arXiv subjects

Emmanuel Dror Farjoun

Publications and source records attributed to Emmanuel Dror Farjoun.

6 recordsLinked to original sources

arXiv2025

Bousfield-Kan completion as a codensity $\infty$-monad

Working in the setting of $\infty$-categories, we develop a general theory of the codensity monad $T_\mathcal{D}$ associated with a full subcategory $\mathcal{D}\subseteq \mathcal{C}$. We show that $T_\mathcal{D}$ has a canonical monad structure (unique up to a contractible space of choices), and characterize it as a terminal monad preserving all objects of $\mathcal{D}$. For a monad $\mathcal{M}$ on an $\infty$-category $\mathcal{C}$, we consider the $\mathcal{M}$-completion functor defined as the totalization of the cosimplicial resolution associated with $\mathcal{M}$. We show that the $\mathcal{M}$-completion functor is the codensity monad associated with the full subcategory of $\mathcal{C}$ spanned by objects that admit a structure of $\mathcal{M}$-algebra. In particular, the $\mathcal{M}$-completion functor is the terminal monad preserving all objects that admit a structure of an $\mathcal{M}$-algebra. This gives a full $\infty$-categorical characterization of the classical Bousfield-Kan $R$-completion functor as the terminal monad on the category of spaces preserving the empty space and all products of Eilenberg-MacLane spaces $K(A,n)$, where $A$ is an $R$-module.

math.AT↗

arXiv2024

A Note on the Non-Existence of Functors

We consider several types of non-existence theorems for functors. For example, there are no nontrivial functors from the category of groups (or the category of pointed sets, or vector spaces) to any small category. Another type of questions that we consider are questions about nonexistence of subfunctors and quotients of the identity functor on the category of groups (or abelian groups). For example, there is no a natural non-trivial way to define an abelian subgroup of a group, or a perfect quotient group of a group. As an auxiliary result we prove that, for any non-trivial subfunctor $F$ of the identity functor on the category of groups, any group can be embedded into a simple group that lies in the essential image of $F.$ The paper concludes with a few questions regarding the non-existence of certain (co-)augmented functors in the $\infty$-category of spaces.

math.CT↗

arXiv2023

Completions and Terminal Monads

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 the category of co-augmented endo-functors. Once extended to infinity categories, this gives, for example, a complete characterization of the well-known Bousfield-Kan R-homology completion. In addition, we note that an idempotent pro-completion tower can be associated with any co-augmented endo functor M, whose limit is the terminal monad that preserves the closure of ImM, the image of M, under finite limits. We conclude that some basic properties of the homological completion tower of a space can be formulated and proved for general monads over any category with limits, and characterized as universal

math.CT↗

arXiv2014

Homotopy colimits of nilpotent spaces

We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield-Kan homology completion tower z_k X whose terms we prove are all X-cellular for any X. As straightforward consequences, we show that if X is K-acyclic and nilpotent for a given homology theory K, then so are all its Postnikov sections, and that any nilpotent space for which the space of pointed self-maps map_*(X,X) is "canonically" discrete must be aspherical.

math.AT↗

arXiv2013

Conditionally flat functors on spaces and groups

Consider an extension of groups 1 -> K -> G -> Q -> 1 which enjoys the property that the quotient by the lower central series Gamma_{c+1} produces another extension 1 -> K/ Gamma_{c+1} K -> G /Gamma_{c+1} G -> Q / Gamma_{c+1} Q -> 1, of nilpotent groups of class c. We say that the extension is Gamma_{c+1}-flat. Let us pull back the original extension along any homomorphism X -> Q. Does the pullback extension enjoy the same Gamma_{c+1}-flatness property? To answer this question we consider not only quotients by the lower central series, but any localization functor in the category of groups. In fact we start by studying the analogous question for spaces, where we replace extensions by fibration sequences. We prove that the only homotopical localization functors which behave well under pull-backs are nullifications. In the category of groups, nullifications also enjoy this property, and so do all epireflections arising from a variety of groups. In particular the answer to the question about the nilpotent quotients is positive.

math.AT↗

arXiv2012

Spaces of sections of Banach algebra bundles

Suppose that $B$ is a $G$-Banach algebra over $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, $X$ is a finite dimensional compact metric space, $ζ: P \to X$ is a standard principal $G$-bundle, and $A_ζ= Γ(X, P \times_G B)$ is the associated algebra of sections. We produce a spectral sequence which converges to $π_*(GL_o A_ζ) $ with [E^2_{-p,q} \cong \check{H}^p(X ; π_q(GL_o B)).] A related spectral sequence converging to $\K_{*+1}(A_ζ)$ (the real or complex topological $K$-theory) allows us to conclude that if $B$ is Bott-stable, (i.e., if $ π_*(GL_o B) \to \K_{*+1}(B)$ is an isomorphism for all $*>0$) then so is $A_ζ$.

math.OA↗