169 research outputs found

    Variations on Algebra: monadicity and generalisations of equational theories

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    Dedicated to Rod Burstal

    Gabriel-Ulmer duality for topoi and its relation with site presentations

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    Let Îș\kappa be a regular cardinal. We study Gabriel-Ulmer duality when one restricts the 2-category of locally Îș\kappa-presentable categories with Îș\kappa-accessible right adjoints to its locally full sub-2-category of Îș\kappa-presentable Grothendieck topoi with geometric Îș\kappa-accessible morphisms. In particular, we provide a full understanding of the locally full sub-2-category of the 2-category of Îș\kappa-small cocomplete categories with Îș\kappa-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 Îș\kappa-presentable Grothendieck topoi with geometric Îș\kappa-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 Îș\kappa-ary sites in the sense of Shulman [37].Comment: 25 page

    Behavior of Quillen (co)homology with respect to adjunctions

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    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

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    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

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    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

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    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

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    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 Ab,SSet,Cat\mathbf{Ab},\mathbf{SSet},\mathbf{Cat} and Met\mathbf{Met}. 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 V\mathcal V so that (a) ⇔\Leftrightarrow (b) holds; (2) give sufficient conditions on V\mathcal V so that (a) ⇔\Leftrightarrow (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

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    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 Set\mathbf{Set}. In this paper, we generalize this phenomenon to locally finitely presentable categories using partial Horn logic. For each locally finitely presentable category A\mathscr{A}, we define an "algebraic concept" relative to A\mathscr{A}, which will be called an A\mathscr{A}-relative algebraic theory, and show that A\mathscr{A}-relative algebraic theories are equivalent to finitary monads on A\mathscr{A}. Finally, we generalize Birkhoff's variety theorem for classical algebraic theories to our relative algebraic theories.Comment: 34 page
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