An Adjoint Characterization of the Category of Sets

Rosebrugh, Robert; Wood, Richard J.

doi:10.2307/2161031

Cited by 5 publications

(6 citation statements)

References 6 publications

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“…Remark 3.5. By a result of Rosebrugh and Wood [RW94], the category of finite sets is characterized amongst locally finite categories by the existence of the five left adjoints to its Yoneda embedding k → y k : Fin → Fin Fin op . The adjoint sextuple displayed in (4) is just the observation that these six functors restrict to the subcategory Dir.…”

Section: The Categories Poly and Dirmentioning

confidence: 99%

Dirichlet Polynomials form a Topos

Spivak,

Myers

2020

Preprint

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One can think of power series or polynomials in one variable, such as P(y) 2y 3 + y + 5, as functors from the category Set of sets to itself; these are known as polynomial functors. Denote by Poly Set the category of polynomial functors on Set and natural transformations between them. The constants 0, 1 and operations +, × that occur in P(y) are actually the initial and terminal objects and the coproduct and product in Poly Set .Just as the polynomial functors on Set are the copresheaves that can be written as sums of representables, one can express any Dirichlet series, e.g. ∞ n 0 n y , as a coproduct of representable presheaves. A Dirichlet polynomial is a finite Dirichlet series, that is, a finite sum of representables n y . We discuss how both polynomial functors and their Dirichlet analogues can be understood in terms of bundles, and go on to prove that the category of Dirichlet polynomials is an elementary topos.

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Section: The Categories Poly and Dirmentioning

confidence: 99%

Dirichlet Polynomials form a Topos

Spivak,

Myers

2020

Preprint

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“…Rosebrugh and Wood [14] have defined an analogue of this notion for arbitrary categories rather than just posets 1 . A locally small category E is totally distributive if there exist adjunctions…”

Section: Completely Distributive Lattices Totally Distributive Catego...mentioning

confidence: 99%

“…The aim of this paper is to establish certain connections between the work of Marmolejo, Rosebrugh, and Wood [14,13] on totally distributive categories and two other bodies of work on distinct topics: Firstly, that of Johnstone and Joyal [4,7] on injective toposes and continuous categories, and secondly, that of Kelly-Lawvere [8] and Kennett-Riehl-Roy-Zaks [9] on essential localizations and essential subtoposes. One of our observations, 1.5.9 (2), when taken together with a theorem of Kelly-Lawvere which we recall in 1.5.6, yields a concrete combinatorial description of all totally distributive categories with a small set of generators.…”

Section: Introductionmentioning

confidence: 99%

“…We adopt the foundational conventions of [6] (and [4,7]), since our only use of the stronger foundational assumptions of [17,16,18,14,13] is made in finally deducing our main results (1.5.9) as strengthened variants of propositions which precede them. We let CAT represent the meta-2-category of categories, functors, and natural transformations (see [6], 1.1.1), and we let CAT be its full sub-(meta)-2-category consisting of locally small categories.…”

Section: Introductionmentioning

confidence: 99%

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Totally distributive toposes

Lucyshyn-Wright

2012

Journal of Pure and Applied Algebra

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A locally small category E is totally distributive (as defined by Rosebrugh-Wood) if there exists a string of adjoint functors t -| c -| y, where y : E --> E^ is the Yoneda embedding. Saying that E is lex totally distributive if, moreover, the left adjoint t preserves finite limits, we show that the lex totally distributive categories with a small set of generators are exactly the injective Grothendieck toposes, studied by Johnstone and Joyal. We characterize the totally distributive categories with a small set of generators as exactly the essential subtoposes of presheaf toposes, studied by Kelly-Lawvere and Kennett-Riehl-Roy-Zaks.Comment: Now includes extended result: The lex totally distributive categories with a small set of generators are exactly the injective Grothendieck toposes; Made changes to abstract and intro to reflect the enhanced result; Changed formatting of diagram

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“…see [22,24,28] for discussions of such categories (called totally distributive categories there). The motivation of this paper originates from a famous theorem in the theory of semigroups that reveals the closed relationship between regular relations (i.e., regular arrows in the category Rel of sets and relations) and (cd) lattices.…”

Section: Introductionmentioning

confidence: 99%

Regularity vs. constructive complete (co)distributivity

Lai,

Shen

2016

Preprint

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It is well known that a relation ϕ between sets is regular if, and only if, Kϕ is completely distributive (cd), where Kϕ is the complete lattice consisting of fixed points of the Kan adjunction induced by ϕ. For a small quantaloid Q, we investigate the Q-enriched version of this classical result, i.e., the regularity of Q-distributors versus the constructive complete distributivity (ccd) of Q-categories, and prove that "the dual of Kϕ is (ccd) =⇒ ϕ is regular =⇒ Kϕ is (ccd)" for any Q-distributor ϕ. Although the converse implications do not hold in general, in the case that Q is a commutative integral quantale, we show that these three statements are equivalent for any ϕ if, and only if, Q is a Girard quantale.

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An Adjoint Characterization of the Category of Sets

Abstract: Abstract. If a category B with Yoneda embedding Y: B -» CAT(B°P, set) has an adjoint string, U -\V -\W -\ X -\Y , then B is equivalent to set.

Cited by 5 publications

References 6 publications

Dirichlet Polynomials form a Topos

Dirichlet Polynomials form a Topos

Totally distributive toposes

Regularity vs. constructive complete (co)distributivity

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