nLab combinatorial spectrum

Idea

Combinatorial spectra is to spectra of topological spaces as a simplicial sets are to topological spaces:

Combinatorial spectra are graded sets that behaves like a set of simplices constituting a space, but such that the simplices are not just in non-negative degree $n \in \mathbb{N}$ but in all integral degrees $n \in \mathbb{Z}$, hence including negative degrees.

Definition

A combinatorial spectrum is (Kan 63, def. 4.1)

• 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

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

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

The sub-spectrum

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

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

$$S^n$$

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 $\Lambda^{k}_j[n-k]$ for $0 \leq j \leq k$ 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

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

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

Properties

Model category structure

The category of combinatorial spectra admits a model structure (Brown 73), whose cofibrations are monomorphisms and weak equivalences are maps that induce isomorphisms on homotopy groups.

Generating (acyclic) cofibrations are given by inclusions of horns respectively boundaries into stable simplices, as defined in the previous section, in complete analogy with the usual Kan-Quillen model structure on simplicial sets. (Acyclic) fibrations are maps satisfying the corresponding lifting properties.

This is indeed a model structure for spectra, related by a zig-zag of Quillen equivalences to the Bousfield-Friedlander model structure on sequential spectra (Bousfield-Friedlander 78, section 2.5). See also below.

Smash product

In (Brown 73, Appendix A) is defined a smash product of spectra on the homotopy category of combinatorial spectra. The main idea is to define the smash product of a stable $m$-simplex and a stable $n$-simplex to be the stable $(m+n)$-simplex whose face maps are defined using the face maps of the two simplices involved using a formula that somewhat resembles the formula for the differential of the tensor product of two chain complexes.

Whether or not it is possible to introduce a symmetric monoidal smash product on the category of combinatorial spectra obtaining a monoidal model category that is Quillen equivalent to the monoidal model category of (say) symmetric simplicial spectra is currently an open problem.

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. In (Bousfield-Friedlander 78, section 2.5) this is lifted to a zig-zag of Quillen equivalence between the model structure for Kan’s combinatorial spectra and the Bousfield-Friedlander model structure on sequential spectra in sSet to

I don’t know whether anyone has gone back to treat combiantorial from a “modern” standpoint, such as by putting a monoidal 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 ($A_\infty$ or $E_\infty$) 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.

Stable Dold-Kan correspondence

Combinatorial spectra internal to abelian groups (Kan 63, def. 51) are equivalent to chain complexes (Kan 63, prop.5.8). See at stable Dold-Kan correspondence for more on this.

Related concepts

• (∞,Z)-category

References

An early reference is

• Daniel Kan, Semisimplicial spectra, Illinois J. Math. Volume 7, Issue 3 (1963), 463-478. (Euclid)

see also at stable Dold-Kan correspondence.

The definition is recalled in part II, section 7 of

• Kenneth Brown, Abstract homotopy theory and generalized sheaf cohomology, 1973

Discussion of the relation (equivalence of homotopy categories) to the Bousfield-Friedlander model structure on sequential spectra in simplicial sets is in

• Aldridge Bousfield, Eric Friedlander, section 2.5 of Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (pdf)

See also

• Ruian Chen, Igor Kriz and Ales Pultr, Kan’s combinatorial spectra and their sheaves revisited, Theory and Applications of Categories, Vol. 32, 2017, No. 39, pp 1363-1396. (TAC)