In the first part of this paper we study fibrations of $(\infty,2)$-categories. We give a simple characterization of such fibrations in terms of a certain square being a pullback, and apply this to show that in some cases $(\infty,2)$-categories of functors and partially (op)lax transformations preserve fibrations. We also describe free fibrations of $(\infty,2)$-categories, including in the case where we only ask for (co)cartesian lifts of specified 1- and 2-morphisms in the base, and describe the right adjoint to pullback from fibrations to such partial fibrations along an arbitrary functor. In the second part of the paper we apply these results to study colimits and Kan extensions of $(\infty,2)$-categories. Most notably, we give a fibrational description of both partially (op)lax and weighted (co)limits of $(\infty,2)$-categories and construct partially lax Kan extensions. Among other results, we also include a model-independent version of cofinality for $(\infty,2)$-categories and briefly consider presentable $(\infty,2)$-categories, characterizing them as accessible localizations of presheaves of $\infty$-categories.
We define and discuss lax and weighted colimits of diagrams in $\infty$-categories and show that the coCartesian fibration associated to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free Cartesian fibration associated to a a functor of $\infty$-categories. As an application of these results, we prove that lax representable functors are preserved under exponentiation, and also that the total space of a presentable Cartesian fibration between $\infty$-categories is presentable, generalizing a theorem of Makkai and Paré to t…
In this paper we study maximal chains in certain lattices constructed from powers of chains by iterated lax colimits in the $2$-category of posets. Such a study is motivated by the fact that in lower dimensions, we get some familiar combinatorial objects such as Dyck paths and Kreweras walks.
We consider categories of posets with $\mathfrak{C}$-valued structure sheaves for any category $\mathfrak{C}$ and see how they possess poset-indexed lax colimits that are both easy to describe and "weakly equivalent" to their ordinary colimits in a certain sense. We employ this construction to study descent problems on schematic spaces -- a particular scheme-like kind of ringed poset -- , proving a general Seifert-Van Kampen Theorem for their étale fundamental group that recovers and generalizes the homonym result for schemes to the topology of flat monomorphisms. The techniques are general en…
We provide, among other things: (i) a Bousfield--Kan formula for colimits in $\infty$-categories (generalizing the 1-categorical formula for a colimit as a coequalizer of maps between coproducts); (ii) $\infty$-categorical generalizations of Barwick--Kan's Theorem B$_n$ and Dwyer--Kan--Smith's Theorem C$_n$ (regarding homotopy pullbacks in the Thomason model structure, which themselves vastly generalize Quillen's Theorem B); and (iii) an articulation of the simultaneous and interwoven functoriality of colimits (or dually, of limits) for natural transformations and for pullback along maps of di…
Like categories, small 2-categories have well-understood classifying spaces. In this paper, we deal with homotopy types represented by 2-diagrams of 2-categories. Our results extend to homotopy colimits of 2-functors lower categorical analogues that have been classically used in algebraic topology and algebraic K-theory, such as the Homotopy Invariance Theorem (by Bousfield and Kan), the Homotopy Colimit Theorem (Thomason), Theorems A and B (Quillen), or the Homotopy Cofinality Theorem (Hirschhorn).
In this mainly expository note, we state a criterion for when a left Kan extension of a lax monoidal functor along a strong monoidal functor can itself be equipped with a lax monoidal structure, in a way that results in a left Kan extension in MonCat. This belongs to the general theory of algebraic Kan extensions, as developed by Melliès-Tabareau, Koudenburg and Weber, and is very close to an instance of a theorem of Koudenburg. We find this special case particularly important due to its connections with the theory of graded monads.
Given a marked $\infty$-category $\mathcal{D}^{\dagger}$ (i.e. an $\infty$-category equipped with a specified collection of morphisms) and a functor $F: \mathcal{D} \to \mathbb{B}$ with values in an $\infty$-bicategory, we define $\operatorname{colim}^{\dagger} F$, the marked colimit of $F$. We provide a definition of weighted colimits in $\infty$-bicategories when the indexing diagram is an $\infty$-category and show that they can be computed in terms of marked colimits. In the maximally marked case $\mathcal{D}^{\sharp}$, our construction retrieves the $\infty$-categorical colimit of $F$ in …
We introduce partially lax limits of infinity-categories, which interpolate between ordinary limits and lax limits. Most naturally occurring examples of lax limits are only partially lax; we give examples arising from enriched categories and operads. Our main result is a formula for partially lax limits and colimits in terms of the Grothendieck construction. This generalizes a formula of Lurie for ordinary limits and of Gepner-Haugseng-Nikolaus for fully lax limits.
