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Atrium > Mathematics, Physics & Philosophy: Diffeological Spaces, Frolicher Spaces, and Universal Categories

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  1.  
    • CommentRowNumber1.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 15th 2010
    • (edited Dec 15th 2010)

    In a discussion in the comments to my answer to David Carchedi’s question on MO about diffeological spaces, Thomas Nikolaus asked:

    Does this mean, that Fröhlicher Spaces are the universal cocompletion of Man subject to the requirement that the inclusion preserves all colimits?

    I would like to prove this (albeit for Hausdorff Frölicher spaces), but my problem is that being a novice at category theory, I don’t really know what one would have to prove here. Thomas suggested:

    Let CoCat coCoCat_{co} be the (bi-)category of cocomplete categories together with cocontinous functors. There is the forgetful functor CoCat co→Cat coCoCat_{co} \to Cat_{co} where Cat coCat_{co} denotes the (bi-)category of categories with cocontinous functors. Then it would be left adjoint to this functor.

    This discussion is partly a place for Thomas and me to continue our discussion (as the comments to an MO answer are rapidly getting unusable for this), and partly for anyone else to easily follow along. In particular, if anyone can help me understand what would need to be proved here, that would be very useful.

    Incidentally, I would say “completion and cocompletion”.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeDec 15th 2010

    Without thinking about it too deeply, this would seem to be related to exploring the connection between ind−Manind-Man and the category of Frolicher spaces. Here I am thinking of ManMan as being a subcategory of the cat. of Frolicher spaces. I doubt that the linkage is very strong but it may be worth thinking about. Referring to ind-object, the colimits would need to be filtered ones, not completely general ones.

  3. @Tim: Note that ind−Manind - Man is the free completion under filtered colimits (and the free cocompletion under all colimits is Psh(Man)). But what we are looking for is a cocompletion that respects colimits that do already exist in ManMan…. @Andrew: Let me exand on my comment above. I think we should check the following thing: Given a cocomplete category D, we should check whether the postcomposition with the inclusion Man→FroMan \to Fro induces an equivalence

    [Fro,D] co→[Man,D] co [Fro, D]_{co} \to [Man,D]_{co}

    where [−,−] co[-,-]_{co} denotes the category of colimit preserving functors.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeDec 15th 2010

    @Thomas I realise that, and that is why I said things in the way I did. There is a fairly ’concrete’ construction that I heard of from Andrée Ehresmann (but is ’well known’) that gives the cocompletion yet preserves the existing colimits (up to iso of course), i.e. the free conservative cocompletion. She was looking at it in exactly the sort of situation you are considering. I cannot however find the construction :-(.

  5. And by FroFro I mean Hausdorff Fröhlicher Spaces, as Andrew pointed out before.

    • CommentRowNumber6.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 15th 2010

    Let me try to spell this out a little more concretely (as I hinted above, abstract nonsense makes my head spin!). Is the following what we want to prove?

    Suppose that F:Man→CF \colon Man \to C is a functor to a complete, co-complete category that preserves all limits and colimits. Then there is a functor F^:HFro→C\hat{F} \colon HFro \to C such that F=F^ιF = \hat{F} \iota, where ι:Man→HFro\iota \colon Man \to HFro is the “standard” embedding.

    By “preserves all (co)limits” I mean that whenever a (co)limit of a diagram exists in ManMan then FF of the (co)limit is the (co)limit of FF of the diagram (I should read the pages continuous functor and cocontinuous functor to check that that’s what those two terms mean …).

    • CommentRowNumber7.
    • CommentAuthorThomas Nikolaus
    • CommentTimeDec 15th 2010
    • (edited Dec 15th 2010)

    @Tim: Hm, I know of a construction that goes back to Kelly for a category CC. Therefore you have to take first the category C^:=[C op,Set] con\hat C:=[C^op,Set]_{con} of continous functors (i.e. limit preserving, note that limits in C opC^op are colimits in CC). Then you close the image of the Yoneda embedding C→C^C \to \hat C under colimits. And that should be the conservative cocompletion. What I have not mentioned is that there are many kinds of Size-Issues one has to deal with here (sometimes it might not be possible, i.e. the conservative cocompletion does not exist).

