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Lecture

Derived functors: Two technical lemmas

In course

MATH-436: Homotopical algebra

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous examples of model categories and their applications in ...

Description

This lecture focuses on two technical lemmas crucial for proving the Fundamental Theorem of homotopical algebra. The first lemma establishes the homotopy equivalence of two objects, while the second lemma involves a Quillen pair and morphisms between cofibrant and fibrant objects. The instructor presents detailed proofs for both lemmas, setting the stage for the upcoming proof of the Fundamental Theorem.

Official source

Ontological neighbourhood

Mathematics

Topology: Algebraic topology

Related concepts (32)

Homotopy

In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (həˈmɒtəpiː, ; ˈhoʊmoʊˌtoʊpiː, ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces.

Homological algebra

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of .

Homotopical connectivity

In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of n-connectedness generalizes the concepts of path-connectedness and simple connectedness. An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is n-connected (or n-simple connected) if its first n homotopy groups are trivial.

Derived functor

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. It was noted in various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations. Suppose we are given a covariant left exact functor F : A → B between two A and B.

Quillen adjunction

In homotopy theory, a branch of mathematics, a Quillen adjunction between two C and D is a special kind of adjunction between that induces an adjunction between the Ho(C) and Ho(D) via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen. Given two closed model categories C and D, a Quillen adjunction is a pair (F, G): C D of adjoint functors with F left adjoint to G such that F preserves cofibrations and trivial cofibrations or, equivalently by the closed model axioms, such that G preserves fibrations and trivial fibrations.

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Explores the homotopy category of chain complexes and the relation between quasi-isomorphisms and chain homotopy equivalences.

Derived Functors in Homotopical Algebra