nLab
combinatorial spectrum

Idea

A combinatorial spectrum is to a spectrum of topological spaces as a simplicial set is to a topological space: it is a graded set that behaves like a set of simplices constituting a space, where the special property is that the simplices are not just in non-negative degree $n\in \mathbb{N}$ but in all integral degrees $n\in \mathbb{Z}$.

Definition

A combinatorial spectrum is

• a sequence $E=\{E_n{\}}_{n\in \mathbb{Z}}$ of pointed sets

• equipped for each $n\in \mathbb{Z}$ and $i\in \mathbb{N}$ with

  • morphisms of pointed sets $d_i:E_n\to E_{n-1}$ called face maps;

  • morphisms of pointed sets $s_i:E_n\to E_{n+1}$ called degeneracy maps

• such that

  • the usual simplicial identities are satisfied;

  • each simplex has only finitely many faces different from the point of $E_{n-1}$: i.e. for every $x\in E_n$ there are only finitely many $i\in \mathbb{N}$ for which $d_i(x)$ is not the point.

Examples

The standard simplicial sets corresponding to the standard simplices have their analogs for simplicial spectra. . The difference is that regarded as a spectrum the $k$-simplex may sit in any degree $n\in \mathbb{N}$, not necessarily in degree $k$.

The $(k+1)$-simplex in degree $n$. For each integer $n\in \mathbb{Z}$ and $k\in \mathbb{N}$ there is a spectrum

$${\Delta }^k[n-k]$$\Delta^{k}[n-k]

which is generated from a single element $x\in E_n$ subject to the relation that $d_{i}x=*$ for $i>k$. So this is something with $(k+1)$-faces, hence looking like a $k$-simplex, but sitting in degree $n$.

The sub-spectrum

$$\partial ({\Delta }^k[n-k])$$\partial (\Delta^{k}[n-k])

of ${\Delta }^k[n-k]$ generated by the faces $d_{i}x$. This is the boundary of the $k$-simplex in degree $n$.

The spectrum

$$S^n$$S^n

generated by a single simplex $x\in E_n$ subject to the relation $d_{i}x=0$ for all $i$. This is the $n$-sphere as a spectrum.

The sub-spectrum ${\Lambda }_j^k[n-k]$ for $0\le j\le k$ is the sub-spectrum of ${\Delta }^k[n-k]$ generated from all the faces $d_{i}x$ except $d_{j}x$. This is the $j$th horn of the $k$-simplex in degree $n$. Compare with the horn of a simplex.

As for simplices, there are canonical horn inclusion morphisms of combinatorial spectra

$${\Lambda }_j^k[n-k]\hookrightarrow {\Delta }^k[n-k].$$\Lambda^k_j[n-k] \hookrightarrow \Delta^k[n-k] \,.

A condition entirely analogous to the Kan fibration condition on Kan simplicial sets leads to the notion of Kan combinatorial spectrum.

Relationship to other spectra

From the perspective of a combinatorial spectrum, an “intuitive spectrum” is supposed to be some sort of space-like object having “cells in all integer dimensions,” while a “space” (or simplicial set) has cells only in nonnegative dimensions. The traditional definitions of spectra approximate this intuition by using a sequence of spaces $\{E_n\}$ with maps $E_n\to \Omega E_{n+1}$ or $\Sigma E_n\to E_{n+1}$, where we think of the space $E_n$ as being “shifted down by $n$ dimensions.” Thus, for instance, the $(-2)$-cells of the spectrum can come from 0-cells of $E_2$, or 1-cells in $E_3$, or 2-cells in $E_4$, etc. The structure maps $\Sigma E_n\to E_{n+1}$ support this intuition, since the suspension $\Sigma$ shifts things up by one dimension; thus it maps the $k$-cells of $E_n$ into the $k+1$-cells of $E_{n+1}$.

In fact, this can be made precise: starting from a spectrum of simplicial sets, in the sense of a sequence of spaces with maps $\Sigma E_n\to E_{n+1}$, one can construct a combinatorial spectrum by “piecing together” the cells in all dimensions. This construction can be found in Kan’s original article; it provides an equivalence of homotopy theories between combinatorial spectra and ordinary spectra built from simplicial sets.

I don’t know whether anyone has gone back to treat these from a “modern” standpoint, such as by putting a model category structure on combinatorial spectra. They do seem less interesting and useful from a modern standpoint, because no one has ever managed to give them a smash product which is associative and unital on the point-set level; thus they don’t provide a good framework for talking about ($A_{\mathrm{\infty }}$ or $E_{\mathrm{\infty }}$) ring spectra, module spectra, and other aspects of brave new algebra?. It’s also not clear how hard anyone has tried, though. Presumably one would have to modify the definition by incorporating the “symmetries” somehow, as is done for example by passing from ordinary simplicial-set spectra to symmetric spectra.

References

• Dan Kan, “Semisimplicial spectra”

Part II, section 7 of

• Ken Brown, Abstract homotopy theory and generalized sheaf cohomology

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Discussion

A previous version of this entry triggered the following discussion:

Mike: Are you sure about that last condition? I remember a condition more like “for each $x\in E_n$ there is some finite $m<n$ 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 $m<n$ 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 $m<n$ is sort of saying that each simplex comes from $X_n$ for some $n<\mathrm{\infty }$. 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 $\mathrm{xyz}$s. Surely some people out there will already be looking at functors ${\Delta }^{\mathrm{op}}\to \mathrm{Spectra}$ 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.