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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2012

    I have added at combinatorial spectrum the missing bibliographical information for Kan’s original article.

    While doing so I noticed old forgotten discussion sitting there, which hereby I move from there to here:

    — begin forwarded discussion —

    A previous version of this entry triggered the following discussion:

    +–{: .query} Mike: Are you sure about that last condition? I remember a condition more like “for each xE nx\in E_n there is some finite m<nm \lt n such that all faces of xx in E mE_m are the basepoint.

    Urs: on the bottom of page 437 in the reference by Brown it says: “each simplex of EE has only finitely many faces different from **”.

    I see that my original phrasing reflected this only very imprecisely. I have tried to improve that now. But it also seems that this condition m<nm \lt n which you mention is not implied by Brown(?) In particular, it seems this condition does not harmoize with the fact that nn may be negative.

    But this looks like the condition which does appear in the definition of the nn-simplex spectra (next page of Brown). I have added that in the list of examples now.

    Another question: what’s the established term for these things here? I made up both “combinatorial spectrum” and “simplicial spectrum” after reading Brown’s article, which just calls this “spectrum” without qualification. I am tending to think that “simplicial spectrum” would be a good term.

    Related to that: what’s a more recent good reference on these combinatorial version of spectra?

    Mike: I was remembering a condition like that from Kan’s original article “Semisimplicial spectra,” which I unfortunately don’t have access to a copy of right now. I think the idea is that a spectrum of this sort is built out of a naive prespectrum of simplicial sets (that is, a sequence of based simplicial sets X nX_n with maps ΣX nX n+1\Sigma X_n \to X_{n+1}) by making the kk-simplices of X nX_n into (kn)(k-n)-simplices in the spectrum. I thought the condition on m<nm\lt n is sort of saying that each simplex comes from X nX_n for some n<n\lt \infty. But possibly my memory is just wrong.

    Since Kan’s original term was “semisimplicial spectrum” back when “semisimplicial set” meant what we now call a “simplicial set,” it’s hard to argue with “simplicial spectrum.” As far as I know, however, no algebraic topologist has really thought seriously about these things for quite some time, probably due largely to the appearance of symmetric monoidal categories of spectra (EKMM SS-modules, orthogonal spectra, symmetric spectra, etc.) of which there is no known analogue for this sort of spectra. It’s kind of a shame, I think, since these spectra give a really good intuition of “an object with kk-cells for all kk\in\mathbb{Z}.” I spent a little while once trying to come up with a version of these that would have a symmetric monoidal smash product, maybe starting with simplicial symmetric spectra instead of naive prespectra, but I failed.

    Urs: thanks, very useful. That’s a piece of information that I was looking for.

    Yes, this combinatorial spectrum is nicely suggestive of a \mathbb{Z}-category. It seems surprising that there shouldn’t be a symmetric monoidal product on that. What goes wrong?

    Concerning terminology: now that I thought about it I feel that “simplicial spectrum” may tend to be misleading, as it collides with the use of “simplicial xyz” as a simplicial object internal to the category of xyzxyzs. Surely some people out there will already be looking at functors Δ opSpectra\Delta^{op} \to Spectra and call them “simplicial spectra” (?)

    Mike: Yes, you’re quite right that “simplicial spectrum” should probably be reserved for a simplicial object in spectra; I wasn’t thinking. What we really need is a name for the shape category that arises here, analogous to “simplex category,” “cube category,” and so on. Like “spectrix category.” Then combinatorial spectra would be “spectricial sets.” (I’m only half joking.)

    The thing that goes wrong with the symmetric monoidal product is, as far as I can tell, sort of the same thing that goes wrong for naive prespectra: there are automorphisms that don’t get taken into account. But it’s possible that no one has just been clever enough.

    =–

    — end forwarded discussion —

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeMar 31st 2012

    Thanks. Of course, my memory was playing tricks on me during that discussion; the definition in Kan’s article is also exactly what is on the page.

  1. @Mike’s “spectricial sets”: As far as I understand, combinatorial spectra are not simply diagrams of shape the category of stable simplices, but only certain such diagrams satisfying an additional finiteness condition. So even if we were to refer to the category of stable simplices as the “spectrix category”, the term “spectricial set” would be misleading. Or is there perhaps a better shape category which somehow encodes the finiteness condition??

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJun 2nd 2014

    That’s one of the reasons that my comment was half joking.

