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KAMPS AND T. PORTER Appendix For convenience we include here a sketch of Crans description of Gray categories as algebraic structures (cf. [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.

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

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