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

An early reference seems to be

• Dan Kan, Semisimplicial spectra, Illinois J. Math. Volume 7 (1963), 463-478.

The definition is recalled in part II, section 7 of

• Ken Brown, Abstract homotopy theory and generalized sheaf cohomology