    • CommentRowNumber5.
    • CommentAuthorTim Campion
    • CommentTimeDec 19th 2016
    • (edited Dec 20th 2016)

    It seems to me that the category SS of combinatorial spectra is the category of Set *Set_\ast-valued presheaves on a small category DD. Here, DD consists of “shifts of simplices”.

    Let D D_\infty denote the site for the obvious Set *Set_\ast-valued presheaf category into which SS embeds, i.e. there is an object for each integer, and morphisms are generated by the usual simplicial maps and identities. Let me define DD as a full subcategory of SS, which in turn I view as a full subcategory of [D op,Set *][D_\infty^{op},Set_\ast]. For each nn \in \mathbb{Z}, d{1}d \in \mathbb{N} \cup \{-1\}, there is an object Δ d[n]\Delta^d[n] of DD, defined by Δ d[n](m)=Δ([m+n],[d]) +\Delta^d[n](m) = \Delta([m+n],[d])_+. Here I’m using brackets in the standard way to denote objects in the (augmented) simplex category, with the convention that Δ([p],[d])=\Delta([p],[d]) = \emptyset if p<1p\lt -1. So Δ d[n]\Delta^d[n] has a top-dimensional nondegenerate, non-basepoint simplex in dimension dnd-n; its nondegenerate, non-basepoint simplices in degree dnkd-n-k correspond to the codimension-kk faces of the standard dd-simplex. In degree n-n, we have d+1d+1 0-dimensional simplices corresponding to the the vertices of the standard dd-simplex, and every face map applied to them results in a basepoint. Likewise, the “higher face maps” applied to a face of Δ d[n]\Delta^d[n] (the ones not specified in the above formula) are all the basepoint. In [D op,Set *][D_\infty^{op},Set_\ast], the object Δ d[n]\Delta^d[n] corepresents a very natural functor: Hom(Δ d[n],X)={xX(n) i 0 i kx=*fori j>dj}Hom(\Delta^d[n],X) = \{x \in X(n) \mid \partial_{i_0} \cdots \partial_{i_k} x = \ast \,\text{for}\, i_j \gt d-j\}. Say that an element of such a hom-set (for some d,nd,n) is a simplex of finite dimension. Note that XSX \in S iff every simplex of XX is of finite dimension.

    I claim that the density comonad for DD in [D op,Set *][D_\infty^{op},Set_\ast] is idempotent, with fixed point category SS: any colimit of objects of DD has simplices all of finite dimension, and conversely the canonical colimit comparison map for an object of SS is surjective because every simplex in an object of SS has finite dimension, and it is injective because every face map between these simplices is witnessed by a map in DD. The coreflection from [D op,Set *][D_\infty^{op},Set_\ast] to SS throws away those simplices of infinite dimension. The comparison functor from SS to [D op,Set *][D^{op},Set_\ast] is an equivalence. This can be seen because the objects of DD are retracts of representables in [D op,Set][D_\infty^{op},Set] and the inclusion functor from DD to the Cauchy completion of D D_\infty is fully faithful. So SS is equivalent to [D op,Set *][D^{op},Set_\ast].

    Assuming I haven’t made a mistake and SS is really a presheaf category, it becomes tempting to define a symmetric monoidal structure on it via Day convolution…

    • CommentRowNumber6.
    • CommentAuthorTim Campion
    • CommentTimeDec 19th 2016
    • (edited Dec 19th 2016)

    As a first guess, maybe we want to define Δ d[n]Δ d[n]=(Δ d×Δ d)[n+n]\Delta^d[n] \wedge \Delta^{d'}[n'] = (\Delta^d \times \Delta^{d'})[n+n']. Here I’m using the obvious suspension-like functor [n][n], which shifts the degree of everything down by nn, so ought to act like nn-fold desuspension. I’m writing Δ d\Delta^{d} for Δ d[0]\Delta^{d}[0]. The product of simplices is meant to suggest the formula (Δ d×Δ d)(m)=((Δ d×Δ d) m) +(\Delta^d \times \Delta^{d'})(m) = ((\Delta^d \times \Delta^{d'})_{m})_+ where on the right hand side I mean the simplicial set Δ d×Δ d\Delta^d \times \Delta^{d'}. I think this defines a symmetric monoidal structure on shifts of finite products of simplices, which extends by Day convolution to a symmetric monoidal structure on S=[D op,Set *]S = [D^{op},Set_\ast]. I think this might be the correct smash product, but that will take some work to check.