    @Andrew: Yes, something like this. But I would relax the equality F=F^ιF=\hat F \iota to a natural isomorphism and additonally ask the functor F^ \hat F to preserve all limits and colimit too. Btw: Is it true that each Fröhlicher space is a colimit (limit) of manifolds (within Fröhlicher spaces)?

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2010

    A quick remark on a somewhat different aspect, relating back to Dave’s original question:

    it seems that a crucial formal difference between FroehlicherFroehlicher and DiffeologicalDiffeological is that the latter is locally cartesian closed, the former not.

    If we start with the subcategory Manifolds↪Sh(CartSp)Manifolds \hookrightarrow Sh(CartSp) and then somehow close it off under first internal homs, second “local internal homs”, what would one get?

    • CommentRowNumber9.
    • CommentAuthorDavidCarchedi
    • CommentTimeDec 15th 2010
    @Thomas: Every DIFFEOLOGICAL space is a colimit of manifolds, and Frolicher spaces are a reflective subcategory. So, if $X$ is a Frolicher space, then considered as a diffeological space, $X$ is the colimit of a diagram $M_i$ of manifolds, and by applying the reflector, $L:DiffSP \to Frolicher$, $LX=X$ is the colimit of $LM_i=M_i$.
    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeDec 15th 2010

    What’s the exact definition of “Man” being used here? Are manifolds required to be Hausdorff and/or second countable? I presume they are all finite-dimensional. It makes a difference regarding whether Man is essentially small or not, which is in general a big deal when looking for (co)completions.

    • CommentRowNumber11.
    • CommentAuthorThomas Nikolaus
    • CommentTimeDec 15th 2010
    • (edited Dec 15th 2010)

    @David: Ah okay, I see. I wonder whether the same argument also applies to the category [C op,Set] con[C^op,Set]_{con} from my last post, because then the closure of the image would be the whole category. Maybe I remember the Kelly-construction wrong…

    @Mike: I would say hausdorff,so it should be essentially small, because we can embed each manifold into an R^n (is it not ess. small otherwise?)….

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeDec 15th 2010

    I think second countability is also important for essential smallness, since otherwise a discrete space with arbitrarily large cardinality would be a 0-dimensional manifold. I’m not sure whether Hausdorff is necessary for essential smallness, although together with second countability it is of course sufficient, by the embedding theorem.

    • CommentRowNumber13.
    • CommentAuthorDavidCarchedi
    • CommentTimeDec 15th 2010
    Nonetheless, Hausdorff is necessary since there are examples of colimits of non-Hausdorff manifolds, that exist, which are not preserved under the embedding into Frolicher spaces. Andrew provided me with an example before.
    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeDec 16th 2010

    Okay, so “Man” means Hausdorff and second countable. In particular, it’s essentially small, which means that there shouldn’t be any size issues in discussing (co)completions of it.

    For a small category C and some class Φ of colimit diagrams in C, the “free cocompletion of C preserving the colimits in Φ” can be constructed as the full subcategory of the presheaf category of C containing those presheaves which preserve the colimits in Φ (i.e. take them to limits in Set). Its universal property is then that cocontinuous functors out of it are equivalent to functors out of C which preserve the colimits in Φ. I don’t know the original source for this; it might be Kelly.

    • CommentRowNumber15.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 16th 2010

    Couple of responses to minor points:

    1. Every Frölicher space is both a limit and colimit of manifolds. In fact, one doesn’t need anything like the full category of manifolds. Let (X,C,F)(X,C,F) be a Frölicher space. Let 𝒞\mathcal{C} be the category with one object for each curve c∈Cc \in C and a morphism g:c→c′g \colon c \to c' is a smooth function g:ℝ→ℝg \colon \mathbb{R} \to \mathbb{R} such that c′g=cc' g = c. Define a functor 𝒞→Fro\mathcal{C} \to Fro by sending each object to ℝ\mathbb{R} and sending each morphism to itself. Then (X,C,F)(X,C,F) is the colimit of this functor. Limits are similar, except that we use FF and our morphisms go the other way.

      (I suspect that the proof of HFroHFro being the universal limit-colimit category would follow fairly easily from this, but I’d need to write down the details to be sure)

    2. This means that the debate about second countable is somewhat academic since all one really needs to consider is the monoid C ∞(ℝ,ℝ)C^\infty(\mathbb{R},\mathbb{R}) viewed as a one-object category. I suspect that that satisfies all the possible “smallness” conditions, though I’m no expert!

