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, 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 $\backslash mathcal\; \backslash rightarrow\; \backslash mathcal\; \backslash times\; \backslash mathcal$ is given by $\backslash Delta(a)\; =\; \backslash langle\; a,a\; \backslash rangle$, 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 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)
* ...

$\backslash mathcal$: a product $a\; \backslash times\; b$ is a universal arrow from $\backslash Delta$ to $\backslash langle\; a,b\; \backslash rangle$. 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 $\backslash mathcal$, 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 ...

$\backslash mathcal^\backslash mathcal$, 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 $\backslash mathcal$, there is a constant diagram $\backslash Delta\_a\; :\; \backslash mathcal\; \backslash to\; \backslash mathcal$ that maps every object in $\backslash mathcal$ to $a$ and every morphism in $\backslash mathcal$ to $1\_a$. The diagonal functor $\backslash Delta\; :\; \backslash mathcal\; \backslash rightarrow\; \backslash mathcal^\backslash mathcal$ assigns to each object $a$ of $\backslash mathcal$ the diagram $\backslash Delta\_a$, and to each morphism $f:\; a\; \backslash rightarrow\; b$ in $\backslash mathcal$ 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 ...

$\backslash eta$ in $\backslash mathcal^\backslash mathcal$ (given for every object $j$ of $\backslash mathcal$ by $\backslash eta\_j\; =\; f$). