28 research outputs found
Gabriel-Ulmer duality for topoi and its relation with site presentations
Let be a regular cardinal. We study Gabriel-Ulmer duality when one
restricts the 2-category of locally -presentable categories with
-accessible right adjoints to its locally full sub-2-category of
-presentable Grothendieck topoi with geometric -accessible
morphisms. In particular, we provide a full understanding of the locally full
sub-2-category of the 2-category of -small cocomplete categories with
-colimit preserving functors arising as the corresponding 2-category of
presentations via the restriction. We analyse the relation of these
presentations of Grothendieck topoi with site presentations and we show that
the 2-category of locally -presentable Grothendieck topoi with
geometric -accessible morphisms is a reflective sub-bicategory of the
full sub-2-category of the 2-category of sites with morphisms of sites
genearated by the weakly -ary sites in the sense of Shulman [37].Comment: 25 page
Monads with arities and their associated theories
After a review of the concept of "monad with arities" we show that the
category of algebras for such a monad has a canonical dense generator. This is
used to extend the correspondence between finitary monads on sets and Lawvere's
algebraic theories to a general correspondence between monads and theories for
a given category with arities. As application we determine arities for the free
groupoid monad on involutive graphs and recover the symmetric simplicial nerve
characterisation of groupoids.Comment: New introduction; Section 1 shortened and redispatched with Section
2; Subsections on symmetric operads (3.14) and symmetric simplicial sets
(4.17) added; Bibliography complete
Accessibility and presentability in 2-categories
We outline a definition of accessible and presentable objects in a 2-category
endowed with a Yoneda structure; this perspective suggests a
unified treatment of many "Gabriel-Ulmer like" theorems (like the classical
Gabriel-Ulmer representation for locally presentable categories, Giraud
theorem, and Gabriel-Popescu theorem), asserting how presentable objects arise
as reflections of generating ones. In a 2-category with a Yoneda structure, two
non-equivalent definitions of presentability for can in
principle be given: in the most interesting, it is generally false that all
presheaf objects are presentable; this leads to the
definition of a Gabriel-Ulmer structure, i.e. a Yoneda structure rich enough to
concoct Gabriel-Ulmer duality and to make this asymmetry disappear. We end the
paper with a roundup of examples, involving classical (set-based and enriched),
low dimensional and higher dimensional category theory.Comment: 28 pages, revised versio
Healthiness from Duality
Healthiness is a good old question in program logics that dates back to
Dijkstra. It asks for an intrinsic characterization of those predicate
transformers which arise as the (backward) interpretation of a certain class of
programs. There are several results known for healthiness conditions: for
deterministic programs, nondeterministic ones, probabilistic ones, etc.
Building upon our previous works on so-called state-and-effect triangles, we
contribute a unified categorical framework for investigating healthiness
conditions. We find the framework to be centered around a dual adjunction
induced by a dualizing object, together with our notion of relative
Eilenberg-Moore algebra playing fundamental roles too. The latter notion seems
interesting in its own right in the context of monads, Lawvere theories and
enriched categories.Comment: 13 pages, Extended version with appendices of a paper accepted to
LICS 201
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
Lawvere theories enriched over a general base
AbstractWe generalise the correspondence between Lawvere theories and finitary monads on Set in two ways. First, we allow our theories to be enriched in a category V that is locally finitely presentable as a symmetric monoidal closed category: symmetry is convenient but not necessary. And second, we allow the arities of our theories to be finitely presentable objects of a locally finitely presentable V-category A. We call the resulting notion that of a Lawvere A-theory. We extend the correspondence for ordinary Lawvere theories to one between Lawvere A-theories and finitary V-monads on A. We illustrate this with examples leading up to that of the Lawvere Cat-theory for cartesian closed categories, i.e., the Set-enriched theory on the category Cat for which the models are all small cartesian closed categories. We also briefly investigate change-of-base
String diagram rewrite theory II: Rewriting with symmetric monoidal structure
Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras