Tim Porter algebraic models for homotopy n-types

Algebraic models for homotopy n-types:

The problem of the classification of homotopy $n$-types was one that had been formulated by J. H. C. Whitehead in his fundamental work on Combinatorial Homotopy? in about 1949. The idea of a homotopy $n$-type is that of a space for which all the homotopy groups of order higher than n are trivial. It was known that 1-types corresponded to groups. With Saunders Mac Lane, Whitehead (1950) gave a neat class of algebraic models for 2-types, namely crossed modules.

In the 1970s there was work by various researchers, notably R. Brown (Bangor) and P.J. Higgins (Durham) on ways of calculating the 2-type, and also on crossed complexes which modelled a restricted class of homotopy types including the 2-types. This shed more light on the area, but there were few other significant advances on this specific problem until Loday (1982) published a proof of a general case involving algebraic objects that he called cat n groups. These modelled (n+1)-types and extended the crossed modules in an interesting way.

Unfortunately there were some details wrong in his proof and it was also quite complicated, using various technical tools from algebraic topology. Even when the holes in Loday’s proof were patched, the result was not functorial in its construction. It was functorial ‘up to homotopy’, which was fine for the result as such but not so good for use outside that. The problem of the functorial modelling of homotopy n-types in detail was important for Grothendieck’s conjectural correspondence between (n+1)-types and n-stacks and for that theory, there was also a need for a detailed homotopy theory of the models. Following the example of crossed modules, this would require an extension of cat n groups to some type of complex, extending the crossed complexes of Brown and Higgins. Porter’s contributions to this list of problems were included the following:

• In the late 1980s, (but published in Topology 1993), Porter simplified, and corrected Loday’s proof, by avoiding his repeated use of the passage to and fro between the algebra and the topology. The result used some basic group theory together with some fairly simple simplicial homotopy and simplicial group theory to give a proof that not only gave the result in general but was functorial and related the structures in neighbouring dimensions, so that crossed n-cubes, the algebraic models for n+1-types, could be considered as morphisms of crossed (n-1)-cubes. The direct construction even gave the quotient of such a morphism as being the model for the n-type whilst the kernel was the (n+1)th homotopy group. (A related proof was published by Bullejos, Cegarra and Duskin slightly later. It uses more simplicial methods. Both approaches have been generalised to give new methods for some technically and conceptually hard areas of non-Abelian homological algebra.)

• Porter’s proof of Loday’s theorem used simplicial groups in an essential way. The algebraic structure had been investigated by Pilar Carrasco in 1987. The theory used for Loday’s theorem had a clear generalisation to a wider range of homotopy types. Within a homotopy n-type there will usually be some non-trivial Whitehead products, but as the homotopy groups are trivial from dimension n+1 onwards, this structure was bound to be trivial above that dimension. For the case of n=2, it was known that requiring all Whitehead products to be trivial (above dimension 2) led to the class of spaces modelled by the crossed complexes of Brown and Higgins, and for which there was a beautifully structured homotopy theory. The next step in understanding homotopy types from the bottom above (as opposed to, say, the invariants studied by stable homotopy theory) therefore seemed to be to examine the algebraic models that combined both a structure as revealed by Loday and a crossed complex type structure above dimension n. With ex-student Ehlers, Porter gave a complete description of the simplicial group(oid)s that corresponded to such homotopy types. Somewhat surprisingly, each of these classes formed a variety in the category of all simplicial groupoids, and moreover the reflection functor from all simplicial groupoids to this reflexive subcategory was given by an explicit formula generalising both that given by Porter’s construction of Loday’s models, and by Carrasco’s model for crossed complexes. The varieties corresponding to each dimension are nested with the extra structure at each level being explicitly given. (Current research by Porter in collaboration with Carassco is to investigate the possibility of using this technology to produce a non-Abelian tensor product in this context, with the aim of application in both algebraic topology and homological algebra.)

• The general method of Porter’s proof of Loday’s theorem has important consequences in homological algebra. This is due to its simplicity and hence its ubiquity as it can be applied in many contexts. For instance, the classical Hopf formula for the cohomology of groups looks very like the formula that results from low dimensional versions of that method. Higher dimensional versions of the Hopf formula, due to Brown and Ellis, show this even more clearly. With Donadze and Inassaridze, Porter used the explicit methods of that construction to set up a general theory of n-fold Čech derived functors, and illustrated the methods of this theory to generalise further the Brown–Ellis higher Hopf formulae.

• Porter working with a student, Mutlu, continued the analysis of the structure of simplicial groups started by Carrasco. Together they revealed a new set of pairings in the Moore complex of a simplicial group, generalising the Peiffer pairing for 2-crossed modules given by Conduché. Their methods give new tools for the treatment of structure akin to the Whitehead product, but also suggest explicit combinatorial links between these topologically motivated structures and structure known from the study of higher category theory. (This theory has been extended by workers in the non-Abelian homological algebra research group at Louvain-la-Neuve.)