All $(\infty,1)$-toposes have strict univalent universes

Shulman, Michael

doi:10.48550/arxiv.1904.07004

Cited by 28 publications

(44 citation statements)

References 69 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In order to prove this theorem, it is necessary to develop a substantial number of results, many of which are of independent interest. As we are working in homotopy type theory, which has models 1 in all ∞-toposes ( [8], [9], [14]), our results hold in more general situations than the homotopy theory of spaces. To achieve this greater generality, all of our proofs and constructions must be homotopy invariant, all of our arguments must be constructive (avoiding the law of excluded middle and the axiom of choice), and we cannot make use of Whitehead's theorem, which does not hold in this generality.…”

Section: Introductionmentioning

confidence: 61%

Localization in Homotopy Type Theory

Christensen¹,

Opie²,

Rijke³

et al. 2018

Preprint

View full text Add to dashboard Cite

We study localization at a prime in homotopy type theory, using self maps of the circle. Our main result is that for a pointed, simply connected type X, the natural map X → X (p) induces algebraic localizations on all homotopy groups. In order to prove this, we further develop the theory of reflective subuniverses. In particular, we show that for any reflective subuniverse L, the subuniverse of L-separated types is again a reflective subuniverse, which we call L ′ . Furthermore, we prove results establishing that L ′ is almost left exact. We next focus on localization with respect to a map, giving results on preservation of coproducts and connectivity. We also study how such localizations interact with other reflective subuniverses and orthogonal factorization systems. As key steps towards proving the main theorem, we show that localization at a prime commutes with taking loop spaces for a pointed, simply connected type, and explicitly describe the localization of an Eilenberg-Mac Lane space K(G, n) with G abelian. We also include a partial converse to the main theorem.

show abstract

Section: Introductionmentioning

confidence: 61%

Localization in Homotopy Type Theory

Christensen¹,

Opie²,

Rijke³

et al. 2018

Preprint

View full text Add to dashboard Cite

show abstract

“…Our handling of universes has a model in ∞-toposes following Shulman [14]. It differs from that of the HoTT book [15], and Coq [4], in that we don't assume cumulativity, and it agrees with that of Agda [3].…”

Section: Underlying Formal Systemmentioning

confidence: 99%

Injective types in univalent mathematics

Escardó¹

2019

Preprint

View full text Add to dashboard Cite

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.

show abstract

“…sheaves on Sch that take values in a reasonable ∞-category A, thus naturally leads to the emerging field of homotopy type theory [Uni13]. In fact, it is now known [Shu19] that much of homotopy type theory has a model in an arbitrary ∞-topos.…”

Section: Introductionmentioning

confidence: 99%

Yoneda's lemma for internal higher categories

Martini¹

2021

Preprint

View full text Add to dashboard Cite

We develop some basic concepts in the theory of higher categories internal to an arbitrary ∞topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal categories. Contents 1. Introduction Motivation Main results Related work Acknowledgment 2. Preliminaries 2.1. General conventions and notation 2.2. Set theoretical foundations 2.3. ∞-topoi 2.4. Universe enlargement 2.5. Factorisation systems 3. Categories in an ∞-topos 3.1. Simplicial objects in an ∞-topos 3.2. Categories in an ∞-topos 3.3. Functoriality and base change 3.4. The (∞, 2)-categorical structure of Cat(B) 3.5. Cat(S)-valued sheaves on an ∞-topos 3.6. Objects and morphisms 3.7. The universe for groupoids 3.8. Fully faithful and essentially surjective functors 3.9. Subcategories 4. Groupoidal fibrations and Yoneda's lemma 4.1. Left fibrations 4.2. Slice categories 4.3. Initial functors 4.4. Covariant equivalences 4.5. The Grothendieck construction 4.6. Yoneda's lemma References

show abstract

All $(\infty,1)$-toposes have strict univalent universes

Cited by 28 publications

References 69 publications

Localization in Homotopy Type Theory

Localization in Homotopy Type Theory

Injective types in univalent mathematics

Yoneda's lemma for internal higher categories

Contact Info

Product

Resources

About