    (I’m thinking about Δ d[n]\Delta^d[n] as a copy of S nS^{-n} which has been “fleshed-out” to have dimension dd larger. I’m not sure if that’s what a cell in a spectrum is supposed to be like?)

    I think this monoidal product at least doesn’t fall afoul of Lewis’s impossibility theorem. The Σ Ω \Sigma^\infty \dashv \Omega^\infty adjunction comes from the inclusion ΔD\Delta \to D, [d]Δ d[0][d] \mapsto \Delta^d[0]. I think this is a lax/colax monoidal adjunction, but it should escape Lewis’s theorem because Σ S 0\Sigma^\infty S^0 is “Δ + \Delta^\infty_+” whereas the monoidal unit is just Δ 0[0]\Delta^0[0], which is not isomorphic, just weakly equivalent.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeDec 20th 2016

    Very interesting! I don’t have time to think deeply about it right now, but nothing jumps out at me as wrong.

    • CommentRowNumber8.
    • CommentAuthorKarol Szumiło
    • CommentTimeDec 20th 2016

    I admit that I haven’t read your proposed construction in detail. However, I want to mention one potential obstruction for the category of combinatorial spectra to be a category of presheaves. Namely, why is it complete? The most obvious construction of the product of an infinite family (X i)(X_i) won’t work if there is no universal bound for the number of non-basepoint faces of a cell (in a given dimension). Perhaps if you could explicitly describe how products in your presheaf category translate back to the standard description of the category of combinatorial spectra, it would help me digest your approach.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeDec 20th 2016

    I think the “obvious construction” would be the product in the presheaf category [D op,Set *][D_\infty^{op},Set_\ast], and Tim’s presheaf category is coreflective therein, so its products are obtained by coreflecting the ambient ones. I think what that amounts to is discarding all the cells with infinitely many nonbasepoint faces.

    • CommentRowNumber10.
    • CommentAuthorTim Campion
    • CommentTimeDec 20th 2016
    • (edited Dec 20th 2016)

    Actually, wondering whether SS was complete or cocomplete was my initial question – I thought it seemed likely given that Ken Brown was able to build a model structure on SS, and this seemed surprising from the description of SS. I mucked around building colimits by hand, realized that SS was actually closed under colimits in [D op,Set *][D_\infty^{op},Set_\ast], wondered whether SS might be coreflective in [D op,Set *][D_\infty^{op},Set_\ast], recalled that under Vopenka’s principle it must be – and must in fact be locally presentable. From there, the description of the coreflector and the natural choice of generator are not so hard to see, and it’s natural to ask whether one has a presheaf category. But I love how these Vopenka’s principle results can help shape one’s thinking so nicely!

    [As a side note – I don’t know so many coreflective subcategories of presheaf categories. Are they all themselves presheaf categories?]

    Also, I’m now noticing that the proposed smash product violates another one of Lewis’ axioms: Ω Σ X\Omega^\infty \Sigma^\infty X is not weakly equivalent to colim nΩ nΣ nXcolim_n \Omega^n \Sigma^n X. Well – I guess this axiom is independent of the choice of smash product. I think this is okay because you only expect this hold after deriving these functors, right? After all, these functors have been used in the literature by Brown.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeDec 20th 2016

    I don’t remember all of Lewis’s axioms, but I think all or most of them are things that must hold at the derived level; the no-go theorem is that you can’t get them all to also hold simultaneously at the point-set level.

    • CommentRowNumber12.
    • CommentAuthorTim Campion
    • CommentTimeDec 22nd 2016
    • (edited Dec 22nd 2016)

    I think I was mistaken, and now I suspect that SS is not a presheaf category. It does seem to be locally finitely presentable, though. I think most of what I claimed was basically correct, except for the actual formulas, and moreover except for the assertion that the objects Δ d[n]\Delta^d[-n] were retracts of representables. The more I think about this, the more I’m confused about the interpretation of an element xX(n)x \in X(n) as a sort of simplex of integer dimension.

