Diagonal functor
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In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, 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 ...
as well as morphisms. 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) ...
\mathcal: a product a \times b is a universal arrow from \Delta to \langle a,b \rangle. The arrow comprises the projection maps. More generally, given a small
index category 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 is ...
\mathcal, 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 ...
\mathcal^\mathcal, 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 a in \mathcal, there is a constant diagram \Delta_a : \mathcal \to \mathcal that maps every object in \mathcal to a and every morphism in \mathcal to 1_a. The diagonal functor \Delta : \mathcal \rightarrow \mathcal^\mathcal assigns to each object a of \mathcal the diagram \Delta_a, and to each morphism f: a \rightarrow b in \mathcal 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 ...
\eta in \mathcal^\mathcal (given for every object j of \mathcal by \eta_j = f). Thus, for example, in the case that \mathcal 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 \mathcal \rightarrow \mathcal \times \mathcal is recovered. Diagonal functors provide a way to define
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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 \mathcal : \mathcal \rightarrow \mathcal, a natural transformation \Delta_a \to \mathcal (for some object a of \mathcal) 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 \mathcal. These cones and their factorizations correspond precisely to the objects and morphisms of the comma category (\Delta\downarrow\mathcal), and a limit of \mathcal is a terminal object in (\Delta\downarrow\mathcal), 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 ...
\Delta \rightarrow \mathcal. 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 \mathcal is an initial object in the comma category (\mathcal\downarrow\Delta), i.e., a universal arrow \mathcal \rightarrow \Delta. If every functor from \mathcal to \mathcal has a limit (which will be the case if \mathcal 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 \mathcal^\mathcal to \mathcal. 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 \mathcal \rightarrow \mathcal \times \mathcal 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 A ''pushout'' is a student who leaves their school before graduation, through the encouragement of the school. A student who leaves of their own accord (e.g., to work or care for a child), rather than through the action of the school, is consider ...
, 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 = \operatorn ...


References

* * Category theory {{cattheory-stub