And Still Another Definition of ∞-categories

The aim of this paper is to present a simplified version of the notion of ∞-groupoid developed by Grothendieck in “Pursuing Stacks” and to introduce a definition of ∞-categories inspired by Grothendieck’s approach. Introduction The precise definition of Grothendieck ∞-groupoids [5, sections 1-13] has been presented in [7]. In this paper, we give a slightly simplified version of this notion, and a variant leading to a definition of (weak) ∞-categories [8], very close to Batanin’s operadic definition [2]. The precise relationship between these two notions is investigated by Ara in [1]. The basic intuition leading to the definition of a ∞-groupoid is presented as follows by Grothendieck (for a ∞-groupoid F , with set of i-cells Fi): “Intuitively, it means that whenever we have two ways of associating to a finite family (ui)i∈I of objects of an ∞-groupoid, ui ∈ Fn(i), subjected to a “standard” set of relations on the ui’s, an element of some Fn, in terms of the ∞-groupoid structure only, then we have automatically a “homotopy” between these built-in in the very structure of the ∞-groupoid, provided it makes at all sense to ask for one . . . ” [5, section 9]. This leads him to the notion of coherator, category C endowed with a “universal ∞-cogroupoid”, a ∞-groupoid being a presheaf on C satisfying some left exactness conditions, improperly called Segal conditions in the literature. In particular, Grothendieck ∞-groupoids define an algebraic structure species, and the category of ∞-groupoids is locally presentable. The notion of a ∞-groupoid depends on the choice of a coherator. Two different coherators give rise in general to non-equivalent categories of ∞-groupoids. Nevertheless, the two notions of∞-groupoid are expected to be equivalent in some subtler way. Grothendieck illustrates this fact as follows: “Roughly saying, two different mathematicians, working independently on the conceptual problem I had in mind, assuming they both wind up with some explicit definition, will almost certainly get non-equivalent definitions – namely with non-equivalent categories of (set-valued, say) ∞-groupoids! And, secondly and as importantly, that this ambiguity however is an irrelevant one. To make this point a little clearer, I could say that a third mathematician, informed of the work of both, will readily think out a functor or rather a pair of functors, associating to any structure of Mr. X one of Mr. Y and conversely, in such a way that by composition of the two, we will associate to a X-structure (T say) another T ′, which will not be isomorphic to T of course, but endowed with a canonical ∞-equivalence (in the sense of Mr. X ) T ' ∞ T ′, and the same on the Mr. Y side. Most probably, a fourth mathematician, faced with the same situation as the third, will get his own pair of functors to reconcile Mr. X 2000 Mathematics Subject Classification. 18A30, 18B40, 18C10, 18C30, 18C35, 18D05, 18D50, 18E35, 18G50, 18G55, 55P10, 55P15, 55Q05, 55U35, 55U40.