Higher Category Theory

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10 domain-specific abstractions whose origin domain is Higher Category Theory.

• 2-group — A monoidal groupoid in which every object has a weak inverse, categorifying the notion of a group.
• 2-Yoneda lemma — A bicategorical generalization of Yoneda identifying pseudonatural transformations from a representable pseudofunctor to F with the category F assigns to the representing object.
• Accessible quasi-category — An infinity-category equivalent to the closure of a small infinity-category under kappa-filtered colimits for some regular cardinal kappa.
• Globular set — A sequence of sets of n-cells with source and target maps satisfying globularity equations, forming the presheaf carrier for many higher-category structures.
• Joyal Model Structure — The model structure on simplicial sets whose cofibrations are monomorphisms, weak equivalences are categorical equivalences, and fibrant objects are quasi-categories.
• Opposite simplicial set — The simplicial set obtained by precomposing with the order-reversing automorphism of the simplex category, extending categorical arrow reversal to higher categorical models.
• Quasi-category — A simplicial set satisfying every inner horn-filling condition, modeling an infinity-category with composition coherent up to higher homotopy.
• Simplicially enriched category — A category whose hom-objects are simplicial sets and whose composition and identities are simplicial maps, encoding higher homotopies between morphisms.
• Tetracategory — A weak four-dimensional categorical structure in which composition and coherence extend tricategorical cells by one dimension rather than holding strictly.
• Weak n-category — Organize cells through dimension n with composition and units that satisfy associativity and unitality up to coherently related higher cells rather than by strict equality.