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By Kamps K.H., Porter T.

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Math. , Providence, 1998, pp. 99–116. [V] Vogt, R. : Homotopy limits and colimits, Math. Z. 134 (1973), 11–52.

Structures 7 (1999), 227–260. : Pursuing stacks, Manuscript, 1984. Gray, J. : Formal Category Theory: Adjointness for 2-Categories, Lecture Notes in Math. 391, Springer, Berlin, 1974. Guin-Walery, D. : Obstruction a` l’excision en K-th´eorie alg´ebrique, In: Evanston Conference, 1980, Lecture Notes in Math. 854, Springer, New York, 1981, pp. 179–216. Hardie, K. , Kamps, K. H. and Kieboom, R. : A homotopy 2-groupoid of a Hausdorff space, Appl. Categ. Structures 8 (2000), 209–234. 2-GROUPOID ENRICHMENTS IN HOMOTOPY THEORY 409 [JS] Joyal, A.

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’. This is to handle triples of composable maps in G, g1 g2 g3 x0 −→ x1 −→ x2 −→ x3 404 K. H. KAMPS AND T. PORTER and might be called a 3-cocycle condition. ) We are, however, approaching the limits of what is known at this point, so need to look where we are.