the other proof that was recently supplanted based on a similar suggestion of Karol.
(Perhaps the overall approach is original with me – I can affirm that I’ve not seen it elsewhere. But it would be odd if no one came up with it before, since it’s a simple idea piggybacking on a beautiful observation of Joyal.)
When I wake up more, I’ll be interested to read the rest of what you wrote.
I think that Proposition 2.36, together with Proposition 2.30 and the fact that coproducts are preserved, in these notes of Strickland establishes that pullbacks (and thus in particular equalisers) preserve colimits in the category of compactly generated weak Hausdorff spaces. I have not worked through the proof myself, though.
I too have not seen your approach taken elsewhere. Like many things, it is simple once one has seen it, but there are several little nuggets of creativity there that I feel one might easily overlook with regard to coming up with it in the first place, so I can believe that no-one has come up with it before!
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