By Kamps K.H., Porter T.

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We have not touched on the abstract homotopy that can be done within a (G2 , ⊗)-category except in the construction of c(CB ), but throughout the intuition we have pushed to the fore has been homotopic and geometric rather than categorical, not that the deep categorical insights and methods are irrelevant, but merely since they are deep and, therefore, can seem incomprehensible to a less categorically minded reader interested in trying to evaluate this part of higher dimensional algebra for its potential usefulness or interest as a research area.

Cr]), but for convenience have inverted the interchange 3-cell, adjusting throughout. A Gray category C consists of collections C0 of objects, C1 of arrows, C2 of 2-arrows and C3 of 3-arrows together with • functions sn , tn : Ci −→ Cn for all 0 n < i 3 called n-source and n-target, • functions n : Cn+1 sn ×tn Cn+1 −→ Cn+1 for all 0 n < 3 called vertical composition, • functions n : Ci sn ×tn Cn+1 −→ Ci and n : Cn+1 sn ×tn Ci −→ Ci for all 0 n 1, n + 1 < i 3, called whiskering, • a function 0 : C2 s0 ×t0 C2 −→ C3 , called horizontal composition, and • functions id− : Ci −→ Ci+1 for all 0 i 2, called identity, such that (i) C is a globular set, (ii) for every C, C ∈ C0 , the collection of elements of C with 0-source C and 0-target C forms a 2-category C(C, C ) with n-composition in C(C, C ) given by n+1 and identities given by id− , (iii) for every g : C −→ C in C1 and every C and C in C0 , − 0 g is a 2-functor, C(C , C ) −→ C(C , C ) and g 0 − is a 2-functor C(C, C ) −→ C(C, C ), (iv) for every C in C0 and every C, C in C0 , − 0 idC , is equal to the identity functor on C(C , C ) and idC 0 − is equal to the identity functor on C(C, C ), (v) for every γ , δ ∈ C2 with t0 (γ ) = s0 (δ), γ : f ⇒ f and δ : g ⇒ g , s2 (δ t2 (δ 0 0 γ ) = (δ γ ) = (g f) 0 γ) 0 1 1 (g (δ 0 0 γ ), f) and δ 0 γ is an iso − 3-arrow, (It may help to draw the diagrams to identify what this axiom states.

MP] Mutlu, A. : Iterated Peiffer pairings in the Moore complex of a simplicial group, Appl. Categ. Struct. 9 (2001), 111–130. : Abstract homotopy theory: the interaction of category theory and homotopy theory, Preprint, 2001. : Homotopy types of strict 3-groupoids, e-print math. CT/9810059, Toulouse, 1998. html. : Categorical structures, In: Handbook of Algebra, Vol. 1, Elsevier, Amsterdam, 1996, pp. 529–577. : The role of Batanin’s monoidal globular categories, In: E. Getzler et al. (eds), Higher Category Theory.

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

by George

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