    Which leads me to the more interesting point; Urs wrote:

    it seems that a crucial formal difference between FroehlicherFroehlicher and DiffeologicalDiffeological is that the latter is locally cartesian closed, the former not.

    to which one should add that in order to become locally cartesian closed, DiffeologicalDiffeological has to mess up the colimits from ManifoldsManifolds. So we could think of a hierarchy:

    (ℝ,C ∞(ℝ,ℝ))→Man→HFro→Fro→Diff→Sh(Cart)→PSh(Cart) (\mathbb{R},C^\infty(\mathbb{R},\mathbb{R})) \to Man \to HFro \to Fro \to Diff \to Sh(Cart) \to PSh(Cart)

    Up to HFroHFro, there’s no loss of categorical information (as far as I know): each embeds in the next without destroying anything already there. Once we go beyond HFroHFro, we start destroying stuff. So for me, the real question is why I should bother going beyond HFroHFro? Why is the gain of locally cartesian enough to outweigh the loss of colimit-preserving? And if the gain of locally cartesian is big enough to justify this loss, why stop there? Since DiffDiff is only a quasi-topos and Sh(Cart)Sh(Cart) a topos, why not go the whole way to Sh(Cart)Sh(Cart)? Or even to Psh(Cart)Psh(Cart) since then I don’t have to bother with “sheaffffffifffffication”. Only, if a topos is the Ultimate Goal, what’s wrong with Sh(ℝ,C ∞)Sh(\mathbb{R},C^\infty)? Why do I need to throw in all higher dimensional spaces? So the above diagram just seems overly complicated and could be simplified to

    (ℝ,C ∞(ℝ,ℝ))→Man→HFro→Sh(ℝ,C ∞) (\mathbb{R},C^\infty(\mathbb{R},\mathbb{R})) \to Man \to HFro \to Sh(\mathbb{R},C^\infty)

    Chopping at the tree even further down, I’m still to be convinced that sheaves are the Right Thing in this context. They feel too topological to me. There should be another construction that can be done in similar generality that correctly captures the essence of smoothness and which would (finally) release Frölicher spaces from their sets.

    • CommentRowNumber16.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 16th 2010

    Incidentally, the example that David Carchedi refers to is on the page examples of Frölicher spaces.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeDec 16th 2010
    • (edited Dec 16th 2010)

    to which one should add that in order to become locally cartesian closed, DiffeologicalDiffeological has to mess up the colimits from ManifoldsManifolds .

    One should also ask: why would one want to keep colimits of manifolds? What is good about them?

    One way to understand the passage from manifolds to Lie groupoids, stacks on manifolds and further is as a way to correct the colimits . Here “correct” means “make them interact properly with taking cohomology”. From a modern perpectve we see that this is really a tautology: it turns out that all recipes for defining cohomology really construct hom-spaces in ∞\infty-toposes, and so “correcting colimits” of manifolds, of schemes, etc., just means to regard them as objects in these ∞\infty-toposes, so that then manifestly they interact correctly with taking cohomology, because this then is just taking the hom.

    There is a dual story about correcting limits by passing to the ∞\infty-topos over the “geometric envelope” (derived geometry) of the original site.

    Much if not all the renewed interest in diffeological spaces (after the topic had been rather dormant) came from thinking about smooth groupoids. If one goes this route all the way, one wants to speak of ∞\infty-groupoids modeled on the geometry of manifolds. It does not seem to me that there is a good way to do this except by passing to the ∞\infty-sheaf ∞\infty-topos over that category.

    Your question why not go even further to the presheaves has the following answer: we want the “smooth ∞\infty-groupoids” to have underlying bare ∞\infty-groupoids. There are two ways to do this: either forget the smooth structure, or form the geometric realization Π:Sh ∞(Manifolds)→∞Grpds\Pi : Sh_\infty(Manifolds) \to \infty Grpds. The latter operation we want to be such that it exhibits a smooth ∞\infty-groupoid as indeed a smooth refinement of the corresponding bare thing. For instance a manifold regarded as a 0-truncated smooth ∞\infty-groupoid should have a geometric realization that is equivalent to its underlying topological space. This is precisely the point that makes us localize the ∞\infty-presheaves at the Cech-covers.

