169 research outputs found
Variations on Algebra: monadicity and generalisations of equational theories
Dedicated to Rod Burstal
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
Behavior of Quillen (co)homology with respect to adjunctions
This paper aims to answer the following question: Given an adjunction between
two categories, how is Quillen (co)homology in one category related to that in
the other? We identify the induced comparison diagram, giving necessary and
sufficient conditions for it to arise, and describe the various comparison
maps. Examples are given. Along the way, we clarify some categorical
assumptions underlying Quillen (co)homology: cocomplete categories with a set
of small projective generators provide a convenient setup.Comment: Minor corrections. To appear in Homology, Homotopy and Application
Enriched Lawvere Theories for Operational Semantics
Enriched Lawvere theories are a generalization of Lawvere theories that allow
us to describe the operational semantics of formal systems. For example, a
graph enriched Lawvere theory describes structures that have a graph of
operations of each arity, where the vertices are operations and the edges are
rewrites between operations. Enriched theories can be used to equip systems
with operational semantics, and maps between enriching categories can serve to
translate between different forms of operational and denotational semantics.
The Grothendieck construction lets us study all models of all enriched theories
in all contexts in a single category. We illustrate these ideas with the
SKI-combinator calculus, a variable-free version of the lambda calculus.Comment: In Proceedings ACT 2019, arXiv:2009.0633
Multitensor lifting and strictly unital higher category theory
In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result -- the lifting theorem for multitensors -- enables us to see the Gray tensor product of 2-categories and the Crans tensor product of Gray categories as part of this framework. We define weak n-categories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak n-categories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)-categories with strict units
Multitensor lifting and strictly unital higher category theory
International audienceIn this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result - the lifting theorem for multitensors - enables us to see the Gray tensor product of 2-categories and the Crans tensor product of Gray categories as part of this framework. We define weak n-categories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak n-categories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)-categories with strict units
Flat vs. filtered colimits in the enriched context
The importance of accessible categories has been widely recognized; they can
be described as those freely generated in some precise sense by a small set of
objects and, because of that, satisfy many good properties. More specifically
finitely accessible categories can be characterized as: (a) free cocompletions
of small categories under filtered colimits, and (b) categories of flat
presheaves on some small category. The equivalence between (a) and (b) is what
makes the theory so general and fruitful.
Notions of enriched accessibility have also been considered in the literature
for various bases of enrichment, such as
and . The problem in
this context is that the equivalence between (a) and (b) is no longer true in
general. The aim of this paper is then to:
(1) give sufficient conditions on so that (a)
(b) holds;
(2) give sufficient conditions on so that (a)
(b) holds up to Cauchy completion;
(3) explore some examples not covered by (1) or (2).Comment: Revised version: major changes to the introduction, added some words
at the beginning of Sect. 3 and 4. To appear on Advances in Mathematic
Birkhoff's variety theorem for relative algebraic theories
An algebraic theory, sometimes called an equational theory, is a theory
defined by finitary operations and equations, such as the theories of groups
and of rings. It is well known that algebraic theories are equivalent to
finitary monads on . In this paper, we generalize this phenomenon
to locally finitely presentable categories using partial Horn logic. For each
locally finitely presentable category , we define an "algebraic
concept" relative to , which will be called an
-relative algebraic theory, and show that -relative
algebraic theories are equivalent to finitary monads on . Finally,
we generalize Birkhoff's variety theorem for classical algebraic theories to
our relative algebraic theories.Comment: 34 page
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