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In
category theory Category theory is a general theory of mathematical structures and their relations. It 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 ...
, a branch of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a presheaf on a
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
C is a
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
F\colon C^\mathrm\to\mathbf. If C is the
poset In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
of
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s in a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, interpreted as a category, then one recovers the usual notion of
presheaf In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the d ...
on a topological space. A
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
of presheaves is defined to be a
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
of functors. This makes the collection of all presheaves on C into a category, and is an example of a
functor category In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object i ...
. It is often written as \widehat = \mathbf^ and it is called the category of presheaves on C. A functor into \widehat is sometimes called a profunctor. A presheaf that is
naturally isomorphic In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
to the contravariant
hom-functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between object (category theory), objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applicati ...
Hom(–, ''A'') for some
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an a ...
''A'' of C is called a representable presheaf. Some authors refer to a functor F\colon C^\mathrm\to\mathbf as a \mathbf-valued presheaf.


Examples

* A
simplicial set In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "n ...
is a Set-valued presheaf on the simplex category C=\Delta. * A directed multigraph is a presheaf on the category with two elements and two parallel morphisms between them i.e. C = (E \overset V). * An arrow category is a presheaf on the category with two elements and one morphism between them. i.e. C = (E \overset V). * A right group action is a presheaf on the category created from a group G, i.e. a category with one element and invertible morphisms.


Properties

* When C is a
small category In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
, the functor category \widehat=\mathbf^ is
cartesian closed In category theory, a Category (mathematics), category is Cartesian closed if, roughly speaking, any morphism defined on a product (category theory), product of two Object (category theory), objects can be naturally identified with a morphism defin ...
. * The poset of
subobject In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory ...
s of P form a
Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' call ...
, whenever P is an object of \widehat=\mathbf^ for small C. * For any morphism f:X\to Y of \widehat, the pullback functor of subobjects f^*:\mathrm_(Y)\to\mathrm_(X) has a
right adjoint In mathematics, specifically category theory, 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 k ...
, denoted \forall_f, and a left adjoint, \exists_f. These are the universal and existential quantifiers. * A locally small category C embeds fully and faithfully into the category \widehat of set-valued presheaves via the
Yoneda embedding In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a ...
which to every object A of C associates the
hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between object (category theory), objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applicati ...
C(-,A). *The category \widehat admits small
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2009 ...
and small
colimit In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions s ...
s. See
limit and colimit of presheaves In category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topolog ...
for further discussion. * The density theorem states that every presheaf is a colimit of representable presheaves; in fact, \widehat is the
colimit In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions s ...
completion of C (see #Universal property below.)


Universal property

The construction C \mapsto \widehat = \mathbf(C^, \mathbf) is called the colimit completion of ''C'' because of the following
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
: ''Proof'': Given a presheaf ''F'', by the density theorem, we can write F =\varinjlim y U_i where U_i are objects in ''C''. Then let \widetilde F = \varinjlim \eta U_i, which exists by assumption. Since \varinjlim - is functorial, this determines the functor \widetilde: \widehat \to D. Succinctly, \widetilde is the left
Kan extension Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan, who constructed certain (Kan) extensions us ...
of \eta along ''y''; hence, the name "Yoneda extension". To see \widetilde commutes with small colimits, we show \widetilde is a left-adjoint (to some functor). Define \mathcalom(\eta, -): D \to \widehat to be the functor given by: for each object ''M'' in ''D'' and each object ''U'' in ''C'', :\mathcalom(\eta, M)(U) = \operatorname_D(\eta U, M). Then, for each object ''M'' in ''D'', since \mathcalom(\eta, M)(U_i) = \operatorname(y U_i, \mathcalom(\eta, M)) by the Yoneda lemma, we have: :\begin \operatorname_D(\widetilde F, M) &= \operatorname_D(\varinjlim \eta U_i, M) = \varprojlim \operatorname_D(\eta U_i, M) = \varprojlim \mathcalom(\eta, M)(U_i) \\ &= \operatorname_(F, \mathcalom(\eta, M)), \end which is to say \widetilde is a left-adjoint to \mathcalom(\eta, -). \square The proposition yields several corollaries. For example, the proposition implies that the construction C \mapsto \widehat is functorial: i.e., each functor C \to D determines the functor \widehat \to \widehat.


Variants

A presheaf of spaces on an ∞-category ''C'' is a contravariant functor from ''C'' to the
∞-category of spaces In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy theory speaking, that the ∞-groupoids are space (mathematics), spaces. One version of the hypothesis was claimed to be proved in the 1991 paper by M ...
(for example, the nerve of the category of
CW-complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
es.) It is an
∞-category In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a Category (ma ...
version of a presheaf of sets, as a "set" is replaced by a "space". The notion is used, among other things, in the ∞-category formulation of Yoneda's lemma that says: C \to \widehat is fully faithful (here ''C'' can be just a
simplicial set In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "n ...
.) A copresheaf of a category ''C'' is a presheaf of ''Cop''. In other words, it is a covariant functor from ''C'' to ''Set''.


See also

*
Topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notio ...
*
Category of elements In category theory, a branch of mathematics, the category of elements of a presheaf is a category associated to that presheaf whose objects are the elements of sets in the presheaf. It and its generalization are also known as the Grothendieck cons ...
* Simplicial presheaf (this notion is obtained by replacing "set" with "simplicial set") * Presheaf with transfers


Notes


References

* * * *


Further reading

* * *{{nlab, id=free+cocompletion, title=Free cocompletion *Daniel Dugger
Sheaves and Homotopy Theory
th
pdf file
provided by nlab. Functors Sheaf theory Topos theory