Algebraic Kan extensions along morphisms of internal algebra classifiers

Weber, Mark

doi:10.1515/tmj-2016-0006

Cited by 14 publications

(25 citation statements)

References 19 publications

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“…The present contribution exploits the objective method initiated in [21], establishing the Faà di Bruno formula as the homotopy cardinality of an equivalence of groupoids, but takes a further abstraction step, which leads to a more general formula and a much simpler proof. We achieve this by leveraging some recent advances in category theory: on one hand, 2-categorical perspectives on operads and related structures discussed in [51,52,53], and on the other hand, the theory of decomposition spaces [23,24]. With these tools, the proof of the equivalence of groupoids ends up being rather neat, emerging naturally from general principles.…”

Section: Outline Of Results and Proof Ingredientsmentioning

confidence: 99%

“…Proof Proposition 4.6.5 of states that a strictly cartesian 2‐natural transformation between polynomial 2‐functors (between strict slices of

Grpd

) is also homotopy cartesian if just the codomain 2‐functor is familial , which essentially means that it preserves fibrations. On the other hand, familiality is implied by the following slight variation of [, Theorem 4.4.5], whose notation we use freely: in the proof given in , the condition that the lowershriek component is a fibration is not needed.…”

Section: Monad Adjunctions and The Relative (Two‐sided) Bar Constructionmentioning

confidence: 99%

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Faà di Bruno for operads and internal algebras

Kock

Weber

2018

J. London Math. Soc.

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For any coloured operad sans-serifR, we prove a Faà di Bruno formula for the ‘connected Green function’ in the incidence bialgebra of sans-serifR. This generalises on one hand the classical Faà di Bruno formula (dual to composition of power series), corresponding to the case where sans-serifR is the terminal reduced operad, and on the other hand the Faà di Bruno formula for P‐trees of Gálvez–Kock–Tonks (P a finitary polynomial endofunctor), which corresponds to the case where sans-serifR is the free operad on P. Following Gálvez–Kock–Tonks, we work at the objective level of groupoid slices, hence all proofs are ‘bijective’: the formula is established as the homotopy cardinality of an explicit equivalence of groupoids, in turn derived from a certain two‐sided bar construction. In fact we establish the formula more generally in a relative situation, for algebras of one polynomial monad internal to another. This covers in particular nonsymmetric operads (for which the terminal reduced case yields the noncommutative Faà di Bruno formula of Brouder–Frabetti–Krattenthaler).

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Section: Outline Of Results and Proof Ingredientsmentioning

confidence: 99%

“…Proof Proposition 4.6.5 of states that a strictly cartesian 2‐natural transformation between polynomial 2‐functors (between strict slices of

Grpd

Section: Monad Adjunctions and The Relative (Two‐sided) Bar Constructionmentioning

confidence: 99%

Faà di Bruno for operads and internal algebras

Kock

Weber

2018

J. London Math. Soc.

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“…It follows from the general property of classifiers for polynomial monads that this square is a pullback (cf. the proof of Theorem 5.7.2 in [Web16] or proof of Theorem 5.15 from [BB17]).…”

Section: Internal Algebra Classifiersmentioning

confidence: 99%

Homotopy theory of algebras of substitudes and their localisation

Batanin¹,

White²

2020

Preprint

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We study the category of algebras of substitudes (also known to be equivalent to the regular patterns of Getzler) equipped with a (semi)model structure lifted from the model structure on the underlying presheaves. We are especially interested in the case when the model structure on presheaves is a Cisinski style localisation with respect to a proper Grothendieck fundamental localiser. For example, for W = W∞ the minimal fundamental localiser, the local objects in such a localisation are locally constant presheaves, and local algebras of substitudes are exactly algebras whose underlying presheaves are locally constant.We investigate when this localisation has nice properties. We single out a class of such substitudes which we call left localisable and show that the substitudes for n-operads, symmetric, and braided operads are in this class. As an application we develop a homotopy theory of higher braided operads and prove a stabilisation theorem for their W k -localisations. This theorem implies, in particular, a generalisation of the Baez-Dolan Stabilisation Hypothesis for higher categories.

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“…The second of these, concerning colax Talgebras, generalises the main result of [Kou15a], which describes creation of algebraic Kan extensions in the case of a double category K. The latter in turn is a generalisation of a result by Getzler, on the lifting of pointwise left Kan extensions along symmetric monoidal enriched functors, that was given in [Get09], where it was used to obtain a coherent way of freely generating many types of generalised operad. In [Web15] a variant of Theorem 7.7, considered in the setting of 2-categories, is used to obtain algebraic Kan extensions along 'morphisms of internal algebra classifiers'. Theorem 7.8 can also be regarded as an extension of Kelly's result on 'doctrinal adjunction', given in [Kel74], as we shall see in the next section.…”

Section: Creativity Of the Forgetful Functors U : T -Alg → Kmentioning

confidence: 99%

A double-dimensional approach to formal category theory

Koudenburg¹

2015

Preprint

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Whereas formal category theory is classically considered within a 2-category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of hypervirtual double categories, a notion extending that of virtual double category (also known as fc-multicategory) by adding cells with nullary target.This paper starts by introducing the notion of hypervirtual double category, followed by describing its basic theory and its relation to other types of double category. After this the notion of 'weak' Kan extension within a hypervirtual double category is considered, together with three strengthenings. The first of these generalises Borceux-Kelly's notion of Kan extension along enriched functors, the second one generalises Street's notion of pointwise Kan extension in 2-categories, and the third is a combination of the other two; these stronger notions are compared. The notion of yoneda embedding is then considered in a hypervirtual double category, and compared to that of a good yoneda structure on a 2-category; the latter in the sense of Street-Walters and Weber. Conditions are given ensuring that a yoneda embedding y : A → A defines A as the free small cocompletion of A, in a suitable sense.In the second half we consider formal category theory in the presence of algebraic structures. In detail: to a monad T on a hypervirtual double category K several hypervirtual double categories T -Alg (v,w) of T -algebras are associated, one for each pair of types of weak coherence satisfied by the Talgebras and their morphisms respectively. This is followed by the study of the creation of, amongst others, left Kan extensions by the forgetful functors T -Alg (v,w) → K. The main motivation of this paper is the description of conditions ensuring that yoneda embeddings in K lift along these forgetful functors, as well as ensuring that such lifted algebraic yoneda embeddings again define free small cocompletions, now in T -Alg (v,w) . As a first example we apply the previous to monoidal structures on categories, hence recovering Day convolution of presheaves and Im-Kelly's result on free monoidal cocompletion, as well as obtaining a "monoidal Yoneda lemma". MotivationCentral to classical category theory is the Yoneda lemma which, for a locally small category A, describes the position of the representable presheaves A(-, x) within * Part of this paper was written during a visit to Macquarie University, while its main results formed the subject of a series of talks at the Australian Category Seminar. I would like to thank

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Algebraic Kan extensions along morphisms of internal algebra classifiers

Cited by 14 publications

References 19 publications

Faà di Bruno for operads and internal algebras

Faà di Bruno for operads and internal algebras

Homotopy theory of algebras of substitudes and their localisation

A double-dimensional approach to formal category theory

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