Presheaf (category theory)

In , a branch of mathematics, a presheaf on a is a functor . If is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space. A morphism of presheaves is defined to be a natural transformation of functors. This makes the collection of all presheaves on into a category, and is an example of a . It is often written as . A functor into is sometimes called a profunctor. A presheaf that is naturally isomorphic to the contravariant hom-functor Hom(–, A) for some A of C is called a representable presheaf. Some authors refer to a functor as a -valued presheaf. A simplicial set is a Set-valued presheaf on the . When is a , the functor category is cartesian closed. The poset of subobjects of form a Heyting algebra, whenever is an object of for small . For any morphism of , the pullback functor of subobjects has a right adjoint, denoted , and a left adjoint, . These are the universal and existential quantifiers. A locally small category embeds fully and faithfully into the category of set-valued presheaves via the Yoneda embedding which to every object of associates the hom functor . The category admits small and small colimits. See limit and colimit of presheaves for further discussion. The states that every presheaf is a colimit of representable presheaves; in fact, is the colimit completion of (see #Universal property below.) The construction is called the colimit completion of C because of the following universal property: Proof: Given a presheaf F, by the , we can write where are objects in C. Then let which exists by assumption. Since is functorial, this determines the functor . Succinctly, is the left Kan extension of along y; hence, the name "Yoneda extension". To see commutes with small colimits, we show is a left-adjoint (to some functor). Define to be the functor given by: for each object M in D and each object U in C, Then, for each object M in D, since by the Yoneda lemma, we have: which is to say is a left-adjoint to .

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (19)

Related courses (1)

Related lectures (33)

Hom functor

In mathematics, specifically in , hom-sets (i.e. sets of morphisms between ) give rise to important functors to the . These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics. Let C be a (i.e. a for which hom-classes are actually sets and not proper classes). For all objects A and B in C we define two functors to the as follows: {| class=wikitable |- ! Hom(A, –) : C → Set ! Hom(–, B) : C → Set |- | This is a covariant functor given by: Hom(A, –) maps each object X in C to the set of morphisms, Hom(A, X) Hom(A, –) maps each morphism f : X → Y to the function Hom(A, f) : Hom(A, X) → Hom(A, Y) given by for each g in Hom(A, X).

Representable functor

In mathematics, particularly , a representable functor is a certain functor from an arbitrary into the . Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings. From another point of view, representable functors for a category C are the functors given with C. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory.

Presheaf (category theory)

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

Torsion: Hom Functoriality and Exact Sequences MATH-211: Group Theory

Explores the concept of torsion in group theory, focusing on its Hom functoriality and its role in exact sequences.

Untitled MATH-436: Homotopical algebra

Group Actions: Equivariance and Functors

Explores equivariance in group actions and functors, demonstrating its importance in group theory.