Locally Cartesian Closed Quasicategories from Type Theory

We prove that the quasicategories arising from models of Martin-L\"of type theory via simplicial localization are locally cartesian closed.Comment: 21 pages; to appear in J. Topolog

Quasicategories of Frames of Cofibration Categories

We show that the quasicategory of frames of a cofibration category, introduced by the second-named author, is equivalent to its simplicial localization.Comment: 22 page

The Simplicial Model of Univalent Foundations (after Voevodsky)

We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to obtain coherence. We then construct a (weakly) universal Kan fibration, and use it to exhibit a model in simplicial sets. Lastly, we introduce the Univalence Axiom, in several equivalent formulations, and show that it holds in our model. As a corollary, we conclude that Martin-L\"of type theory with one univalent universe (formulated in terms of contextual categories) is at least as consistent as ZFC with two inaccessible cardinals.Comment: 50 pages. V5: final journal version, to appear in Journal of the European Mathematical Society; no change in theorem numbering. Homotopy-theoretic portions appear also in the note "Univalence in Simplicial Sets", arXiv:1203.255

Homotopical inverse diagrams in categories with attributes

We define and develop the infrastructure of homotopical inverse diagrams in categories with attributes. Specifically, given a category with attributes $C$ and an ordered homotopical inverse category $I$, we construct the category with attributes $C^I$ of homotopical diagrams of shape $I$ in $C$ and Reedy types over these; and we show how various logical structure ($\Pi$-types, identity types, and so on) lifts from $C$ to $C^I$. This may be seen as providing a general class of diagram models of type theory. In a companion paper "The homotopy theory of type theories" (arXiv:1610.00037), we apply the present results to construct semi-model structures on categories of contextual categories.Comment: v3: various minor revisions for publication version; no change in theorem numberin

A cubical model for (∞,n)(\infty, n)-categories

We propose a new model for the theory of $(\infty,n)$-categories (including the case $n=\infty$) in the category of marked cubical sets with connections, similar in flavor to complicial sets of Verity. The model structure characterizing our model is shown to be monoidal with respect to suitably defined (lax and pseudo) Gray tensor products; in particular, these tensor products are both associative and biclosed. Furthermore, we show that the triangulation functor to pre-complicial sets is a left Quillen functor and is strong monoidal with respect to both Gray tensor products.Comment: submitted; 38 pages; v2 minor revision

Univalent categories and the Rezk completion

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of "category" for which equality and equivalence of categories agree. Such categories satisfy a version of the Univalence Axiom, saying that the type of isomorphisms between any two objects is equivalent to the identity type between these objects; we call them "saturated" or "univalent" categories. Moreover, we show that any category is weakly equivalent to a univalent one in a universal way. In homotopical and higher-categorical semantics, this construction corresponds to a truncated version of the Rezk completion for Segal spaces, and also to the stack completion of a prestack.Comment: 27 pages, ancillary files contain formalized proofs in the proof assistant Coq; v2: version for publication in Math. Struct. in Comp. Sci., incorporating suggestions by referees and Voevodsk

Homotopy limits in type theory

Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.Comment: 33 pages; v3: theorem numbering changed since v2 to match journal versio

Univalence in Simplicial Sets

We present an accessible account of Voevodsky's construction of a univalent universe of Kan fibrations.Comment: 13 pages. Not intended for publication. An extended version appears as "The Simplicial Model of Univalent Foundations (after Voevodsky)", arXiv:1211.2851, presenting the logical as well as the homotopy-theoretic aspects of the mode

Reply to Comment on "Non-monotonicity in the Quantum-Classical Transition: Chaos Induced by Quantum Effects"

Response to comment by Finn et al.Comment: 1 page, 1 figure, response to arxiv: 0903.341

Cubical models of (∞,1)(\infty, 1)-categories

We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We show that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor. As an application, we show that cubical quasicategories admit a convenient notion of a mapping space, which we use to characterize the weak equivalences between fibrant objects in our model structure as DK-equivalences.Comment: 98 pages; submitted for publication, comments still welcome; v2 significant changes, including two new section