Related papers: A note on the definition of derived functors

We define the Grothendieck-Witt category over a fixed ground ring. In order to study the structure of this category, we introduce the general theory of Gysin functors and their associated categories of correspondences. The latter…

Taking symmetric powers of varieties can be seen as a functor from the category of varieties to the category of varieties with an action by the symmetric group. We study a corresponding map between the Grothendieck groups of these…

Representations over diagrams of abelian categories unify quite a few notions appearing widely in literature such as representations of categories, presheaves of modules over categories, representations of species, etc. In this series of…

Let A be an algebra with a countable basis and let B be, say, a Frechet algebra that contains A as a dense subalgebra. This embedding induces a functor from the derived category of B-modules to the derived category of A-modules. In many…

This paper develops two theories, the geometric theory of derived Grassmannians (and flag schemes) and the algebraic theory of derived Schur (and Weyl) functors, and establishes their connection, a derived generalization of the…

We are checking the closed categories beginning with the category of sets and ending with the category of categories. The novelty is a generalizing the notion of adjoint functors to the joint pair of functors in the category of directed…

Motivated by the Grothendieck construction, we study the functorialities of the comma construction for strict $\omega$-categories. To state the most general functorialities, we use the language of Gray $\omega$-categories, that is,…

We study fiber functors on Tannakian categories which are equipped with a grading or a filtration. Our goal is to give a comprehensive set of foundational results about such functors. A main result is that each filtration on a fiber functor…

Given a cartesian closed category $\mathcal{V}$, we introduce an internal category of elements $\int_\mathcal{C} F$ associated to a $\mathcal{V}$-functor $F\colon \mathcal{C}^{\mathrm{op}}\to \mathcal{V}$. When $\mathcal{V}$ is extensive,…

A covariant functor from the category of the complex tori to the category of the Effros-Shen algebras is constructed. The functor maps isomorphic complex tori to the stably isomorphic Effros-Shen algebras. Our construction is based on the…

Voevodsky's derived category of motives is the main arena today for the study of algebraic cycles and motivic cohomology. In this paper we study whether the inclusions of three important subcategories of motives have a left or right…

The category of contexts underlying a model of Martin-L\"of type theory with Unit-, $\Sigma$-, and $\Pi$-types need not be locally Cartesian closed, but is necessarily a $\pi$-clan. We exploit this $\pi$-clan structure to build the theory…

This article is first in a series of papers where we reprove the statements in constructing the Enhanced Operation Map and the abstract six-functor formalism developed by Liu-Zheng. In this paper, we prove a theorem regarding constructing…

The well-known difficulties arising in a classification which is not set-theoretically trivial---involving what is sometimes called a non-smooth quotient---have been overcome in a striking way in the theory of operator algebras by the use…

We generalize the adjunction between the functors $Rf_*$ and $f^!$ of derived categories of quasi-coherent sheaves for proper morphisms $f\colon X \to Y$ of Noetherian schemes to the following situation: Let $f$ be a finite type morphism…

We describe a procedure for constructing morphisms in additive categories, combining Auslander's concept of a morphism determined by an object with the existence of flat covers. Also, we show how flat covers are turned into projective…

We study finite-dimensional groups definable in models of the theory of real closed fields with a generic derivation (also known as CODF). We prove that any such group definably embeds in a semialgebraic group. We extend the results to…

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…

We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen's small object argument). The necessity of such a generalization arose with appearance of several…

The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…