    When one has this, one can do things like find smooth refinements of the standard objects of interest in homotopy theory. For instance one shows that for GG a Lie group, the one-object Lie groupoid *//G*//G maps under Π\Pi indeed to the classifying space BGB G, hence is indeed a smooth refinement of that classifying space. And so on: one finds that there are smooth refinements of the Whitehead towers of Lie groups in terms of higher smooth Lie groupoids, and so on.

    All these constructions revolve around the notion of cohomology. Making things interact well with passing to Lie group cohomology, Lie groupoid cohomology, etc. All this naturally lives in the ∞\infty-topos.

    So if one is interested in cohomological quesstion, the topos perspective is mandatory. It seems. But still, one will be interested in seeing if in special cases one happens to be working just in a small tame corner of the big wild ∞\infty-topos. So therefore it is important to have a good idea of how it is filtered by tame/wild degree into subcategories of tamer objects.

    Only, if a topos is the Ultimate Goal, what’s wrong with Sh(ℝ,C ∞)Sh(\mathbb{R},C^\infty)? Why do I need to throw in all higher dimensional spaces?

    If you want to describe 1-dimensional smooth geometry, then Sh(ℝ,C ∞)Sh(\mathbb{R}, C^\infty) is the right topos to go to. But certain higher dimensional objects do not exist here. Such as the object Ω 2(−)\Omega^2(-). So then also there is for instance only degree 1 Deligne cohomology in this topos, nothing higher dimensional.

    • CommentRowNumber18.
    • CommentAuthorThomas Nikolaus
    • CommentTimeDec 16th 2010
    • (edited Dec 16th 2010)

    Mike: Do there arise Problems if we let Φ\Phi be the class of alle colimits?

    Additionally the inclusion of CC into the conservative cocompletion will preserve limits. Thats a fact I have always wondered about. It is clear when you construct it using the Yoneda embedding but has anyone an abstract reason for this behaviour?

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeDec 16th 2010

    This means that the debate about second countable is somewhat academic since all one really needs to consider is the monoid C ∞(ℝ,ℝ)C^\infty(\mathbb{R},\mathbb{R}) viewed as a one-object category.

    Not exactly, I don’t think. Just knowing that A⊂B⊂C and every object of C is a colimit of objects in A (hence also in B) doesn’t imply that saying “C is a universal cocompletion of A” (in some sense) is equivalent to “C is a universal cocompletion of B” (in the same sense). For instance, *⊂FinSet⊂Set, and Set is the free cocompletion of *, but not the free cocompletion of FinSet.

    So the above diagram just seems overly complicated

    Well, “colimit-preserving” is not a binary property. A functor can preserve some colimits, or more or fewer colimits than another functor. Presumably the embedding of Man into Fro preserves some colimits, and probably more colimits than the embedding into Diffeological, which probably preserves more colimits than the embedding into Sh(Cart), which certainly preserves more colimits than the embedding into PSh(Cart). So it’s a tradeoff between how many colimits we want to preserve and how much extra good categorical structure we want.

    So maybe we should first ask “which colimits that exist in Man do we really care about preserving?” If the answer is “all of them”, then maybe we should stick with HFro. But if the colimits that get destroyed in Fro can be dismissed as “pathological,” then maybe it’s okay to go up to Fro and get whatever benefits it has. And so on. In each case, the cocomplete category we embed into has its own opinions about what the “correct” colimits should be. For instance, the perspective of Sh(Cart) is that the only colimits in Man that we really care about are the colimits of open covers, and that all others it’s better to “throw in freely.”

    Do there arise Problems if we let Φ be the class of all colimits?

    I don’t know; I’m only sure it works if Φ\Phi is a small set. But if C is a small category, then it seems unlikely that we would be interested in colimits of a size larger than C itself?

    • CommentRowNumber20.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 17th 2010
    • (edited Dec 17th 2010)

    Mike:

    For instance, *⊆FinSet⊆Set* \subseteq FinSet \subseteq Set, and SetSet is the free cocompletion of **, but not the free cocompletion of FinSetFinSet.

