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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 $x\in E_n$ there is some finite $m \lt n$ such that all faces of $x$ in $E_m$ are the basepoint.
Urs: on the bottom of page 437 in the reference by Brown it says: “each simplex of $E$ 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 $m \lt n$ which you mention is not implied by Brown(?) In particular, it seems this condition does not harmoize with the fact that $n$ may be negative.
But this looks like the condition which does appear in the definition of the $n$-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 $X_n$ with maps $\Sigma X_n \to X_{n+1}$) by making the $k$-simplices of $X_n$ into $(k-n)$-simplices in the spectrum. I thought the condition on $m\lt n$ is sort of saying that each simplex comes from $X_n$ for some $n\lt \infty$. 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 $S$-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 $k$-cells for all $k\in\mathbb{Z}$.” 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 $xyz$s. Surely some people out there will already be looking at functors $\Delta^{op} \to Spectra$ 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 —
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.
@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??
That’s one of the reasons that my comment was half joking.
It seems to me that the category $S$ of combinatorial spectra is the category of $Set_\ast$-valued presheaves on a small category $D$. Here, $D$ consists of “shifts of simplices”.
Let $D_\infty$ denote the site for the obvious $Set_\ast$-valued presheaf category into which $S$ embeds, i.e. there is an object for each integer, and morphisms are generated by the usual simplicial maps and identities. Let me define $D$ as a full subcategory of $S$, which in turn I view as a full subcategory of $[D_\infty^{op},Set_\ast]$. For each $n \in \mathbb{Z}$, $d \in \mathbb{N} \cup \{-1\}$, there is an object $\Delta^d[n]$ of $D$, defined by $\Delta^d[n](m) = \Delta([m+n],[d])_+$. Here I’m using brackets in the standard way to denote objects in the (augmented) simplex category, with the convention that $\Delta([p],[d]) = \emptyset$ if $p\lt -1$. So $\Delta^d[n]$ has a top-dimensional nondegenerate, non-basepoint simplex in dimension $d-n$; its nondegenerate, non-basepoint simplices in degree $d-n-k$ correspond to the codimension-$k$ faces of the standard $d$-simplex. In degree $-n$, we have $d+1$ 0-dimensional simplices corresponding to the the vertices of the standard $d$-simplex, and every face map applied to them results in a basepoint. Likewise, the “higher face maps” applied to a face of $\Delta^d[n]$ (the ones not specified in the above formula) are all the basepoint. In $[D_\infty^{op},Set_\ast]$, the object $\Delta^d[n]$ corepresents a very natural functor: $Hom(\Delta^d[n],X) = \{x \in X(n) \mid \partial_{i_0} \cdots \partial_{i_k} x = \ast \,\text{for}\, i_j \gt d-j\}$. Say that an element of such a hom-set (for some $d,n$) is a simplex of finite dimension. Note that $X \in S$ iff every simplex of $X$ is of finite dimension.
I claim that the density comonad for $D$ in $[D_\infty^{op},Set_\ast]$ is idempotent, with fixed point category $S$: any colimit of objects of $D$ has simplices all of finite dimension, and conversely the canonical colimit comparison map for an object of $S$ is surjective because every simplex in an object of $S$ has finite dimension, and it is injective because every face map between these simplices is witnessed by a map in $D$. The coreflection from $[D_\infty^{op},Set_\ast]$ to $S$ throws away those simplices of infinite dimension. The comparison functor from $S$ to $[D^{op},Set_\ast]$ is an equivalence. This can be seen because the objects of $D$ are retracts of representables in $[D_\infty^{op},Set]$ and the inclusion functor from $D$ to the Cauchy completion of $D_\infty$ is fully faithful. So $S$ is equivalent to $[D^{op},Set_\ast]$.
Assuming I haven’t made a mistake and $S$ is really a presheaf category, it becomes tempting to define a symmetric monoidal structure on it via Day convolution…
As a first guess, maybe we want to define $\Delta^d[n] \wedge \Delta^{d'}[n'] = (\Delta^d \times \Delta^{d'})[n+n']$. Here I’m using the obvious suspension-like functor $[n]$, which shifts the degree of everything down by $n$, so ought to act like $n$-fold desuspension. I’m writing $\Delta^{d}$ for $\Delta^{d}[0]$. The product of simplices is meant to suggest the formula $(\Delta^d \times \Delta^{d'})(m) = ((\Delta^d \times \Delta^{d'})_{m})_+$ where on the right hand side I mean the simplicial set $\Delta^d \times \Delta^{d'}$. 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 $S = [D^{op},Set_\ast]$. I think this might be the correct smash product, but that will take some work to check.
