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 but in all integral degrees .
A combinatorial spectrum is (Kan 63, def. 4.1)
a sequence of pointed sets
equipped for each and with
the usual simplicial identities are satisfied;
each simplex has only finitely many faces different from the point of : i.e. for every there are only finitely many for which is not the point.
The standard simplicial sets corresponding to the standard simplices have their analogs for simplicial spectra. . The difference is that regarded as a spectrum the -simplex may sit in any degree , not necessarily in degree .
The -simplex in degree . For each integer and there is a spectrum
which is generated from a single element subject to the relation that for . So this is something with -faces, hence looking like a -simplex, but sitting in degree .
of generated by the faces . This is the boundary of the -simplex in degree .
generated by a single simplex subject to the relation for all . This is the -sphere as a spectrum.
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.
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 -simplex and a stable -simplex to be the stable -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.
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 with maps or , where we think of the space as being “shifted down by dimensions.” Thus, for instance, the -cells of the spectrum can come from 0-cells of , or 1-cells in , or 2-cells in , etc. The structure maps support this intuition, since the suspension shifts things up by one dimension; thus it maps the -cells of into the -cells of .
In fact, this can be made precise: starting from a spectrum of simplicial sets, in the sense of a sequence of spaces with maps , 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 ( or ) 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.
An early reference is
see also at stable Dold-Kan correspondence.
The definition is recalled in part II, section 7 of