Categorical Homotopy Theory

Riehl, Emily

doi:10.1017/cbo9781107261457

Cited by 242 publications

(121 citation statements)

References 44 publications

(1 reference statement)

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“…For the reader's convenience, we review the basics of the theory of weighted limits in §5.1. A more thorough treatment can be found in [14] or [22]. In §5.3, we establish a correspondence between projective cofibrant simplicial functors and certain relative simplicial computads that will be exploited in section 6 to identify projective cofibrant weights.…”

Section: Weighted Limits In Qcat ∞mentioning

confidence: 99%

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Homotopy coherent adjunctions and the formal theory of monads

Riehl

Verity

2016

Advances in Mathematics

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Abstract. In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasi-categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contractible Kan complexes.We extract several simplicial functors from the free homotopy coherent adjunction and show that quasi-categories are closed under weighted limits with these weights. These weighted limits are used to define the homotopy coherent monadic adjunction associated to a homotopy coherent monad. We show that each vertex in the quasi-category of algebras for a homotopy coherent monad is a codescent object of a canonical diagram of free algebras. To conclude, we prove the quasi-categorical monadicity theorem, describing conditions under which the canonical comparison functor from a homotopy coherent adjunction to the associated monadic adjunction is an equivalence of quasi-categories. Our proofs reveal that a mild variant of Beck's argument is "all in the weights"-much of it independent of the quasi-categorical context.

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Section: Weighted Limits In Qcat ∞mentioning

confidence: 99%

“…In this way, we see that qCat ∞ is closed under the formation of homotopy limits. See [22] for more details. 5.2.9.…”

Section: 28mentioning

confidence: 99%

Homotopy coherent adjunctions and the formal theory of monads

Riehl

Verity

2016

Advances in Mathematics

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“…To spell this out more explicitly, the hom‐objects are

double-struckZ

‐graded

F_{2}

‐vector spaces,

Mor false(A, B false) = ⨁_{i \in Z} {prefixMor}_{i} false(A, B false)

endowed with differentials

\partial_{i} : {prefixMor}_{i} false(A, B false) \to {prefixMor}_{i - 1} false(A, B false),

that is, vector space homomorphisms satisfying

\partial_{i - 1} \partial_{i} = 0

and

\partial \circ m = m \circ (\partial \otimes id + id \otimes \partial),

where

m : {prefixMor}_{j} false(B, C false) \otimes {prefixMor}_{i} false(A, B false) \to {prefixMor}_{i + j} false(A, C false)

denotes composition in

scriptC

, which is associative and unital. For more details on enriched categories, see for example . Note that the identity morphisms have degree 0 and lie in the kernel of

\partial

.…”

Section: Preliminaries: Algebraic Structures From Dg Categoriesmentioning

confidence: 99%

Peculiar modules for 4‐ended tangles

Zibrowius

2020

Journal of Topology

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With a 4‐ended tangle T, we associate a Heegaard Floer invariant prefixCFT∂false(Tfalse), the peculiar module of T. Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University, 2011), we prove a glueing formula for this invariant which recovers link Floer homology HFL̂. Moreover, we classify peculiar modules in terms of immersed curves on the 4‐punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson (Preprint, 2016, arXiv:1604.03466v2), we prove general classification results for the category of curved complexes over a marked surface with arc system. This allows us to reinterpret the glueing formula for peculiar modules in terms of Lagrangian intersection Floer theory on the 4‐punctured sphere. We then study some applications: Firstly, we show that peculiar modules detect rational tangles. Secondly, we give short proofs of various skein exact triangles. Thirdly, we compute the peculiar modules of the 2‐stranded pretzel tangles T2n,−false(2m+1false) for n,m>0 using nice diagrams. We then observe that these peculiar modules enjoy certain symmetries which imply that mutation of the tangles T2n,−false(2m+1false) preserves δ‐graded, and for some orientations even bigraded link Floer homology.

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“…Standard reasoning by induction on context thus becomes simple algebraic calculation. A second, crucial notion is cofibrantly generated factorisation systems, a notion from homotopy theory [13,22] which, together with cellularity, allows for a conceptually simple, yet relevant characterisation of well-behaved transition contexts.…”

Section: Contextmentioning

confidence: 99%

Cellular Monads from Positive GSOS Specifications

Hirschowitz

2019

Electron. Proc. Theor. Comput. Sci.

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We give a leisurely introduction to our abstract framework for operational semantics based on cellular monads on transition categories. Furthermore, we relate it for the first time to an existing format, by showing that all Positive GSOS specifications generate cellular monads whose free algebras are all compositional. As a consequence, we recover the known result that bisimilarity is a congruence in the generated labelled transition system.• The treatment of languages with variable binding is significantly more technical than the basic setting [9,8,24].• More importantly, the bialgebraic study of higher-order languages like the λ -calculus or the higherorder π-calculus is only in its infancy [20].This leaves some room for exploring potential alternatives. * Thanks to Jorge Pérez and Jurriaan Rot for the invitation, and to the referees for helpful comments.Lemma 8.2 thus accounts for the factorisation of the original square as on the left in (2): M and R respectively factor as ⋆ M ′

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Categorical Homotopy Theory

Cited by 242 publications

References 44 publications

Homotopy coherent adjunctions and the formal theory of monads

Homotopy coherent adjunctions and the formal theory of monads

Peculiar modules for 4‐ended tangles

Cellular Monads from Positive GSOS Specifications

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