We study the 2-category of elements from an abstract point of view. We generalize to dimension 2 the well-known result that the category of elements can be captured by a comma object that also exhibits a pointwise left Kan extension. For this, we propose an original definition of pointwise Kan extension along a discrete 2-opfibration in the lax 3-category of 2-categories, 2-functors, lax natural transformations and modifications. Such definition uses cartesian-marked lax limits, which are an alternative to weighted 2-limits. We show that a pointwise Kan extension along a discrete 2-opfibration is always a weak one as well. The proof is based on an original generalization of the parametrized Yoneda lemma which is as lax as it can be.
We introduce the notion of an effective Kan fibration, a new mathematical structure that can be used to study simplicial homotopy theory. Our main motivation is to make simplicial homotopy theory suitable for homotopy type theory. Effective Kan fibrations are maps of simplicial sets equipped with a structured collection of chosen lifts that satisfy certain non-trivial properties. This contrasts with the ordinary, unstructured notion of a Kan fibration. We show that fundamental properties of Kan fibrations can be extended to explicit constructions on effective Kan fibrations. In particular, we …
In this paper we consider a notion of pointwise Kan extension in double categories that naturally generalises Dubuc's notion of pointwise Kan extension along enriched functors. We show that, when considered in equipments that admit opcartesian tabulations, it generalises Street's notion of pointwise Kan extension in 2-categories.
We review the concept of a univalent fibration and show by elementary means that every Kan fibration in simplicial sets can be embedded in a univalent Kan fibration.
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…
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.
A relative category is a category with a chosen class of weak equivalences. Barwick and Kan produced a model structure on the category of all relative categories, which is Quillen equivalent to the Joyal model structure on simplicial sets and the Rezk model structure on simplicial spaces. We will prove that the underlying relative category of a model category or even a fibration category is fibrant in the Barwick--Kan model structure.
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 $\infty$-group…
We study Kan extensions in three weakenings of the Eilenberg-Moore double category associated to a double monad, that was introduced by Grandis and Paré. To be precise, given a normal oplax double monad $T$ on a double category $\mathcal K$, we consider the double categories consisting of pseudo $T$-algebras, `weak' vertical $T$-morphisms, horizontal $T$-morphisms and $T$-cells, where `weak' means either `lax', `colax' or `pseudo'. Denoting these double categories by $\mathsf{Alg}_w T$, where w = l, c or ps accordingly, our main result gives, in each of these cases, conditions ensuring that (p…
We develop parametrized generalizations of a number of fundamental concepts in the theory of $\infty$-categories, including factorization systems, free fibrations, exponentiable fibrations, relative colimits and relative Kan extensions, filtered and sifted diagrams, and the universal constructions Ind and $P^Σ$.
Various models of $(\infty,1)$-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an $\infty$-cosmos. In a generic $\infty$-cosmos, whose objects we call $\infty$-categories, we introduce modules (also called profunctors or correspondences) between $\infty$-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from $A$ to $B$ is an $\infty$-category equipped with a left action of $A$ and a right action of $B$, in a suitable se…
This paper contains results from two areas -- formal theory of Kan extensions and concrete categories. The contribution to the former topic is based on the extension of the concept of Kan extension to the cones and we prove that limiting cones create Kan extensions. The latter topic focuses on two significant families of concrete categories over an arbitrary category. Beck categories are defined by preservance properties while newly introduced l-algebraic categories are described by limits of categories of functor algebras. The latter family is shown to be rather natural. The well known Beck's…
There are two main constructions in classical descent theory: the category of algebras and the descent category, which are known to be examples of weighted bilimits. We give a formal approach to descent theory, employing formal consequences of commuting properties of bilimits to prove classical and new theorems in the context of Janelidze-Tholen "Facets of Descent II", such as Bénabou-Roubaud Theorems, a Galois Theorem, embedding results and formal ways of getting effective descent morphisms. In order to do this, we develop the formal part of the theory on commuting bilimits via pseudomonad th…
We use the terms $\infty$-categories and $\infty$-functors to mean the objects and morphisms in an $\infty$-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of fibrant objects. Quasi-categories, Segal categories, complete Segal spaces, marked simplicial sets, iterated complete Segal spaces, $θ_n$-spaces, and fibered versions of each of these are all $\infty$-categories in this sense. Previous work in this series shows that the basic category theory of $\infty$-categories and $\infty$-functors can be developed only in reference to the axioms …
A theory of a derivator version of six-functor-formalisms is developed, using an extension of the notion of fibered multiderivator due to the author. Using the language of (op)fibrations of 2-multicategories this has (like a usual fibered multiderivator) a very neat definition. This definition not only encodes all compatibilities among the six functors but also their interplay with homotopy Kan extensions. One could say: a nine-functor-formalism. This is essential for dealing with (co)descent questions. Finally, it is shown that every fibered multiderivator (for example encoding any kind of de…
In this paper we construct an analogue of Lurie's "unstraightening" construction that we refer to as the "comprehension construction". Its input is a cocartesian fibration $p \colon E \to B$ between $\infty$-categories together with a third $\infty$-category $A$. The comprehension construction then defines a map from the quasi-category of functors from $A$ to $B$ to the large quasi-category of cocartesian fibrations over $A$ that acts on $f \colon A \to B$ by forming the pullback of $p$ along $f$. To illustrate the versatility of this construction, we define the covariant and contravariant Yon…
