Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSo, what is a derived functor?
149
0
0.0
(
0
)
Authors
V. Hinich
Ask ChatGPT about the research
No Arabic abstract
In the context of infinity categories, we rethink the notion of derived functor in terms of correspondences. This is especially convenient for the description of a passage from an adjoint pair (F,G) of functors to a derived adjoint pair (LF,RG). In particular, canonicity of this passage becomes obvious. 2nd version: added comparison to Delignes definition (SGA4) and a discussion of diagrams of derived functors. Introduction rewritten and references added. 3rd version: description of Kan extensions in terms of correspondences more detailed. 4th version: the final version accepted to HHA.
rate research
Read More
For topological spaces $X$ and $Y$, a (not necessarily continuous) function $f:X rightarrow Y$ naturally induces a functor from the category of closed subsets of $X$ (with morphisms given by inclusions) to the category of closed subsets of $Y$. The function $f$ also naturally induces a functor from the category of closed subsets of $Y$ to the category of closed subsets of $X$. Our aim in this expository note is to show that the function $f$ is continuous if and only if the first of the above two functors is a left adjoint to the second. We thereby obtain elementary examples of adjoint pairs (apparently) not part of the standard introductory treatments of this subject.
We show that the hypercohomology of the Chevalley-Eilenberg-de Rham complex of a Lie algebroid L over a scheme with coefficients in an L-module can be expressed as a derived functor. We use this fact to study a Hochschild-Serre type spectral sequence attached to an extension of Lie algebroids.
We show that the comma category $(mathcal{F}downarrowmathbf{Grp})$ of groups under the free group functor $mathcal{F}: mathbf{Set} to mathbf{Grp}$ contains the category $mathbf{Gph}$ of simple graphs as a full coreflective subcategory. More broadly, we generalize the embedding of topological spaces into Steven Vickers category of topological systems to a simple technique for embedding certain categories into comma categories, then show as a straightforward application that simple graphs are coreflective in $(mathcal{F}downarrowmathbf{Grp})$.
Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $infty$-categories. One of our main results is an $infty$-categorical generalization of Freyds classical General Adjoint Functor Theorem. As an application of this result, we recover Luries adjoint functor theorems for presentable $infty$-categories. We also discuss the comparison between adjunctions of $infty$-categories and homotopy adjunctions, and give a treatment of Brown representability for $infty$-categories based on Hellers purely categorical formulation of the classical Brown representability theorem.
Let $mathcal{B}$ be a subcategory of a given category $mathcal{D}$. Let $mathcal{B}$ has monoidal structure. In this article, we discuss when can one extend the monoidal structure of $mathcal{B}$ to $mathcal{D}$ such that $mathcal{B}$ becomes a sub monoidal category of monoidal category $mathcal{D}$. Examples are discussed, and in particular, in an example of loop space, we elaborated all results discussed in this article.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
