2-Groupoid Enrichments in Homotopy Theory and Algebra by Kamps K.H., Porter T.

By Kamps K.H., Porter T.

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E. g. a (G2 , ⊗)-enrichment that can be usefully derived? ) A second potential application is within Grothendieck’s Pursuing Stacks [Gro]. That programme requires the study of actions of n-types on (n − 1)-types. A 3-type is representable by a (G2 , ⊗)-groupoid as we have noted and the category of 2types ‘is’ the ‘category’ of 2-groupoids. An action of a 3-type on a 2-type can thus be studied via (G2 , ⊗)-functors (or lax-versions of them) from a 3-type model to the (G2 , ⊗)-category of 2-groupoids itself.

The ‘obvious’ way to attempt to get around this difficulty is to have, for each pair (g1 , g2 ) of composable arrows in G, a 3-arrow α(g2 , g1 ) : H (g2 0 g1 ) −→ (Y (g2 ) 0 H (g1 )) 1 (H (g2 ) 0 X(g1 )) to replace equality in the ‘functoriality condition’ (∗). Thus now a morphism from X to Y is a triple (α, H, f ) : f assigning 1-arrows of C to objects of G, H assigning 2-arrows of C to 1-arrows of G and α, now, assigning 3-arrows of C to pairs of 1-arrows of G. This handles the composability at levels 1 and 2, but there has to be a condition to replace ‘functoriality’.

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