Lecture

Categories and Functors: An Introduction

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Description

This lecture introduces the fundamental concepts of categories, functors, and natural transformations. It begins with the definition of categories, highlighting the importance of objects and morphisms. The instructor explains groups and monoids, emphasizing their role as categories. The concept of invertibility in categories is discussed, illustrating how morphisms can be invertible. The lecture then transitions to the construction of new categories from existing ones, using examples such as posets and digraphs. The instructor elaborates on the relationship between categories and functors, detailing how functors map between categories while preserving structure. The distinction between forgetful functors and inclusion functors is made clear, showcasing their applications in various mathematical contexts. The lecture concludes with a discussion on presheaves and their significance in category theory, providing canonical examples to illustrate these concepts. Overall, this lecture serves as a foundational overview of category theory, essential for further studies in mathematics and related fields.

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Related concepts (32)

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.

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.

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.

Full and faithful functors

In , a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a fully faithful functor. Explicitly, let C and D be () and let F : C → D be a functor from C to D. The functor F induces a function for every pair of objects X and Y in C. The functor F is said to be faithful if FX,Y is injective full if FX,Y is surjective fully faithful (= full and faithful) if FX,Y is bijective for each X and Y in C.

Monad (category theory)

In , a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a in the of endofunctors of some fixed category. An endofunctor is a functor mapping a category to itself, and a monad is an endofunctor together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories.

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