We show that categories of modules over a ring in homotopy type theory (HoTT) satisfy the
internal versions of the AB axioms from homological algebra. The main subtlety lies in …

Homotopy limits and colimits are homotopical replacements for the usual limits and colimits
of category theory, which can be approached either using classical explicit constructions or …

The notion of logos was introduced by Boardman and Vogt in their work on homotopy
invariant algebraic structures [BV]. The following notes are a collection of assertions on …

The goal of this work is to develop a functional transfer of properties between a module A
and the category* M* E of right modules over its endomorphism ring* E that is more sensitive …

The basic notions of category theory, such as limit, adjunction, and orthogonality, all involve
assertions of the existence and uniqueness of certain arrows. Weak notions arise when one …

The search, of almost a century long, for a unified axiomatic framework for establishing
homomorphism theorems of classical algebra (such as Noether isomorphism theorems and …

We show that, over an arbitrary commutative ring, the localizations of the categories of dg
categories, of unital and of strictly unital A∞ categories with respect to the corresponding …

Homotopy coends are an example of an indexed homotopy colimit. In this expository note,
we give a definition of homotopy coend for arbitrary unenriched model categories and show …

The category of unital modules over a ring with identity element is characterized as a
cocomplete abelian category with a progenerator. More generally, every cocomplete abelian …

An elementary notion of homotopy can be introduced between arrows in a cartesian closed
category $ E $. The input is a finite-product-preserving endofunctor $\Pi_0 $ with a natural …