Homomorphisms of higher categories

Abstract

We describe a construction that to each algebraically specified notion of higher-dimensional category associates a notion of homomorphism which preserves the categorical structure only up to weakly invertible higher cells. The construction is such that these homomorphisms admit a strictly associative and unital composition. We give two applications of this construction. The first is to tricategories; and here we do not obtain the trihomomorphisms defined by Gordon, Power and Street, but only something equivalent in a suitable sense. The second is to Batanin's weak omega-categories.

40 pages; v2: hand-waving arguments replaced by proofs; v3: final journal version

References in corpus (7)

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Cited by in corpus (19)

• A Quillen model structure for Gray-categories
• Algebraic weak factorisation systems I: accessible AWFS
• Algebraic model structures
• The Frobenius Condition, Right Properness, and Uniform Fibrations
• Algebraic weak factorisation systems II: categories of weak maps
• Weak omega-categories from intensional type theory
• On the construction of functorial factorizations for model categories
• Pointed homotopy and pointed lax homotopy of 2-crossed module maps
• Multitensor lifting and strictly unital higher category theory
• The Coalgebraic Structure of Cell Complexes
• Mapping Spaces of Gray-Categories
• Weakly invertible cells in a weak $\omega$-category
• Computads for weak $\omega$-categories as an inductive type
• CaTT contexts are finite computads
• $\omega$-weak equivalences between weak $\omega$-categories
• A Syntax for Strictly Associative and Unital $\infty$-Categories
• Computads for generalised signatures
• Elimination of quotients in various localisations of premodels into models
• Hom weak $\omega$-categories of a weak $\omega$-category