    But isn’t SetSet the universal colimit preserving cocompletion of FinSetFinSet?

    So it’s a tradeoff between how many colimits we want to preserve and how much extra good categorical structure we want.

    Agreed. But up to now, I’ve had the impression that no-one else has noticed that it is a trade and not simply an augmentation.

    For instance, the perspective of Sh(Cart)Sh(Cart) is that the only colimits in ManMan that we really care about are the colimits of open covers, and that all others it’s better to “throw in freely.”

    Which brings us nicely to the key issue (that I keep coming back to): smoothness is tighter than open covers. Open covers are about topology, but smootheology is finer than that. It sits someway between a pointwise property and a germwise property. Exactly where, I’m not sure. A simple example is to consider the concatenation of two paths. If we think about continuity, we merely need that they coincide at the end points. If we think diffeologically, then we need them to be stationary near the end point. But the smootheological viewpoint is that they need to coincide and all their derivatives match up - somewhere in the middle.

    And so to Urs:

    If you want to describe 1-dimensional smooth geometry, then Sh(ℝ,C ∞)Sh(\mathbb{R},C^\infty) is the right topos to go to. But certain higher dimensional objects do not exist here. Such as the object Ω 2(−)\Omega^2(-). So then also there is for instance only degree 1 Deligne cohomology in this topos, nothing higher dimensional.

    Which, frankly, nicely illustrates the paucity of the sheaf construction. Because Ω 2(−)\Omega^2(-) is obtainable from (ℝ,C ∞)(\mathbb{R},C^\infty) simply by iterating Ω 1(−)\Omega^1(-). Higher dimension is an illusion, it just means “do it more than once”. For example, if there’s something that is only detectable by ℝ 2\mathbb{R}^2 then we use exponentiation to detect it by using ℝ\mathbb{R} twice:

    Hom(ℝ 2,X)≅Hom(ℝ,Hom(ℝ,X)) Hom(\mathbb{R}^2, X) \cong \Hom(\mathbb{R}, Hom(\mathbb{R},X))

    Sheaves don’t detect this, that’s true. But then I’ve yet to be convinced that something that extends topology is the right way to extend smootheology.

    I need to digest the rest of your answer a little more; but even accepting the importance of topoi, that doesn’t mean that I have to accept sheaves as the only way to generate a topos.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeDec 17th 2010

    it just means “do it more than once”.

    Not sure what you mean concerning Ω 2(−)\Omega^2(-). What would you take as the replacement of the sheaf of 2-forms in $Sh((\mathbb{R}, C^\infty))S?

    The sheaf U↦Ω 2(ℝ×U)U \mapsto \Omega^2(\mathbb{R} \times U) is nontrivial on (ℝ,C ∞)(\mathbb{R}, C^\infty), but that’s not quite what plays the role of Ω 2(−)\Omega^2(-).

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeDec 17th 2010
    • (edited Dec 17th 2010)

    Which brings us nicely to the key issue (that I keep coming back to): smoothness is tighter than open covers.

    We might have to formalize the argument more to make progres..To me it seems that there is no contradiction or issue here: I think

    1. the site itself encodes the kind of geometry;

    2. the Grothendieck topology on it encodes what the underlying ordinary topology of these geometries is (that’s why it’s called Grothendieck topology ! )

    So to me it makes perfect sense to

    1. pick the category of smooth test spaces, such as to encode smooth geometry;

    2. equip it with the open-cover Grothendieck topology, such as to encode the interaction with underlying topology.

    I can formalize this to some extent: as I mentioned in my earlier message, the choice of the open cover coverage on smooth manifolds is precisely the choice that ensures that the functor Π:Sh ∞(CartSp)→∞Grpd≃Top\Pi : Sh_\infty(CartSp) \to \infty Grpd \simeq Top left adjoint to the “discrete smooth space” functor Disc:∞Grpd→Sh∞(CartSo)Disc : \infty Grpd \to Sh\infty(CartSo) sends a smooth manifold correctly to its underlying topological space.

    What is true, and which I imagine is the point you care about, is that in Sh ∞(CartSp)Sh_\infty(CartSp) are objects that do not look smooth. It seems to me that the discussion is much about whether one should just identify those objects that do “look smooth” in the full topos, or if we should entirely restrict attention to them.

