73 research outputs found
A type theory for synthetic -categories
We propose foundations for a synthetic theory of -categories
within homotopy type theory. We axiomatize a directed interval type, then
define higher simplices from it and use them to probe the internal categorical
structures of arbitrary types. We define Segal types, in which binary
composites exist uniquely up to homotopy; this automatically ensures
composition is coherently associative and unital at all dimensions. We define
Rezk types, in which the categorical isomorphisms are additionally equivalent
to the type-theoretic identities - a "local univalence" condition. And we
define covariant fibrations, which are type families varying functorially over
a Segal type, and prove a "dependent Yoneda lemma" that can be viewed as a
directed form of the usual elimination rule for identity types. We conclude by
studying homotopically correct adjunctions between Segal types, and showing
that for a functor between Rezk types to have an adjoint is a mere proposition.
To make the bookkeeping in such proofs manageable, we use a three-layered
type theory with shapes, whose contexts are extended by polytopes within
directed cubes, which can be abstracted over using "extension types" that
generalize the path-types of cubical type theory. In an appendix, we describe
the motivating semantics in the Reedy model structure on bisimplicial sets, in
which our Segal and Rezk types correspond to Segal spaces and complete Segal
spaces.Comment: 78 pages; v2 has minor updates inspired by discussions at the
Mathematics Research Community on Homotopy Type Theory; v3 incorporates many
expository improvements suggested by the referee; v4 is the final journal
version to appear in Higher Structures with a more precise syntax for our
type theory with shape
Kan extensions and the calculus of modules for -categories
Various models of -categories, including quasi-categories,
complete Segal spaces, Segal categories, and naturally marked simplicial sets
can be considered as the objects of an -cosmos. In a generic
-cosmos, whose objects we call -categories, we introduce
modules (also called profunctors or correspondences) between
-categories, incarnated as as spans of suitably-defined fibrations with
groupoidal fibers. As the name suggests, a module from to is an
-category equipped with a left action of and a right action of ,
in a suitable sense. Applying the fibrational form of the Yoneda lemma, we
develop a general calculus of modules, proving that they naturally assemble
into a multicategory-like structure called a virtual equipment, which is known
to be a robust setting in which to develop formal category theory. Using the
calculus of modules, it is straightforward to define and study pointwise Kan
extensions, which we relate, in the case of cartesian closed -cosmoi,
to limits and colimits of diagrams valued in an -category, as
introduced in previous work.Comment: 84 pages; a sequel to arXiv:1506.05500; v2. new results added, axiom
circularity removed; v3. final journal version to appear in Alg. Geom. To
Coalgebraic models for combinatorial model categories
We show that the category of algebraically cofibrant objects in a
combinatorial and simplicial model category A has a model structure that is
left-induced from that on A. In particular it follows that any presentable
model category is Quillen equivalent (via a single Quillen equivalence) to one
in which all objects are cofibrant.Comment: 12 pages; v2: final journal version with minor improvements suggested
by the refere
Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions
Consider a diagram of quasi-categories that admit and functors that preserve
limits or colimits of a fixed shape. We show that any weighted limit whose
weight is a projective cofibrant simplicial functor is again a quasi-category
admitting these (co)limits and that they are preserved by the functors in the
limit cone. In particular, the Bousfield-Kan homotopy limit of a diagram of
quasi-categories admit any limits or colimits existing in and preserved by the
functors in that diagram. In previous work, we demonstrated that the
quasi-category of algebras for a homotopy coherent monad could be described as
a weighted limit with projective cofibrant weight, so these results immediately
provide us with important (co)completeness results for quasi-categories of
algebras. These generalise most of the classical categorical results, except
for a well known theorem which shows that limits lift to the category of
algebras for any monad, regardless of whether its functor part preserves those
limits. The second half of this paper establishes this more general result in
the quasi-categorical setting: showing that the monadic forgetful functor of
the quasi-category of algebras for a homotopy coherent monad creates all limits
that exist in the base quasi-category, without further assumption on the monad.
This proof relies upon a more delicate and explicit analysis of the particular
weight used to define quasi-categories of algebras.Comment: 33 pages; a sequel to arXiv:1306.5144 and arXiv:1310.8279; v3: final
journal version with updated internal references to the new version of
"Homotopy coherent adjunctions and the formal theory of monads
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