In
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, ca ...
, 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 ...
, the diagonal functor
is given by
, which maps
objects
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 ai ...
as well as
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s. This
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
can be employed to give a succinct alternate description of the product of objects ''within'' the
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) ...
: a product
is a universal arrow from
to
. The arrow comprises the projection maps.
More generally, given a
small
Small may refer to:
Science and technology
* SMALL, an ALGOL-like programming language
* Small (anatomy), the lumbar region of the back
* ''Small'' (journal), a nano-science publication
* <small>, an HTML element that defines smaller text ...
index category , one may construct the
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 in t ...
, the objects of which are called
diagrams
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three- ...
. For each object
in
, there is a
constant diagram that maps every object in
to
and every morphism in
to
. The diagonal functor
assigns to each object
of
the diagram
, and to each morphism
in
the
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 ...
in
(given for every object
of
by
). Thus, for example, in the case that
is a
discrete category In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms:
:hom''C''(''X'', ''X'') = {id''X''} for all objects ''X''
:hom''C''(''X'', ''Y'') = ∅ for all objects ''X'' ≠ '' ...
with two objects, the diagonal functor
is recovered.
Diagonal functors provide a way to define
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 2019 ...
and
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 such ...
s of diagrams. Given a
diagram
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three ...
, a natural transformation
(for some object
of
) is called a
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
for
. These cones and their factorizations correspond precisely to the objects and morphisms of the
comma category
In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become obj ...
, and a limit of
is a terminal object in
, i.e., a
universal arrow
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 fr ...
. Dually, a
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 such ...
of
is an initial object in the comma category
, i.e., a universal arrow
.
If every functor from
to
has a limit (which will be the case if
is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
), then the operation of taking limits is itself a functor from
to
. The limit functor is the
right-adjoint of the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor.
For example, the diagonal functor
described above is the left-adjoint of the binary
product functor and the right-adjoint of the binary
coproduct functor. Other well-known examples include the
pushout, which is the limit of the
span
Span may refer to:
Science, technology and engineering
* Span (unit), the width of a human hand
* Span (engineering), a section between two intermediate supports
* Wingspan, the distance between the wingtips of a bird or aircraft
* Sorbitan ester ...
, and the
terminal object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
, which is the limit of the
empty category
In linguistics, an empty category, which may also be referred to as a covert category, is an element in the study of syntax that does not have any phonological content and is therefore unpronounced.Kosta, Peter, and Krivochen, Diego Gabriel. ''Elim ...
.
See also
*
Diagram (category theory)
In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets i ...
*
Cone (category theory)
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well.
Definition
Let ''F'' : ''J'' → ''C'' be a diagram in ...
*
Diagonal morphism
In category theory, a branch of mathematics, for any object a in any category \mathcal where the product a\times a exists, there exists the diagonal morphism
:\delta_a : a \rightarrow a \times a
satisfying
:\pi_k \circ \delta_a = \operatorna ...
References
*
*
Category theory
{{cattheory-stub