Lecture

Functor Categories: (Co)Limits and Simplicial Sets

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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 delves into the computation of (co)limits in functor categories, with a focus on simplicial sets. The instructor explains the process of calculating limits and colimits in categories, starting with small categories J and I. The lecture covers the calculation of limi, &, and colimi I, with examples such as the case J = Two. The slides also discuss the history of simplicial sets and their significance in homotopical algebra. Furthermore, the lecture explores the concept of equalizers and coequalizers of simplicial maps, emphasizing the importance of faces and degeneracies in defining them.

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Topology: Algebraic topology

Related concepts (32)

Limit (category theory)

In , a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as , and inverse limits. The of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, s and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize.

Coequalizer

In , a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary . It is the categorical construction to the equalizer. A coequalizer is a colimit of the diagram consisting of two objects X and Y and two parallel morphisms f, g : X → Y. More explicitly, a coequalizer of the parallel morphisms f and g can be defined as an object Q together with a morphism q : Y → Q such that q ∘ f = q ∘ g.

Functor category

In , a branch of mathematics, a functor category is a category where the objects are the functors and the morphisms are natural transformations between the functors (here, is another object in the category). Functor categories are of interest for two main reasons: many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable; every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.

Functor

In mathematics, specifically , a functor is a mapping between . Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which is applied.

Simplicial set

In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and . Formally, a simplicial set may be defined as a contravariant functor from the to the . Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization.

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