Given a 2-category $\mathcal{A}$, a $2$-functor $\mathcal{A} \overset {F} {\longrightarrow} \mathcal{C}at$ and a distinguished 1-subcategory $Σ\subset \mathcal{A}$ containing all the objects, a $σ$-cone for $F$ (with respect to $Σ$) is a lax cone such that the structural $2$-cells corresponding to the arrows of $Σ$ are invertible. The conical $σ$-limit is the universal (up to isomorphism) $σ$-cone. The notion of $σ$-limit generalises the well known notions of pseudo and lax limit. We consider the fundamental notion of $σ$-filtered} pair $(\mathcal{A}, \, Σ)$ which generalises the notion of 2-f…
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…
Using the language of double categories we generalise a classical result on finite-product-preserving left Kan extensions, by Adámek and Rosický, to one on left Kan extensions that preserve algebraic structures defined by `suitable' colax-idempotent 2-monads, as well as obtain two related results. To be precise, by `suitable' 2-monads here we mean ones that extend to normal lax double monads. In an appendix we consider induced algebra structures on presheaf objects.
Building on the foundations in our previous paper, we study Segal conditions that are given by finite products, determined by structures we call cartesian patterns. We set up Day convolution on presheaves in this setting and use it to give conditions under which there is a colimit formula for free algebras and other left adjoints. This specializes to give a simple proof of Lurie's results on operadic left Kan extensions and free algebras for symmetric $\infty$-operads.
We give an introduction to the concept of Kan extensions, and study its relation with the notions of coend and adjoint functors. We state and prove in detail a well known formula to compute Kan extensions by using coends: a certain colimit related to the concept of copower. Finally, we study the tensor product of functors, and its relation with Kan extensions, in order to represent the tensor product of $\mathcal{C}$-modules as a particular case.
The fundamental construction underlying descent theory, the lax descent category, comes with a functor that forgets the descent data. We prove that, in any $2$-category $\mathfrak{A} $ with lax descent objects, the forgetful morphisms create all Kan extensions that are preserved by certain morphisms. As a consequence, in the case $\mathfrak{A} = \mathsf{Cat} $, we get a monadicity theorem which says that a right adjoint functor is monadic if and only if it is, up to the composition with an equivalence, (naturally isomorphic to) a functor that forgets descent data. In particular, within the cla…
For a category $\mathcal E$ with finite limits and well-behaved countable coproducts, we construct a model structure, called the effective model structure, on the category of simplicial objects in $\mathcal E$, generalising the Kan--Quillen model structure on simplicial sets. We then prove that the effective model structure is left and right proper and satisfies descent in the sense of Rezk. As a consequence, we obtain that the associated $\infty$-category has finite limits, colimits satisfying descent, and is locally Cartesian closed when $\mathcal E$ is, but is not a higher topos in general.…
We study the duplicial objects of Dwyer and Kan, which generalize the cyclic objects of Connes. We describe duplicial objects in terms of the decalage comonads, and we give a conceptual account of the construction of duplicial objects due to Bohm and Stefan. This is done in terms of a 2-categorical generalization of Hochschild homology. We also study duplicial structure on nerves of categories, bicategories, and monoidal categories.
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…
This chapter, written for "Stable categories and structured ring spectra," edited by Andrew J. Blumberg, Teena Gerhardt, and Michael A. Hill, surveys the history of homotopical categories, from Gabriel and Zisman's categories of fractions to Quillen's model categories, through Dwyer and Kan's simplicial localizations and culminating in $(\infty,1)$-categories, first introduced through concrete models and later re-conceptualized in a model-independent framework. This reader is not presumed to have prior acquaintance with any of these concepts. Suggested exercises are included to fertilize intui…
