Comprehensive factorisation systems

Berger, Clemens; Kaufmann, Ralph M.

doi:10.1515/tmj-2017-0112

Cited by 12 publications

(25 citation statements)

References 30 publications

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“…→ S and G ′′ ⊲ P ′′ ψ ′′ → S, and the maps (σ ′′−1 ) * and σ ′ * are induced by quasibijections (12), commutes. (iii) For every commutative diagram…”

Section: Markl Operadsmentioning

confidence: 99%

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Koszul duality for operadic categories

Batanin¹,

Markl²

2021

Preprint

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The aim of this sequel to [10] is to set up the cornerstones of Koszul duality and Koszulity in the context of a large class of operadic categories. In particular, we will prove that operads, in the generalized sense of [7], governing important operad-and/or PROP-like structures such as the classical operads, their variants such as cyclic, modular or wheeled operads, and also diverse versions of PROPs such as properads, dioperads, 1 2 PROPs, and still more exotic objects such as permutads and pre-permutads are binary quadratic, and describe their Koszul duals. We then prove, using the results of [11], that operads describing the most common structures are Koszul.

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“…→ S and G ′′ ⊲ P ′′ ψ ′′ → S, and the maps (σ ′′−1 ) * and σ ′ * are induced by quasibijections (12), commutes. (iii) For every commutative diagram…”

Section: Markl Operadsmentioning

confidence: 99%

“…12, 2021] By the left triangle, u is a quasibijection while it belongs to O ord by the right triangle. The uniqueness follows from[10, Corollary 2.6].…”

mentioning

confidence: 99%

Koszul duality for operadic categories

Batanin¹,

Markl²

2021

Preprint

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“…When E = Cat and P is the presheaf construction A → [A op , Set], a functor is P -connected if and only if it is final, and it is a P -covering if and only if it is a discrete fibration. See [BK17] for a generalization to any 'consistent' comprehension scheme.…”

Section: 2mentioning

confidence: 99%

Categorical notions of fibration

Loregian

Riehl

2020

Expositiones Mathematicae

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Fibrations over a category B, introduced to category theory by Grothendieck, encode pseudo-functors B op Cat, while the special case of discrete fibrations encode presheaves B op → Set. A twosided discrete variation encodes functors B op × A → Set, which are also known as profunctors from A to B. By work of Street, all of these fibration notions can be defined internally to an arbitrary 2-category or bicategory. While the two-sided discrete fibrations model profunctors internally to Cat, unexpectedly, the dual two-sided codiscrete cofibrations are necessary to model V-profunctors internally to V-Cat. These notes were initially written by the second-named author to accompany a talk given in the Algebraic Topology and Category Theory Proseminar in the fall of 2010 at the University of Chicago. A few years later, the first-named author joined to expand and improve the internal exposition and external references.

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“…The article is subdivided into four sections, the first two being mostly expositary: Section 1 reviews idempotent semigroups with emphasis on regular and normal bands. We characterise right normal bands among right regular bands using a comprehensive factorisation system [6,34]. This sheds light on an important theorem of Kimura-Yamada [20,35]: every right normal band may be represented as the category of elements of a canonical presheaf on its universal semilattice quotient.…”

Section: Introductionmentioning

confidence: 99%

Stone duality for spectral sheaves and the patch monad

Berger,

Gehrke

2022

Preprint

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We establish a duality between global sheaves on spectral spaces and right distributive bands. This is a sheaf-theoretical extension of classical Stone duality between spectral spaces and bounded distributive lattices.The topology of a spectral space admits a refinement, the so-called patch topology, giving rise to a patch monad on sheaves over a fixed spectral space. Under the duality just mentioned the algebras of this patch monad are shown to correspond to distributive skew lattices.

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Comprehensive factorisation systems

Cited by 12 publications

References 30 publications

Koszul duality for operadic categories

Koszul duality for operadic categories

Categorical notions of fibration

Stone duality for spectral sheaves and the patch monad

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