Kan Fibrations Research Articles

Examples and cofibrant generation of effective Kan fibrations

We will make two contributions to the theory of effective Kan fibrations, which are a more explicit version of the notion of a Kan fibration, a notion which plays a fundamental role in simplicial homotopy theory. We will show that simplicial Malcev algebras are effective Kan complexes and that the effective Kan fibrations can be seen as the right class in an algebraic weak factorization system. In addition, we will introduce two strengthenings of the notion of an effective Kan fibration, the symmetric effective and degenerate-preferring Kan fibrations, and show that the previous results hold for these strengthenings as well.

The simplicial model of Univalent Foundations (after Voevodsky)

We present Voevodsky’s construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to obtain coherence. We then construct a (weakly) universal Kan fibration, and use it to exhibit a model in simplicial sets. Lastly, we introduce the Univalence Axiom, in several equivalent formulations, and show that it holds in our model. As a corollary, we conclude that Martin-Löf type theory with one univalent universe (formulated in terms of contextual categories) is at least as consistent as ZFC with two inaccessible cardinals.

On the homotopy theory for Lie ∞–groupoids, with an application to integrating L∞–algebras

Lie $\infty$-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie $\infty$-groupoids called `Lie $\infty$-groups' by integrating finite type Lie $n$-algebras. In order to study the compatibility between this integration procedure and the homotopy theory of Lie $n$-algebras introduced in the companion paper arXiv:1809.05999, we present a homotopy theory for Lie $\infty$-groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie $\infty$-groupoids do not form a category of fibrant objects (CFO), since the category of manifolds lacks pullbacks. Instead, we show that Lie $\infty$-groupoids form an `incomplete category of fibrant objects' in which the weak equivalences correspond to `stalkwise' weak equivalences of simplicial sheaves. This homotopical structure enjoys many of the same properties as a CFO, such as having, in the presence of functorial path objects, a convenient realization of its simplicial localization. We further prove that the acyclic fibrations are precisely the hypercovers, which implies that many of Behrend and Getzler's results also hold in this more general context. As an application, we show that Henriques' integration functor is an exact functor with respect to a class of distinguished fibrations which we call `quasi-split fibrations'. Such fibrations include acyclic fibrations as well as fibrations that arise in string-like extensions. In particular, integration sends $L_\infty$ quasi-isomorphisms to weak equivalences, quasi-split fibrations to Kan fibrations, and preserves acyclic fibrations, as well as pullbacks of acyclic/quasi-split fibrations.

Univalent completion

We show by elementary means that every Kan fibration in simplicial sets can be embedded in a univalent Kan fibration.

The Frobenius condition, right properness, and uniform fibrations

We develop further the theory of weak factorization systems and algebraic weak factorization systems. In particular, we give a method for constructing (algebraic) weak factorization systems whose right maps can be thought of as (uniform) fibrations and that satisfy the (functorial) Frobenius condition. As applications, we obtain a new proof that the Quillen model structure for Kan complexes is right proper, avoiding entirely the use of topological realization and minimal fibrations, and we solve an open problem in the study of Voevodsky's simplicial model of type theory, proving a constructive version of the preservation of Kan fibrations by pushforward along Kan fibrations. Our results also subsume and extend work by Coquand and others on cubical sets.

Singular chains on topological stacks, I

We extend the functor Sing of singular chains to the category of topological stacks and establish its main properties. We prove that Sing respects weak equivalences and takes a morphism of topological stacks that is both a Serre and a Reedy fibration to a Kan fibration of simplicial sets. When restricted to the category of topological spaces Sing coincides with the usual singular functor.

W-types in homotopy-type theory – CORRIGENDUM

In the article below, Theorem 3.4 requires the additional assumption that A is Kan as well. Indeed, the inductive proof as given only shows that if W(f)<α is a Kan complex, then W(f)<α+1 → A is a Kan fibration.

An algebraic weak factorisation system on 01-substitution sets: a constructive proof

We will construct an algebraic weak factorisation system on the category of 01 substitution sets such that the R-algebras are precisely the Kan fibrations together with a choice of Kan filling operation. The proof is based on Garner's small object argument for algebraic weak factorization systems. In order to ensure the proof is valid constructively, rather than applying the general small object argument, we give a direct proof based on the same ideas. We use this us to give an explanation why the J computation rule is absent from the original cubical set model and suggest a way to fix this.

A Kripke model for simplicial sets

By means of a countermodel we show that the homotopy equivalence of the fibers of a Kan fibration over a connected base cannot be proved constructively.

