30 research outputs found
Homotopy locally presentable enriched categories
We develop a homotopy theory of categories enriched in a monoidal model
category V. In particular, we deal with homotopy weighted limits and colimits,
and homotopy local presentability. The main result, which was known for
simplicially-enriched categories, links homotopy locally presentable
V-categories with combinatorial model V-categories, in the case where has all
objects of V are cofibrant.Comment: 48 pages. Significant changes in v2, especially in the last sectio
Abstract elementary classes and accessible categories
We compare abstract elementary classes of Shelah with accessible categories
having directed colimits
Strongly Complete Logics for Coalgebras
Coalgebras for a functor model different types of transition systems in a
uniform way. This paper focuses on a uniform account of finitary logics for
set-based coalgebras. In particular, a general construction of a logic from an
arbitrary set-functor is given and proven to be strongly complete under
additional assumptions. We proceed in three parts. Part I argues that sifted
colimit preserving functors are those functors that preserve universal
algebraic structure. Our main theorem here states that a functor preserves
sifted colimits if and only if it has a finitary presentation by operations and
equations. Moreover, the presentation of the category of algebras for the
functor is obtained compositionally from the presentations of the underlying
category and of the functor. Part II investigates algebras for a functor over
ind-completions and extends the theorem of J{\'o}nsson and Tarski on canonical
extensions of Boolean algebras with operators to this setting. Part III shows,
based on Part I, how to associate a finitary logic to any finite-sets
preserving functor T. Based on Part II we prove the logic to be strongly
complete under a reasonable condition on T
Tameness in generalized metric structures
We broaden the framework of metric abstract elementary classes (mAECs) in
several essential ways, chiefly by allowing the metric to take values in a
well-behaved quantale. As a proof of concept we show that the result of Boney
and Zambrano on (metric) tameness under a large cardinal assumption holds in
this more general context. We briefly consider a further generalization to
partial metric spaces, and hint at connections to classes of fuzzy structures,
and structures on sheaves
Notions of Lawvere theory
Categorical universal algebra can be developed either using Lawvere theories
(single-sorted finite product theories) or using monads, and the category of
Lawvere theories is equivalent to the category of finitary monads on Set. We
show how this equivalence, and the basic results of universal algebra, can be
generalized in three ways: replacing Set by another category, working in an
enriched setting, and by working with another class of limits than finite
products.
An important special case involves working with sifted-colimit-preserving
monads rather than filtered-colimit-preserving ones.Comment: 27 pages. v2 minor changes, final version, to appear in Applied
Categorical Structure