Types are weak ω‐groupoids

By Benno van den Berg, Richard Garner. 2008

157 citations recorded by the source corpus.

We define a notion of weak ω‐category internal to a model of Martin‐Löf's type theory, and prove that each type bears a canonical weak ω‐category structure obtained from the tower of iterated identity types over that type. We show that the ω‐categories arising in this way are in fact ω‐groupoids.

Subfields: Artificial Intelligence, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2acca20b9515366265884b9c5c0a037ffaf4af75

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In Artificial Intelligence

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Citations in the corpus

Intuitionistic Type Theory (2016, background): “or in sequential form (1) J1 axiom (2) J2 axiom (3) J3 by rule r1 from (1) and (2) (4) J4 axiom (5) J5 by rule r2 from (4) (6) J6 axiom (7) J7 by rule r3 from(5) and (6) (8) J8 by rule r4 from (3) and (7) The latter form is common in mathematical arguments.”
A type theory for synthetic ∞-categories (2017, background): “It then automatically follows, as a meta-theorem, that every type internally admits all the structure of an ∞-groupoid, as shown in [30, 17]; but in practical applications we rarely need more than one or two levels of this structure, and we can just “define it as we go”.”
The simplicial model of Univalent Foundations (after Voevodsky) (2012, unknown): “Other current and recent work on such models includes [GvdB11], [AK11], and [Shu15].”
The Frobenius condition, right properness, and uniform fibrations (2015, background): “uch that morphisms in I/Y induce morphisms of left I ⋔-maps. Remark 2.3. Recall that the Frobenius condition for a weak factorization system states that pullback along a right map preserves left maps [4, 16]. Given a weak factorization system (L,R) on a category E, we have that it satisﬁes the Frobe”