    The whole idea was to let Δ d[n]\Delta^d[-n] coreprepresent (in the Set *Set_\ast-enriched sense) the functor F d,n:X{xX(n)ϕ,ϕ *(x)0Image(ϕ)[d+1,ω)}F_{d,n}: X \mapsto \{x \in X(n) \mid \forall \phi,\, \phi^\ast(x) \neq 0 \Rightarrow Image(\phi) \supseteq [d+1,\omega)\}. Here ϕ\phi ranges over all morphisms into nn in D D_\infty, and I’m using the following description of D D_\infty: its object are a \mathbb{Z}’s worth of copies of ω\omega, and a morphism from ω n\omega_{n'} to ω n\omega_{n} is an order-preserving map ωω\omega \to \omega which is eventually just a shift upward by nnn-n'. So the condition Image(ϕ)[d+1,ω)Image(\phi) \supseteq [d+1,\omega) is a way of saying that ϕ\phi factors as a surjection followed by an injection which misses only elements at or below dd, i.e. it is a combination of degeneracies and “lower” face maps.

    The functor F d,nF_{d,n} is indeed corepresentable, by an object we can call Δ d[n]\Delta^d[-n]. We have Δ d[n]=D (,n) +/K d,n\Delta^d[-n] = D_\infty(-,n)_+/K_{d,n}, where K d,nD (,n)K_{d,n} \subseteq D_\infty(-,n) consists of all nontrivial higher faces: K d,n(k)={ϕD (k,n)Image(ϕ)[d+1,ω)} +K_{d,n}(k) = \{\phi \in D_\infty(k,n) \mid Image(\phi) \nsupseteq [d+1,\omega)\}_+. But this quotient does not split: all of the nonzero simplices of D (k,n) +D_\infty(k,n)_+ have all their faces nonzero, so in fact there are no nonzero maps Δ d[n]D (,n) +\Delta^d[-n] \to D_\infty(-, n)_+.

    I do believe that the objects Δ d[n]\Delta^d [-n] form a dense generator of SS. It’s clear that they form a regular generator of SS because every simplex of an object of SS must fall in the image of some Δ d[n]\Delta^d[-n]. I don’t think that density is quite as straightforward as I originally made it out to be, but if xX(n 0)x \in X(n_0) appears as the image of some face under a map Δ d[n]X\Delta^d[-n] \to X, then I think it can be related to the canonical map Δ d 0[n 0]X\Delta^{d_0}[-n_0] \to X via a span in DXD \downarrow X, passing through Δ d[n 0]\Delta^{d'}[-n_0] for some larger dd'.

    But it’s actually easy to see that SS is not a presheaf category on DD: the objects of DD are not tiny, which they would have to be to correspond to representable presheaves. I think that Hom(Δ d[n],)\Hom(\Delta^d[-n],-) preserves Set *Set_\ast-enriched coproducts and filtered colimits, but not pushouts. For an element of X(n)X(n) might have a nonzero d+1d+1-face which is quotiented to become zero in the pushout, so that a new map from Δ d[n]\Delta^d[-n] may appear in the pushout. For example, there is no map Δ d[n]Δ d+1[n]\Delta^d[-n] \to \Delta^{d+1}[-n] mapping 1 ω1_\omega to 1 ω1_\omega, but we can quotient the codomain to obtain Δ d[n]\Delta^d[-n] and then there will be such a morphism into the quotient. Reflecting on this, it seems unlikely to me that that SS contains any tiny objects other than the 0 object, so it is probably not a presheaf category at all.

    The category DD might still be kind of interesting. It has the following description: Hom(Δ d[n],Δ d[n])={fΔ([d],[d+nn])Image(f)[d+1,d+nn]}Hom(\Delta^{d'}[-n'], \Delta^d[-n]) = \{ f \in \Delta([d'], [d' + n-n']) \mid Image(f) \supseteq [d+1,d'+n-n']\} (note the unfortunate notational clash: [d][d] denotes the totally ordered set {0,,d}\{0,\dots, d\} whereas [d 1,d 2][d_1,d_2] denotes the set {d 1,,d 2}\{d_1,\dots, d_2\}). The composite gfg \circ f is defined by extending gg in the natural way and then composing as in Δ\Delta. To view DD as a Set *Set_\ast-enriched category, we add a disjoint basepoint to the homsets. To view the morphisms of DD as morphisms ω nω n\omega_{n'} \to \omega_n, and thus as maps between quotients of representables on D D_\infty, we extend them in the natural way.

    I suppose there still might be some hope of defining a smash product by extending a promonoidal product on DD via Day convolution?

