Weak n-categories are an environment for non-abelian homological algebra

Sjoerd Crans, Seminari de Topologica 2 February 2001

Homological algebra is built on chain complexes: one takes one's favourite structure, constructs a chain complex (for example by resolutions) and takes its homology. Thus, for non-abelian homological algebra, one needs a non-abelian version of chain complexes.

A first attempt is to take crossed complexes [R. Brown and P. J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981), 233--260].
The category of crossed complexes is equivalent to the category of omega-groupoids [R. Brown and P. J. Higgins, The equivalence of infty-groupoids and crossed complexes, Cahiers Topologie Géom. Différentielle Catég. XXII (1981), 371--386].
The commutativity of the composition operations in an omega-groupoid or omega-category has the consequence that horizontal composition can be expressed in terms of vertical composition and `whiskering'.
Thus there is further room for un-abelianization. To this end, define a no-interchange category (NI-category for short) as a globular set with only ``vertical'' composition and whiskering operations Cp ×Cm Cq -> Cmax {p,q} for min {p,q} = m+1 which are associative and unital. A tas [obss] is a NI-category with further composition operations Cp ×Cm Cq -> Cp+q-m-1 for min {p,q} > m+1 which have to satisfy further axioms. A complete definition so far only exists up to dimension 6.
There are various methods to treat the associativity and unit axioms ``similarly'' - one can use higher operads or opetopes or multitopes or Theta-categories or ... - to arrive at a notion of weak omega-category if one does this to omega-categories or at would-be weak teisi from teisi.

Having progressed this far from chain complexes, there is the danger of having gone too far. One criterion for this is that in Ab the structure proposed for non-abelian homological algebra should reduce to the structure for (abelian) homological algebra one started from; if not, the new structure cannot properly be identified as being within the realm of non-abelian homological algebra---at most, it would go beyond it.
Now consider two composable elements of a tas or weak omega-category C in Ab: 
   f      g
A ---> B ---> C   .
Using the fact that compositions are group homomorphisms one calculates fairly easily that g #0 f = idq-1f - idp+q-1B + idp-1g. So, although horizontal composition lands, by definition, in Cp+q-1, the result is an identity, i.e., it factors through Cmax {p,q}. This also shows that any composition can be expressed in terms of addition; consequently, (binary) compositions again commute.

Now consider an arbitrary configuration of composable elements. One would like to use a similar argument as above in order to express a composition operation of this arity in terms of addition, more precisely, one wants to write this composite as a sum where each term is a composite ``all except possibly one of whose parts are identities''. The difficulty here lies in making this precise. Fortunately, this can actually be made precise, for any arity, and defining a weak omega-category to be normalized if any such composite is equal to (the identity on) the given (possibly) non-identity cell, one can indeed proceed similarly as before to prove the

Theorem  A normalized weak omega-category in Ab is a chain complex.

That this does not work for arbitrary weak omega-categories is not so serious a problem, because a coherence theorem saying that for any weak omega-category there exists a normalized weak omega-category weakly equivalent to it is generally expected.

This talk was based on the paper [teab], from where the results presented here have been taken.

A more extensive abstract of this talk is available as a ps-file.

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