Equipping weak equivalences with algebraic structure

Bourke, John

doi:10.1007/s00209-019-02305-w

Cited by 14 publications

(22 citation statements)

References 33 publications

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“…More generally, consider a finitary signature Ω : N → Set and the associated function The following is a standard result. The case dealing with a set, rather than a family, of morphisms is dealt with in the proof of Theorem 5 of [8] and the generalisation to a family of morphisms is trivial. Proposition 3.4.…”

Section: Iterated Algebraic Injectives and Models Of Cellular Theoriesmentioning

confidence: 99%

“…Morphisms of the category Sink(Y ) of sinks commute with the coprojections from the Y j in the evident manner. Given a functor 4 In fact, if C is combinatorial, neither condition is required -see Theorem 15 of [8]. However for our present purposes we make use of these conditions and follow Nikolaus original arguments closely.…”

Section: Free Iterated Algebraic Injectives and Cellularitymentioning

confidence: 99%

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Iterated algebraic injectivity and the faithfulness conjecture

Bourke¹

2018

Preprint

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We introduce iterated algebraic injectivity and show how to describe Grothendieck ∞-groupoids as iterated algebraic injectives. Our approach is then used to prove the faithfulness conjecture of Maltsiniotis.in which Θ 0 is the initial globular theory, and in which each J n n+1 : T n → T n+1 is obtained by freely adjoining fillers -see Section 2.4. In [19] Maltsiniotis conjectured that each functor J n m : T n → T m for n < m defining a cellular globular theory is faithful. Assuming this conjecture Ara [2] established a sharp correspondence between the weak ω-categories introduced by Maltsiniotis and those of Batanin/Leinster [4,18]. However the faithfulness conjecture, on which the correspondence depends, was left unproven. Using the framework of iterated algebraic injectivity, we prove it in Theorem 5.3.Let us now describe the structure of the paper. In Section 2 we describe some background on globular sets, globular theories and the faithfulness conjecture. In Section 3 we recall algebraic injectivity, introduce iterated algebraic injectivity and the important class of (A, B)-iterated algebraic injectives, which are parametrised by a set of objects A and family of maps B in the base category. In Theorem 3.12 we show that, for a suitable choice of A and B, these capture the models of cellular globular theories. In Section 4.1 we abstract results of Nikolaus [20] relating to the construction of free algebraic injectives, and extend them to our iterated setting. Using these results, we prove the faithfulness conjecture in Section 5. In Section 6, using the framework of [11], we generalise some of the results in Section 3 away from the globular setting -in particular, we show that in general (A, B)-iterated algebraic injectives are the categories of algebras for cellular monads/theories with respect to a class of maps determined by A and B.

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Section: Iterated Algebraic Injectives and Models Of Cellular Theoriesmentioning

confidence: 99%

Section: Free Iterated Algebraic Injectives and Cellularitymentioning

confidence: 99%

Iterated algebraic injectivity and the faithfulness conjecture

Bourke¹

2018

Preprint

Self Cite

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“…The idea of Corollary 7.4 comes from a passing remark due to Jeff Smith and mentioned in [5]. [4,Theorem 14] contains interesting results of the same kind for more general combinatorial model categories.…”

Section: Reminder About Model Categories Of Fibrant Objectsmentioning

confidence: 99%

Homotopy theory of Moore flows (I)

Gaucher

2021

Compositionality

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A reparametrization category is a small topologically enriched symmetric semimonoidal category such that the semimonoidal structure induces a structure of a commutative semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a unique way) as a tensor product of two maps. A Moore flow is a small semicategory enriched over the closed semimonoidal category of enriched presheaves over a reparametrization category. We construct the q-model category of Moore flows. It is proved that it is Quillen equivalent to the q-model category of flows. This result is the first step to establish a zig-zag of Quillen equivalences between the q-model structure of multipointed d-spaces and the q-model structure of flows.

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“…(X, Y : U ) ( j : X → Y) ( f : X → D) ∃(g : Y → D), g • j = f , so that we get an unspecified extension g of f along j. The algebraic injectivity of D is defined by a given section (−) | j of the restriction map (−) • j, following Bourke's terminology (Bourke, 2017). By − -distributivity, this amounts to (X, Y : U ) ( j :…”

Section: F -mentioning

confidence: 99%

Injective types in univalent mathematics

Escardó

2021

Math. Struct. Comp. Sci.

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We investigate the injective types and the algebraically injective types in univalent mathematics, both in the absence and in the presence of propositional resizing. Injectivity is defined by the surjectivity of the restriction map along any embedding, and algebraic injectivity is defined by a given section of the restriction map along any embedding. Under propositional resizing axioms, the main results are easy to state: (1) Injectivity is equivalent to the propositional truncation of algebraic injectivity. (2) The algebraically injective types are precisely the retracts of exponential powers of universes. (2a) The algebraically injective sets are precisely the retracts of powersets. (2b) The algebraically injective (n+1)-types are precisely the retracts of exponential powers of universes of n-types. (3) The algebraically injective types are also precisely the retracts of algebras of the partial-map classifier. From (2) it follows that any universe is embedded as a retract of any larger universe. In the absence of propositional resizing, we have similar results that have subtler statements which need to keep track of universe levels rather explicitly, and are applied to get the results that require resizing.

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Equipping weak equivalences with algebraic structure

Cited by 14 publications

References 33 publications

Iterated algebraic injectivity and the faithfulness conjecture

Iterated algebraic injectivity and the faithfulness conjecture

Homotopy theory of Moore flows (I)

Injective types in univalent mathematics

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