    • CommentRowNumber13.
    • CommentAuthorTim Campion
    • CommentTimeDec 26th 2016
    • (edited Dec 27th 2016)

    The notation of my previous comments is “wrong”. The object I was calling “Δ d[n]\Delta^d[-n]” should be called something more like Δ + d[k]\Delta^d_+[-k] where n=dkn = d-k, since it’s really a kk-fold suspension of Σ Δ + d\Sigma^\infty \Delta^d_+.

    The Set *Set_\ast-enriched category DD is actually pretty nice (I was mistaken before in thinking that it lay in the image of the “disjoint basepoint” functor from ordinary categories). It consists of a \mathbb{Z}’s worth of copies of the augmented simplex category (with disjoint baspoints on the homsets), with objects {Δ + d[k]} d,k\{\Delta^d_+[-k]\}_{d \in \mathbb{N}, k \in \mathbb{Z}}; let’s denote the coface maps as δ i k,d:Δ + d1[k]Δ + d[k]\delta_i^{k,d}: \Delta_+^{d-1}[-k] \to \Delta_+^d[-k] and the codegeneracy maps as σ j k,d:Δ + d+1[k]Δ + d[k]\sigma_j^{k,d}: \Delta_+^{d+1}[-k] \to \Delta_+^d[-k]. In addition there are maps ι k,d:Δ + d+1[(k+1)]Δ + d[k]\iota^{k,d}: \Delta_+^{d+1}[-(k+1)] \to \Delta_+^d[-k], which commute with the face and degeneracy maps in the sense that ιδ i=δ iι\iota \delta_i = \delta_i \iota, ισ j=σ jι\iota \sigma_j = \sigma_j \iota – with the interpretation that δ d+1 k,d=0\delta_{d+1}^{k,d} = 0 and σ d+1 k,d=0\sigma_{d+1}^{k,d} = 0, so that ι k,dδ d+1 k+1,d+1=0\iota^{k,d} \delta^{k+1,d+1}_{d+1} =0 and ι k,dσ d+1 k+1,d+1=0\iota^{k,d} \sigma^{k+1,d+1}_{d+1} = 0.

    So a Set *Set_\ast-valued presheaf on DD consists of a \mathbb{Z}’s worth of pointed, augmented simplicial objects {X k,d=X(Δ + d[k])} k,d\{X_{k,d} = X(\Delta_+^d[-k])\}_{k \in \mathbb{Z}, d \in \mathbb{N}}, along with maps (ι k,d) *:X k,dX k+1,d+1(\iota^{k,d})^\ast : X_{k,d} \to X_{k+1,d+1} satisfying (δ d+1 k,d+1)*(ι k,d) *=0(\delta^{k,d+1}_{d+1})\ast \circ (\iota^{k,d})^\ast = 0 and (σ d+1 k+1,d+1) *(ι k,d) *=0(\sigma^{k+1,d+1}_{d+1})^\ast \circ (\iota^{k,d})^\ast = 0; these can be interpreted as the structure maps of a spectrum. Such a presheaf lies in the category SS if and only if for every k,dk,d the map (ι k,d) *(\iota^{k,d})^\ast is the kernel of (δ d+1 k,d+1) *(\delta^{k,d+1}_{d+1})^\ast, making SS a reflective Set *Set_\ast-enriched subcategory of [D op,Set *][D^{op},Set_\ast]. There is a Σ Ω \Sigma^\infty \dashv \Omega^\infty adjunction where Ω (X) d=X 0,d\Omega^\infty(X)_d = X_{0,d}, and Σ \Sigma^\infty is defined by Kan extension, sending Δ + dΔ + d[0]\Delta^d_+ \mapsto \Delta^d_+[0].

    Unfortunately, the smash product I mentioned above, which is forced if Σ \Sigma^\infty is to be strong monoidal, fails even to be a functor [D op,Set *]×[D op,Set *][D op,Set *][D^{op},Set_\ast] \times [D^{op},Set_\ast] \to [D^{op},Set_\ast] as far as I can see, although it is separately functorial in each variable. So if either SS or [D op,Set *][D^{op},Set_\ast] is to admit a symmetric monoidal smash product, it will have to be fancier.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJan 5th 2017

    This is interesting, but unfortunately I don’t have time to understand it all right now. But if you manage to make it work out, I’ll be interested to hear!

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