(I’m thinking about $\Delta^d[n]$ as a copy of $S^{-n}$ which has been “fleshed-out” to have dimension $d$ 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 $\Sigma^\infty \dashv \Omega^\infty$ adjunction comes from the inclusion $\Delta \to D$, $[d] \mapsto \Delta^d[0]$. I think this is a lax/colax monoidal adjunction, but it should escape Lewis’s theorem because $\Sigma^\infty S^0$ is “$\Delta^\infty_+$” whereas the monoidal unit is just $\Delta^0[0]$, which is not isomorphic, just weakly equivalent.
Very interesting! I don’t have time to think deeply about it right now, but nothing jumps out at me as wrong.
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 $(X_i)$ 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.
I think the “obvious construction” would be the product in the presheaf category $[D_\infty^{op},Set_\ast]$, 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.
Actually, wondering whether $S$ 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 $S$, and this seemed surprising from the description of $S$. I mucked around building colimits by hand, realized that $S$ was actually closed under colimits in $[D_\infty^{op},Set_\ast]$, wondered whether $S$ might be coreflective in $[D_\infty^{op},Set_\ast]$, 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: $\Omega^\infty \Sigma^\infty X$ is not weakly equivalent to $colim_n \Omega^n \Sigma^n X$. 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.
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.
I think I was mistaken, and now I suspect that $S$ 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 $\Delta^d[-n]$ were retracts of representables. The more I think about this, the more I’m confused about the interpretation of an element $x \in X(n)$ as a sort of simplex of integer dimension.
The whole idea was to let $\Delta^d[-n]$ coreprepresent (in the $Set_\ast$-enriched sense) the functor $F_{d,n}: X \mapsto \{x \in X(n) \mid \forall \phi,\, \phi^\ast(x) \neq 0 \Rightarrow Image(\phi) \supseteq [d+1,\omega)\}$. Here $\phi$ ranges over all morphisms into $n$ in $D_\infty$, and I’m using the following description of $D_\infty$: its object are a $\mathbb{Z}$’s worth of copies of $\omega$, and a morphism from $\omega_{n'}$ to $\omega_{n}$ is an order-preserving map $\omega \to \omega$ which is eventually just a shift upward by $n-n'$. So the condition $Image(\phi) \supseteq [d+1,\omega)$ is a way of saying that $\phi$ factors as a surjection followed by an injection which misses only elements at or below $d$, i.e. it is a combination of degeneracies and “lower” face maps.
The functor $F_{d,n}$ is indeed corepresentable, by an object we can call $\Delta^d[-n]$. We have $\Delta^d[-n] = D_\infty(-,n)_+/K_{d,n}$, where $K_{d,n} \subseteq D_\infty(-,n)$ consists of all nontrivial higher faces: $K_{d,n}(k) = \{\phi \in D_\infty(k,n) \mid Image(\phi) \nsupseteq [d+1,\omega)\}_+$. But this quotient does not split: all of the nonzero simplices of $D_\infty(k,n)_+$ have all their faces nonzero, so in fact there are no nonzero maps $\Delta^d[-n] \to D_\infty(-, n)_+$.
I do believe that the objects $\Delta^d [-n]$ form a dense generator of $S$. It’s clear that they form a regular generator of $S$ because every simplex of an object of $S$ must fall in the image of some $\Delta^d[-n]$. I don’t think that density is quite as straightforward as I originally made it out to be, but if $x \in X(n_0)$ appears as the image of some face under a map $\Delta^d[-n] \to X$, then I think it can be related to the canonical map $\Delta^{d_0}[-n_0] \to X$ via a span in $D \downarrow X$, passing through $\Delta^{d'}[-n_0]$ for some larger $d'$.
