Related papers: Bousfield-Kan completion as a codensity $\infty$-m…

We consider the terminal monad among those preserving the objects of a subcategory, and in particular preserving the image of a monad. Several common monads are shown to be uniquely characterized by the property of being terminal objects in…

Consider a diagram of quasi-categories that admit and functors that preserve limits or colimits of a fixed shape. We show that any weighted limit whose weight is a projective cofibrant simplicial functor is again a quasi-category admitting…

Given a monad T on a suitable enriched category B equipped with a proper factorization system (E,M), we define notions of T-completion, T-closure, and T-density. We show that not only the familiar notions of completion, closure, and density…

For a number of locally finitely presentable categories K we describe the codensity monad of the full embedding of all finitely presentable objects into K. We introduce the concept of D-ultrafilter on an object, where D is a "nice"…

Given an operad A of topological spaces, we consider A-monads in a topological category C . When A is an A-infinity-operad, any A-monad K : C -> C can be thought of as a monad up to coherent homotopies. We define the completion functor with…

We prove a number of results of the following common flavor: for a category $\mathcal{C}$ of topological or uniform spaces with all manner of other properties of common interest (separation / completeness / compactness axioms), a group (or…

For a cofibrantly generated Quillen model category, we show that the cofibrant replacement functor constructed using the small object argument admits a cotriple structure. If all acyclic cofibrations are monomorphisms, the fibrant…

We study codensity monads $T$ induced by (mostly small, mostly dense) full subcategories $A \subset K$. These monads behave quite similarly, we show some connections with the Isbell duality, pro-finite objects and compact spaces. We prove…

We extend some classical results of Bousfield on homology localizations and nilpotent completions to a presentably symmetric monoidal stable $\infty$-category $\mathscr{M}$ admitting a multiplicative left-complete $t$-structure. If $E$ is a…

We consider the localization of the $\infty$-category of spaces at the $v_n$-periodic equivalences, the case $n=0$ being rational homotopy theory. We prove that this localization is for $n\geq 1$ equivalent to algebras over a certain monad…

A group completion functor Q is constructed in the category of algebras in simplicial sets over a cofibrant E_n-operad M. It is shown that Q defines a Bousfield-Friedander simplicial model category on M-algebras.

We review the condensation completion of a modular tensor category $\mathcal{C}$, which yields a fusion 2-category $\Sigma\mathcal{C}$ of separable algebras, bimodules over algebras and bimodule maps in $\mathcal{C}$. Physically,…

Let $A$ be a finite-dimensional algebra, and $\mathfrak{M}$ be a $d$-cluster tilting subcategory of mod$A$. From the viewpoint of higher homological algebra, a natural question to ask is when $\mathfrak{M}$ induces a $d$-cluster tilting…

The goal of this paper is to prove coherence results with respect to relational graphs for monoidal monads and comonads, i.e. monads and comonads in a monoidal category such that the endofunctor of the monad or comonad is a monoidal functor…

We develop a general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model categories, extending work of Bousfield-Kan and Bendersky-Thompson for ordinary spaces. This is based on a…

In this paper we prove that for any model category, the Bousfield-Kan construction of the homotopy colimit is the absolute left derived functor of the colimit. This is achieved by showing that the Bousfield-Kan homotopy colimit is moreover…

Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence…

We provide conditions on a monoidal model category $\mathcal{M}$ so that the category of commutative monoids in $\mathcal{M}$ inherits a model structure from $\mathcal{M}$ in which a map is a weak equivalence or fibration if and only if it…

Every right adjoint functor between presentable $\infty$-categories is shown to decompose canonically as a coreflection, followed by, possibly transfinitely many, monadic functors. Furthermore, the coreflection part is given a presentation…

For a (possibly large) realized limit sketch $\mathcal{S}$ such that every $\mathcal{S}$-model is small in a suitable sense we show that the category of cocontinuous functors $\mathsf{Mod}(\mathcal{S}) \to \mathcal{C}$ into a cocomplete…