A model of type theory in simplicial sets: A brief introduction to Voevodsky's homotopy type theory

We describe how to interpret constructive type theory in the topos of simplicial sets where types appear as Kan complexes and families of types as Kan fibrations. Since Kan complexes may be understood as weak higher-dimensional groupoids this model generalizes and extends the (ordinary) groupoid model which was introduced by M. Hofmann and the author about 20 years ago. Finally, we discuss Voevodsky's Univalence Axiom which has been shown to hold in this model. This axiom roughly states that isomorphic types are equal. The type theoretic notion of isomorphism provided by this model coincides with homotopy equivalence of Kan complexes. For this reason it has become common to refer to it as Homotopy Type Theory.

Fibrations of Simplicial Sets

There are infinitely many variants of the notion of Kan fibration that, together with suitable choices of cofibrations and the usual notion of weak equivalence of simplicial sets, satisfy Quillen’s axioms for a homotopy model category. The combinatorics underlying these fibrations is purely finitary and seems interesting both for its own sake and for its interaction with homotopy types. To show that these notions of fibration are indeed distinct, one needs to understand how iterates of Kan’s Ex functor act on graphs and on nerves of small categories.

Simplicial homotopy in semi-abelian categories

AbstractWe study Quillen's model category structure for homotopy of simplicial objects in the context of Janelidze, Márki and Tholen's semi-abelian categories. This model structure exists as soon as is regular Mal'tsev and has enough regular projectives; then the fibrations are the Kan fibrations of S. When, moreover, is semi-abelian, weak equivalences and homology isomorphisms coincide.

Monotone-light factorization for Kan fibrations of simplicial sets with respect to groupoids

We show that the class of second order covering maps of simplicial sets in the sense of R. Brown and G. Janelidze is a part of a factorization system for the class of Kan fibrations of simplicial sets.

On Kan Fibrations for Maltsev Algebras

Abstract We prove that any surjective homomorphism of Maltsev algebras is a Kan fibration.

On simplicial loops and H-spaces

An algebraic loop is a ‘group without associativity’. It holds that a surjective homomorphism of simplicial loops is a Kan fibration, the Moore complex is a loop-valued chain complex with homology the homotopy groups of the simplicial loop, and the simplicial loop is minimal iff the Moore complex is minimal. We show that the minimal simplicial model of a connected H-space X is a simplicial loop. As an application, we give a short proof of a Theorem of Kock, Kristensen and Madsen which represents cohomology theories by the homology of loop-valued cochain functors. Then we show that the Postnikov decomposition of a connected minimal simplicial loop consists of central loop extensions. Using this, we prove a refinement of a Theorem of Iriye and Kono: For a connected 2-antilocal H-space X, there exists a strict commutative loop structure on its minimal model.

Schreier theory for singular extensions of categorical groups and homotopy classification* *

If G is a categorical group, a G-module is defined to be a braided categorical group (A c) together with an action of G on (A,c). In this work we define the notions of singular extension of G by the G-module (A,c) and of 1-cocycle of G with coefficients in (A,c) and we obtain, first, a bijection between the set of equivalence classes of singular extensions of G by (Ac) and the set of equivalence classes of 1-cocycles. Next, we associate to any G-module (Ac) a Kan fibration of simplicial sets ϕ: Ner(GAc)) → Ner(G)and then we show that there is a bijection between the set of equivalence classes of singular extensions of G by (A,c) and Γ[Ner(G,A,c/Ner(G)]the set of fibre homotopy classes of cross-sections of the fibration ϕ.

Classification cohomologique et homotopique des extensions singulières de Gr-catégories (Extensions singulières de Gr-catégories)

If G is a categorical group, a G -module is defined to be a braided categorical group ( A , c ) together with an action of G on ( A , c ) . We associate to any G -module ( A , c ) a Kan simplicial set Ner ( G ( A , c ) ) and a Kan fibration Ner ( G ( A , c ) ) → ϕ Ner ( G ) . In addition, we define the set of equivalence classes of singular extensions of G by ( A , c ) , and also a 1-cohomology set of G with coefficients in ( A , c ) . We construct bijections between these sets, and also with the set of fibre homotopy classes of cross-sections of the fibration φ.

K-Invariants in kan fibrations with fiber K(G, n)

k-Invariants in kan fibrations with fiber K(G, n)

Kan fibrations in the category of simplicial spaces

Kan fibrations in the category of simplicial spaces

An approximation theorem for maps into Kan fibrations

An approximation theorem for maps into Kan fibrations