But it’s actually easy to see that $S$ is not a presheaf category on $D$: the objects of $D$ are not tiny, which they would have to be to correspond to representable presheaves. I think that $\Hom(\Delta^d[-n],-)$ preserves $Set_\ast$-enriched coproducts and filtered colimits, but not pushouts. For an element of $X(n)$ might have a nonzero $d+1$-face which is quotiented to become zero in the pushout, so that a new map from $\Delta^d[-n]$ may appear in the pushout. For example, there is no map $\Delta^d[-n] \to \Delta^{d+1}[-n]$ mapping $1_\omega$ to $1_\omega$, but we can quotient the codomain to obtain $\Delta^d[-n]$ and then there will be such a morphism into the quotient. Reflecting on this, it seems unlikely to me that that $S$ contains any tiny objects other than the 0 object, so it is probably not a presheaf category at all.
The category $D$ might still be kind of interesting. It has the following description: $Hom(\Delta^{d'}[-n'], \Delta^d[-n]) = \{ f \in \Delta([d'], [d' + n-n']) \mid Image(f) \supseteq [d+1,d'+n-n']\}$ (note the unfortunate notational clash: $[d]$ denotes the totally ordered set $\{0,\dots, d\}$ whereas $[d_1,d_2]$ denotes the set $\{d_1,\dots, d_2\}$). The composite $g \circ f$ is defined by extending $g$ in the natural way and then composing as in $\Delta$. To view $D$ as a $Set_\ast$-enriched category, we add a disjoint basepoint to the homsets. To view the morphisms of $D$ as morphisms $\omega_{n'} \to \omega_n$, and thus as maps between quotients of representables on $D_\infty$, 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 $D$ via Day convolution?
The notation of my previous comments is “wrong”. The object I was calling “$\Delta^d[-n]$” should be called something more like $\Delta^d_+[-k]$ where $n = d-k$, since it’s really a $k$-fold suspension of $\Sigma^\infty \Delta^d_+$.
The $Set_\ast$-enriched category $D$ 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 $\{\Delta^d_+[-k]\}_{d \in \mathbb{N}, k \in \mathbb{Z}}$; let’s denote the coface maps as $\delta_i^{k,d}: \Delta_+^{d-1}[-k] \to \Delta_+^d[-k]$ and the codegeneracy maps as $\sigma_j^{k,d}: \Delta_+^{d+1}[-k] \to \Delta_+^d[-k]$. In addition there are maps $\iota^{k,d}: \Delta_+^{d+1}[-(k+1)] \to \Delta_+^d[-k]$, which commute with the face and degeneracy maps in the sense that $\iota \delta_i = \delta_i \iota$, $\iota \sigma_j = \sigma_j \iota$ – with the interpretation that $\delta_{d+1}^{k,d} = 0$ and $\sigma_{d+1}^{k,d} = 0$, so that $\iota^{k,d} \delta^{k+1,d+1}_{d+1} =0$ and $\iota^{k,d} \sigma^{k+1,d+1}_{d+1} = 0$.
So a $Set_\ast$-valued presheaf on $D$ consists of a $\mathbb{Z}$’s worth of pointed, augmented simplicial objects $\{X_{k,d} = X(\Delta_+^d[-k])\}_{k \in \mathbb{Z}, d \in \mathbb{N}}$, along with maps $(\iota^{k,d})^\ast : X_{k,d} \to X_{k+1,d+1}$ satisfying $(\delta^{k,d+1}_{d+1})\ast \circ (\iota^{k,d})^\ast = 0$ and $(\sigma^{k+1,d+1}_{d+1})^\ast \circ (\iota^{k,d})^\ast = 0$; these can be interpreted as the structure maps of a spectrum. Such a presheaf lies in the category $S$ if and only if for every $k,d$ the map $(\iota^{k,d})^\ast$ is the kernel of $(\delta^{k,d+1}_{d+1})^\ast$, making $S$ a reflective $Set_\ast$-enriched subcategory of $[D^{op},Set_\ast]$. There is a $\Sigma^\infty \dashv \Omega^\infty$ adjunction where $\Omega^\infty(X)_d = X_{0,d}$, and $\Sigma^\infty$ is defined by Kan extension, sending $\Delta^d_+ \mapsto \Delta^d_+[0]$.
Unfortunately, the smash product I mentioned above, which is forced if $\Sigma^\infty$ is to be strong monoidal, fails even to be a functor $[D^{op},Set_\ast] \times [D^{op},Set_\ast] \to [D^{op},Set_\ast]$ as far as I can see, although it is separately functorial in each variable. So if either $S$ or $[D^{op},Set_\ast]$ is to admit a symmetric monoidal smash product, it will have to be fancier.
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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