    I mentioned two reasons not to entirely restrict attention to them:

    1. the colimits would then come out wrong for many purposes.

    2. the notion of smooth cohomology breaks down more or less.

    • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeDec 17th 2010

    But isn’t Set the universal colimit preserving cocompletion of FinSet?

    Well, it’s the universal finite-colimit-preserving cocompletion. That’s not surprising: if you first complete * by finite colimits to get FinSet, then complete that by small colimits, but preserving the finite ones, then the universal properties mean that you’d have to get the same thing as directly completing * by small colimits.

    I wouldn’t swear to whether or not the inclusion of FinSet into Set preserves all colimits, or even all small ones. (Yes, FinSet does have plenty of infinite colimits. For instance, if D is any category with a terminal object 1 and F:D→C any diagram, then colim(F) exists and is F(1), regardless of the sizes of C and D. Obviously this sort of colimit will be preserved by any functor, but who knows what other sort of “accidental” colimits might exist in FinSet?)

    So, if you could prove that Man were a free cocompletion of C ∞(R,R)C^\infty(R,R) under some class Φ 2\Phi_2 of colimits which preserves some class of colimits Φ 1\Phi_1, and then that Fro were the free cocompletion of Man under Φ 3\Phi_3-colimits which preserves Φ 2\Phi_2-colimits and Φ 1\Phi_1-colimits, then it would follow that Fro is the free cocompletion of C ∞(R,R)C^\infty(R,R) under Φ 3\Phi_3-colimits which preserves Φ 1\Phi_1-colimits. But I don’t think that one can claim that questions about Fro being a cocompletion of Man and of C ∞(R,R)C^\infty(R,R) are “trivially the same.”

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeDec 17th 2010

    A simple example is to consider the concatenation of two paths. If we think about continuity, we merely need that they coincide at the end points.

    I’m not sure I agree that the “topology” which open covers are about is the same as the “topology” in which we can concatenate paths which coincide at their endpoints. One reason is that we can play a similar game with Top instead of Man (modulo size issues). Top happens to already be cocomplete, but if we decide that only some of its colimits are “correct” then we can freely cocomplete it preserving those colimits. In particular, we might decide that only the open covers are the good colimits in Top, and freely throw in other sorts of colimits, getting a topos Sh(Top). In Sh(Top), we cannot concatenate two paths if they agree only at their endpoints; we need them to agree on some open neighborhoods of their endpoints.

    This does suggest, however, that if you believe that in “topology” one should be able to concatenate paths which agree at their endpoints, then Sh(Top) is not the right place to work—you should work in Top itself, or in some better-behaved cocompletion which preserves the pushout [0,2]=[0,1]⊔ 1[1,2][0,2] = [0,1] \sqcup_1 [1,2]… such as subsequential spaces or Johnstone’s topological topos. On the other hand, the latter is, of course, a topos, and so it can be defined using a different, perhaps odd-looking, notion of “open cover.”

    • CommentRowNumber25.
    • CommentAuthorMike Shulman
    • CommentTimeDec 17th 2010

    I do, however, agree that not every “geometric” or “topological” category is, or wants to be, a topos. In fact, one of the best categories of spaces, namely the category of topoi itself, is not a topos!

    • CommentRowNumber26.
    • CommentAuthorDavidCarchedi
    • CommentTimeDec 18th 2010
    @Mike: Just for fun, have you ever read Johnstone's "On a topological topos"? For some purposes, he fixes that $Top$ is not a topos, and in the process gets geometric realization of simplicial sets purely topos-theoretically.
    • CommentRowNumber27.
    • CommentAuthorMike Shulman
    • CommentTimeDec 18th 2010

    @David: yes; I linked to the nLab page about it up in #24. (-:

    • CommentRowNumber28.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 19th 2010

