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. Nowadays, cate ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a subfunctor is a special type of

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 ...

that is an analogue of a

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

Definition

Let C be a

category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

, and let ''F'' be a contravariant functor from C to the

category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of m ...

Set. A contravariant functor ''G'' from C to Set is a subfunctor of ''F'' if # For all objects ''c'' of C, ''G''(''c'') ⊆ ''F''(''c''), and # For all arrows ''f'': ''c''′ → ''c'' of C, ''G''(''f'') is the

restriction Restriction, restrict or restrictor may refer to: Science and technology * restrict, a keyword in the C programming language used in pointer declarations * Restriction enzyme, a type of enzyme that cleaves genetic material Mathematics and logi ...

of ''F''(''f'') to ''G''(''c''). This relation is often written as ''G'' ⊆ ''F''. For example, let 1 be the category with a single object and a single arrow. A functor ''F'': 1 → Set maps the unique object of 1 to some set ''S'' and the unique identity arrow of 1 to the identity function 1_''S'' on ''S''. A subfunctor ''G'' of ''F'' maps the unique object of 1 to a subset ''T'' of ''S'' and maps the unique identity arrow to the identity function 1_''T'' on ''T''. Notice that 1_''T'' is the restriction of 1_''S'' to ''T''. Consequently, subfunctors of ''F'' correspond to subsets of ''S''.

Remarks

Subfunctors in general are like global versions of subsets. For example, if one imagines the objects of some category C to be analogous to the open sets of a topological space, then a contravariant functor from C to the category of sets gives a set-valued

presheaf In mathematics, a sheaf 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 data could ...

on C, that is, it associates sets to the objects of C in a way that is compatible with the arrows of C. A subfunctor then associates a subset to each set, again in a compatible way. The most important examples of subfunctors are subfunctors of the

Hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...

. Let ''c'' be an object of the category C, and consider the functor . This functor takes an object ''c''′ of C and gives back all of the morphisms ''c''′ → ''c''. A subfunctor of gives back only some of the morphisms. Such a subfunctor is called a

sieve A sieve, fine mesh strainer, or sift, is a device for separating wanted elements from unwanted material or for controlling the particle size distribution of a sample, using a screen such as a woven mesh or net or perforated sheet material. T ...

, and it is usually used when defining

Grothendieck topologies In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is cal ...

Open subfunctors

Subfunctors are also used in the construction of

representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and ...

s on the category of

ringed space In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...

s. Let ''F'' be a contravariant functor from the category of ringed spaces to the category of sets, and let ''G'' ⊆ ''F''. Suppose that this inclusion morphism ''G'' → ''F'' is representable by open immersions, i.e., for any representable functor and any morphism , the fibered product is a representable functor {{nowrap, Hom(−, ''Y'') and the morphism ''Y'' → ''X'' defined by the

Yoneda lemma In mathematics, the Yoneda lemma is arguably the most important 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 (view ...

is an open immersion. Then ''G'' is called an open subfunctor of ''F''. If ''F'' is covered by representable open subfunctors, then, under certain conditions, it can be shown that ''F'' is representable. This is a useful technique for the construction of ringed spaces. It was discovered and exploited heavily by Alexander Grothendieck, who applied it especially to the case of

scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...

s. For a formal statement and proof, see Grothendieck, ''

Éléments de géométrie algébrique The ''Éléments de géométrie algébrique'' ("Elements of algebraic geometry, Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or ''EGA'' for short, is a rigorous treatise, in French language, French, on algebraic ge ...

'', vol. 1, 2nd ed., chapter 0, section 4.5. Functors