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 ( ...
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 ...
. If
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
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
and it is called the category of presheaves on
. A functor into
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
as a
-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 .
* A
directed multigraph is a presheaf on the category with two elements and two parallel morphisms between them i.e.
.
* An
arrow category is a presheaf on the category with two elements and one morphism between them. i.e.
.
* A
right group action is a presheaf on the category created from a group
, i.e. a category with one element and invertible morphisms.
Properties
* When
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
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
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
is an object of
for small
.
* For any morphism
of
, the pullback functor of subobjects
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
, 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
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
of
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 ...
.
*The category
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,
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
(see
#Universal property below.)
Universal property
The construction
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
where
are objects in ''C''. Then let
which exists by assumption. Since
is functorial, this determines the functor
. Succinctly,
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
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
.
The proposition yields several corollaries. For example, the proposition implies that the construction
is functorial: i.e., each functor
determines the functor
.
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:
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 ''C
op''. 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 fileprovided by nlab.
Functors
Sheaf theory
Topos theory