19,461 research outputs found
Identity types and weak factorization systems in Cauchy complete categories
It has been known that categorical interpretations of dependent type theory
with Sigma- and Id-types induce weak factorization systems. When one has a weak
factorization system (L, R) on a category C in hand, it is then natural to ask
whether or not (L, R) harbors an interpretation of dependent type theory with
Sigma- and Id- (and possibly Pi-) types. Using the framework of display map
categories to phrase this question more precisely, one would ask whether or not
there exists a class D of morphisms of C such that the retract closure of D is
the class R and the pair (C, D) forms a display map category modeling Sigma-
and Id- (and possibly Pi-) types. In this paper, we show, with the hypothesis
that C is Cauchy complete, that there exists such a class D if and only if
(C,R) itself forms a display map category modeling Sigma- and Id- (and possibly
Pi-) types. Thus, we reduce the search space of our original question from a
potentially proper class to a singleton.Comment: 14 page
Lifting accessible model structures
A Quillen model structure is presented by an interacting pair of weak
factorization systems. We prove that in the world of locally presentable
categories, any weak factorization system with accessible functorial
factorizations can be lifted along either a left or a right adjoint. It follows
that accessible model structures on locally presentable categories - ones
admitting accessible functorial factorizations, a class that includes all
combinatorial model structures but others besides - can be lifted along either
a left or a right adjoint if and only if an essential "acyclicity" condition
holds. A similar result was claimed in a paper of Hess-Kedziorek-Riehl-Shipley,
but the proof given there was incorrect. In this note, we explain this error
and give a correction, and also provide a new statement and a different proof
of the theorem which is more tractable for homotopy-theoretic applications.Comment: This paper corrects an error in the proof of Corollary 3.3.4 of "A
necessary and sufficient condition for induced model structures"
arXiv:1509.0815
Completion, closure, and density relative to a monad, with examples in functional analysis and sheaf theory
Given a monad T on a suitable enriched category B equipped with a proper
factorization system (E,M), we define notions of T-completion, T-closure, and
T-density. We show that not only the familiar notions of completion, closure,
and density in normed vector spaces, but also the notions of sheafification,
closure, and density with respect to a Lawvere-Tierney topology, are instances
of the given abstract notions. The process of T-completion is equally the
enriched idempotent monad associated to T (which we call the idempotent core of
T), and we show that it exists as soon as every morphism in B factors as a
T-dense morphism followed by a T-closed M-embedding. The latter hypothesis is
satisfied as soon as B has certain pullbacks as well as wide intersections of
M-embeddings. Hence the resulting theorem on the existence of the idempotent
core of an enriched monad entails Fakir's existence result in the non-enriched
case, as well as adjoint functor factorization results of Applegate-Tierney and
Day
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