diagonal functor

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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, cat ...
, 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 $\mathcal \rightarrow \mathcal \times \mathcal$ is given by $\Delta\left(a\right) = \langle a,a \rangle$, which maps objects 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, morphism ...
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 ma ...
can be employed to give a succinct alternate description of the product of objects ''within'' the
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$\mathcal$: 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
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index category $\mathcal$, 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 natura ...
$\eta$ 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'' ≠ ''Y ...
with two objects, the diagonal functor $\mathcal \rightarrow \mathcal \times \mathcal$ 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 colimits 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 thr ...
$\mathcal : \mathcal \rightarrow \mathcal$, 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 ...
for $\mathcal$. 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 objec ...
$\left(\Delta\downarrow\mathcal\right)$, and a limit of $\mathcal$ is a terminal object in $\left(\Delta\downarrow\mathcal\right)$, i.e., a universal arrow $\Delta \rightarrow \mathcal$. Dually, a colimit of $\mathcal$ is an initial object in the comma category $\left(\mathcal\downarrow\Delta\right)$, 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), 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, which is the limit of the span, 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.