Abstract

It has been difficult to see precisely the role played by strict n-categories in the nascent theory of n-categories, particularly as related to n-truncated homotopy types of spaces. We propose to show in a fairly general setting that one cannot obtain all 3-types by any reasonable realization functor 1 from strict 3-groupoids (i.e. groupoids in the sense of [20]). More precisely we show that one does not obtain the 3-type of S 2. The basic reason is that the Whitehead bracket is nonzero. This phenomenon is actually well-known, but in order to take into account the possibility of an arbitrary reasonable realization functor we have to write the argument in a particular way. We start by recalling the notion of strict n-category. Then we look at the notion of strict n-groupoid as defined by Kapranov and Voevodsky [20]. We show that their definition is equivalent to a couple of other natural-looking definitions (one of these equivalences was left as an exercise in [20]). At the end of these first sections, we have a picture of strict 3-groupoids having only one object and one 1-morphism, as being equivalent to abelian monoidal objects (G, +) in the category of groupoids, such that (π0(G), +) is a group. In the case in question, this group will be π2(S 2) = Z. Then comes the main

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