Coalgebraic Algebra and Infinite Loop Spaces
In many areas of topology and related fields one is interested in representable functors, that is, a way of associating to a space X a group (or ring, module, etc) E(X) with, among other properties, the useful one that there is a special space Y with E(X) equal to the set of (homotopy classes of) maps from X to Y. In fact, it’s better than this, the space Y itself has the structure of a group, (ring, module, etc) up to homotopy and all the algebraic properties of E(X) can be recovered from knowing about the maps X to Y. Examples of representable functors are ordinary cohomology (here Y is an `Eilenberg-MacLane’ space) and K-theory (where the representing spaces Y are given by the unitary groups and their classifying spaces), though there are many others of great interest. In the case of E (-) a cohomology theory, the representing space is an infinite loop space, rather special and generally highly applicable objects in the world of homotopy theory.
Understanding the topology of the representing space Y of your favourite representable functor is a powerful tool in understanding the properties and behaviour of the functor itself, for example, giving descriptions of the operations on the functor (its natural symmetries). However, it can be a hard problem, not least because the spaces concerned are usually very big as well as complicated. A good first step is often to try to understand its mod p homology, H*(Y;Fp).
Now the mod p homology of any space is always an Fp vector space. In fact it’s more than that — it’s an Fp cocommutative coalgebra. Elementary messing about with category theory shows that if the representing space Y is a group up to homotopy, then H*(Y,Fp) is a group in the category of Fp coalgebras — otherwise better known as a Hopf algebra. Similarly, if the representable functor takes values in, say, rings, Y will be a ring-space and its homology will be a ring object in the category of coalgebras, an object called by some a Hopf ring, or (better when you start to deal with these in the necessary generality) a coalgebraic ring. Ravenel and Wilson showed the utility of considering this very rich structure when they applied it to describing a number of examples of particular representable functors and then used them to solve, eg, the Conner Floyd conjecture.
My interest in this has become to try to understand the relationship between the properties of the functor and topological properties of the representing spaces — not at the level of individual examples, but globally. There are many cases where related functors have clearly had relationships between their representing spaces, but this is only to be seen by ad-hoc calculation; my aim has been to develop the language to enable explanations at a structural level to be approached.
The task starts by needing to produce the standard tools of homological and commutative algebra in the exotic setting as algebraic objects in categories of coalgebras: just as a ring (in the usual sense) could (longwindedly) be described as a `ring object in the category of sets’, the subject of coalgberaic algebra studies rings, modules, resolutions, derived functors etc, in the category of cocommutative coalgebras.
This can be done [HT1]; moreover, the results can be used to go on and prove topological theorems. In [HT2] we showed that when functors are related by the property of one being `exact’ over the other, then, in the right `coalgebraic algebra’ language, there is a very simple relationship between the homology of their representing objects, and hence of their operation algebras. This automatically gives rise to a number of new families of functors whose representing spaces are now described.
In [H] the topic of completion is studied and again simple descriptions of what happens to the representing spaces of a functor when the functor is completed at an appropriate ideal are deduced. This has a number of consequences. In a project [GHS] with the topologists in Wuppertal, the knowledge of such representing spaces is used to give a new, geometric filtration of the group cohomology of a finite group; in work with Martin Bendersky [BH] these ideas are used to produce computable vn periodic unstable Adams spectral sequences for the first time.
[BH] M.Bendersky and J.Hunton, The Hopf ring for Morava E theory, Proceedings of the Edinburgh Mathematical Society 47 (2004), 513-532.
[GHS] D.Green, J.Hunton and B.Schuster, Chromatic characteristic classes in ordinary group cohomology, Topology, 42 (2003), 243-263.
[HT1] J. Hunton and P.Turner, Coalgebraic algebra, Journal of Pure and Applied Algebra 129 (1998), 297-313.
[HT2] J. Hunton and P.Turner, The homology of spaces representing exact pairs of homotopy functors, Topology 38 (1999), 621-634.
[H] J.Hunton, Complete cohomology theories and the homology of their omega spectra, Topology 41 (2002), 931-943.