Lecture

Trivial Functor: Definition and Properties

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Description

This lecture introduces the trivial functor of a category, defining it as a pair of functions that preserve identities and compositions. The properties of the trivial functor are explored, motivated by the analysis of specific cases. The lecture covers the adjoint functor to the orbits functor, emphasizing its existence conditions and natural isomorphisms.

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Related concepts (26)

Adjoint functors

In mathematics, specifically , adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e.

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.

Category theory

Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in almost all areas of mathematics. In particular, numerous constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories.

Forgetful functor

In mathematics, in the area of , a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure.

Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a ) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the , whose objects are sets and whose arrows are functions. is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent.

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