Algebraic weak factorisation systems II: Categories of weak maps

Bourke, John; Garner, Richard

doi:10.1016/j.jpaa.2015.06.003

Cited by 39 publications

(24 citation statements)

References 24 publications

(54 reference statements)

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“…Let (L, R) be a monoidal algebraic weak factorisation system on a (suitably nice) monoidal category V. Then the cofibrant replacement comonad Q for (L, R) is a monoidal comonad on V. Furthermore, the cofibrant replacement comonad S for an (L, R)-enriched algebraic weak factorisation system (H, M) on a V-category A is a locally Q-weak V-comonad on A. Proposition 3.7 then defines a skew-enrichment of (the underlying category of) A over the skew-monoidal structure on V induced by Q, from which it follows that the co-Kleisli category for S (known as the category of weak maps for (H, M) [BG16]) is enriched over this skew-monoidal structure on V.…”

Section: A(sa B)mentioning

confidence: 99%

“…In our motivating examples, this shows that, for a monoidal algebraic weak factorisation system (L, R) on a monoidal category V and an (L, R)-enriched algebraic weak factorisation system (H, M) on a V-category A, there is a skew-enrichment of A over the skew-monoidal structure on V induced by the cofibrant replacement monoidal comonad for (L, R), for which the weak morphisms are the weak maps for (H, M) (in the sense of [BG16]). …”

Section: Introductionmentioning

confidence: 99%

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Skew-Enriched Categories

Campbell

2017

Appl Categor Struct

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Abstract. This paper introduces a skew variant of the notion of enriched category, suitable for enrichment over a skew-monoidal category, the main novelty of which is that the elements of the enriched hom-objects need not be in bijection with the morphisms of the underlying category. This is the natural setting in which to introduce the notion of locally weak comonad, which is fundamental to the theory of enriched algebraic weak factorisation systems. The equivalence, for a monoidal closed category V, between tensored V-categories and hommed V-actegories is extended to the skew setting and easily proved by recognising both skew V-categories and skew V-actegories as equivalent to special kinds of skew V-proactegory.

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Section: A(sa B)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Skew-Enriched Categories

Campbell

2017

Appl Categor Struct

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“…For example, each awfs on C induces a cofibrant replacement comonad on C by factorising the unique maps out of 0; and if we choose our awfs carefully, then the Kleisli category of this comonad Q-whose maps A B are maps QA → B in the original category-will equip C with a usable notion of weak map. For instance, there is an awfs on the category of tricategories [24] and strict morphisms (preserving all structure on the nose) for which Kl(Q) comprises the tricategories and their trihomomorphisms (preserving all structure up to coherent equivalence); this example and others were described in [23], and will be revisited in the companion paper [13].…”

Section: Introductionmentioning

confidence: 99%

Algebraic weak factorisation systems I: Accessible AWFS

Bourke

Garner

2016

Journal of Pure and Applied Algebra

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116

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Abstract. Algebraic weak factorisation systems (awfs) refine weak factorisation systems by requiring that the assignations sending a map to its first and second factors should underlie an interacting comonad-monad pair on the arrow category. We provide a comprehensive treatment of the basic theory of awfs-drawing on work of previous authors-and complete the theory with two main new results. The first provides a characterisation of awfs and their morphisms in terms of their double categories of left or right maps. The second concerns a notion of cofibrant generation of an awfs by a small double category; it states that, over a locally presentable base, any small double category cofibrantly generates an awfs, and that the awfs so arising are precisely those with accessible monad and comonad. Besides the general theory, numerous applications of awfs are developed, emphasising particularly those aspects which go beyond the non-algebraic situation.

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“…For the sake of uniformity, as expressed in the tables of the Introduction, we are slightly adapting the existing terminology on weak factorization systems, from [2,4,10,14,21]. 'Algebraic factorizations', in the present sense, were introduced in [14] under the name of 'natural weak factorization systems', and studied in [4,5,10] as 'algebraic weak factorization systems'. We also note that a 'functorial realisation of a weak factorization system', in the sense of [21], is intermediate between functorial and algebraic factorizations, in the present sense.…”

Section: Functorial Factorizations As Torsion Operators In Categoriesmentioning

confidence: 99%

From torsion theories to closure operators and factorization systems

Grandis¹,

Janelidze

2020

CGASA

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Torsion theories are here extended to categories equipped with an ideal of 'null morphisms', or equivalently a full subcategory of 'null objects'. Instances of this extension include closure operators viewed as generalised torsion theories in a 'category of pairs', and factorization systems viewed as torsion theories in a category of morphisms. The first point has essentially been treated in [15].

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Algebraic weak factorisation systems II: Categories of weak maps

Cited by 39 publications

References 24 publications

Skew-Enriched Categories

Skew-Enriched Categories

Algebraic weak factorisation systems I: Accessible AWFS

From torsion theories to closure operators and factorization systems

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