    @Andrew #20 and @Mike #24

    This raises an interesting question. In the 2-category of topological stacks, unless one is careful, the pushout which expresses the concatenation of [0,1][0,1] with itself to form [0,2][0,2] doesn’t actually work! (This is in Noohi’s paper’s on topological stacks somewhere) Namely, the pushout in spaces is not preserved by the inclusion functor Top→STopStacksTop \to STopStacks unless one restricts to a interesting sub-2-category where the source and target maps have good local lifting properties (this is expressed by saying for a map a→ba\to b there are covers u→au \to a, v→bv\to b and a Serre/Hurewicz fibration u→vu\to v making the obvious square commute). Since for Lie groupoids the source and target maps are generally taken to be submersions (at least, when we are talking about groupoids internal to the usual category of manifolds, not smooth spaces - not sure what the deal is there), which are locally of the form u×ℝ n→uu \times \mathbb{R}^n \to u, the topological problem is not there for Lie groupoids - but then there are smoothness problems to deal with…

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2010
    • (edited Dec 19th 2010)

    I was thinking that it might be helpful to work out the relation between locally ringed objects in a cohesive (∞,1)(\infty,1)-topos more, since one way to look at Fröhlicher spaces is that they are concrete objects that are equipped with the structure of a (locally) ringed object in a compatible way.

    Let 𝒢\mathcal{G} be a site, or rather a geometry. For the purposes of the present discusssion we’d set 𝒢=Manifolds\mathcal{G} = Manifolds, but nothing in what I say now depends on this. Then then big cohesive (∞,1)(\infty,1)-topos H=Sh ∞(𝒢)\mathbf{H} = Sh_\infty(\mathcal{G}) is among other things the classifying topos for 𝒢\mathcal{G}-valued structure sheaves:

    for 𝒳\mathcal{X} any topos (for instance that of the topological space underlying a Frölicher space or better yet its over-topos as described below) a geometric morphism

    (𝒪⊣k):𝒳→k←𝒪H (\mathcal{O} \dashv k) : \mathcal{X} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{k}{\to}} \mathbf{H}

    determines a 𝒢\mathcal{G}-valued structure sheaf on 𝒳\mathcal{X}: a finite limit preserving functor 𝒢→𝒳\mathcal{G} \to \mathcal{X}.

    Now, the little topos incarnation of every object XX of H\mathbf{H} comes canonically with a 𝒢\mathcal{G}-structure sheaf: the etale geometric morphism

    H/X→X *←X *H. \mathbf{H}/X \stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}} \mathbf{H} \,.

    Notice that the inverse image X *X^* is the functor that sends V∈𝒢↪HV \in \mathcal{G} \hookrightarrow \mathbf{H} to the sheafification of X×VX \times V in H/X\mathbf{H}/X. This is indeed the sheaf of VV-valued functions on XX, regarded as an object in the over-topos: if XX itself is representable then under the equivalence PSh ∞(𝒢)/X≃PSh ∞(𝒢/X)PSh_\infty(\mathcal{G})/X \simeq PSh_\infty(\mathcal{G}/X) we have that X×VX \times V identifies with the presheaf that sends (U→X)↦Hom(U,V)(U \to X) \mapsto Hom(U,V).

    If we think of the objects of 𝒢 op\mathcal{G}^{op} as algebras, we could speak of the assignment

    Spec:(A∈𝒢 op)↦(H/A∈Topos/H) Spec : (A \in \mathcal{G}^{op}) \mapsto (\mathbf{H}/A \in Topos/\mathbf{H})

    as the spectrum functor in this context: it identifies an algebra with the corresponding locally ringed topos.

    Using this, up to size issues every locally 𝒢\mathcal{G}-ringed topos (𝒳,𝒪 𝒳)(\mathcal{X}, \mathcal{O}_{\mathcal{X}}) becomes a presheaf on 𝒢\mathcal{G} by the assignment

    (A∈𝒢 op)↦Topos/H(SpecA,(𝒳,𝒪 X)). (A \in \mathcal{G}^{op}) \mapsto Topos/\mathbf{H}( Spec A , (\mathcal{X}, \mathcal{O}_X)) \,.

    First I thought the ∞\infty-stacks obtained this way ought to be related to the concrete objects of H\mathbf{H}, since they have an “underlying ∞\infty-topos”. But if so, some care needs to be exercises here:

    first of all we have an equivalence H≃(Topos/H) et\mathbf{H} \simeq (Topos/\mathbf{H})_{et}, so that in fact for (𝒳,𝒪 𝒳)(\mathcal{X}, \mathcal{O}_{\mathcal{X}}) of the form (H/X,X *)(\mathbf{H}/X, X^*) we have that the above construction reproduces XX. So every object of H\mathbf{H} would be concrete in this sense.

    However, this is overlooking at least one aspect: the above equivalence holds for ToposTopos regarded as an (∞,1)(\infty,1)-category. But actually a morphism of 𝒢\mathcal{G}-ringed toposes (𝒳,𝒪 𝒳)→(𝒴,𝒪 𝒴)(\mathcal{X}, \mathcal{O}_{\mathcal{X}}) \to (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}}) is a morphism

    𝒳 →f 𝒴 ↘ ⇙ ↙ H \array{ \mathcal{X} &&\stackrel{f}{\to}&& \mathcal{Y} \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{H} }

    in ToposTopos with possibly non-invertible 2-morphism f *𝒪 𝒴→𝒪 𝒳f^* \mathcal{O}_{\mathcal{Y}} \to \mathcal{O}_{\mathcal{X}}.

    One should think about how all this interacts with the extra properties on a cohesive (∞,1)(\infty,1)-topos, in particular with the sub-quasi-(∞,1)(\infty,1)-topos of concrete objects.

    • CommentRowNumber30.
    • CommentAuthorMike Shulman
    • CommentTimeDec 21st 2010

    Does the pushout [0,2]=[0,1]⊔ 1[1,2][0,2] = [0,1] \sqcup_1 [1,2] live in the category of topoi? I think it does live in the category of locales, since sober spaces are a coreflective subcategory of locales, hence closed under colimits. But the embedding of locales in topoi doesn’t preserve all colimits.

    One answer to “which colimits of manifolds do we want to preserve?” might be “those which can be interpreted as colimits of locally smooth-ringed topoi”. (That seems to me to be roughly the sort of structure that the geometry of manifolds classifies, by analogy with how the big Zariski topos classifies locally ringed topoi. Although the analogy would be more precise if we included more “affine C ∞C^\infty-schemes” such as in ThCartSp.) I wonder whether there is a way to make a point into “the C ∞C^\infty germ of a manifold” into a locally smooth-ringed topos in such a way that some pushout [0,2]=[0,1]⊔ 1[1,2][0,2] = [0,1] \sqcup_1 [1,2] of locally smooth-ringed topoi implements “gluing paths together which agree along with all their derivatives?”

    • CommentRowNumber31.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 30th 2016
    • (edited Jan 30th 2016)

    This is a really old discussion, but here’s a vague suggestion that may being smootheology back, and which may, just may, be relevant for getting the internal reals in a sheaf topos to be smooth functions to R. In Andrew’s paper Yet More Smooth Mapping Spaces and Their Smoothly Local Properties, he talks about smoothly local properties. One way to think of this is that instead of taking an open cover {U i}\{U_i\} of a manifold MM, indexed by a set, one has a smoothly-varying family of open sets parameterised by MM itself. There are nice constructions that work better if one takes charts for the manifold after this fashion, rather than a chart using open covers and so on.

    Might it be worth thinking about whether these smoothly local open covers/charts give us a reasonable sheaf topos? I’ve not thought it through at all. Perhaps this coverage is refined by the usual open cover one, but perhaps not. Ideally one would have access to germs of maps in some way, as alluded to above.

    • CommentRowNumber32.
    • CommentAuthorMike Shulman
    • CommentTimeJan 31st 2016

    I don’t remember what all discussions we had about it in the past, but right now my feeling is that it shouldn’t be possible to get the internal reals in a sheaf topos to be smooth functions to ℝ\mathbb{R}, since in the internal logic of a sheaf topos one can define non-smooth functions ℝ→ℝ\mathbb{R}\to\mathbb{R} like the absolute value.

    • CommentRowNumber33.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 31st 2016

    @Mike,

    in the internal logic of a sheaf topos one can define non-smooth functions ℝ→ℝ\mathbb{R}\to\mathbb{R} like the absolute value.

    hmm, that’s mildly (very mildly) surprising at first, but a big spanner in the works!

    • CommentRowNumber34.
    • CommentAuthorMike Shulman
    • CommentTimeJan 31st 2016

    I think it emphasizes the same point Andrew had earlier, that constructive logic and sheaves are about continuity, but smoothness is something different.

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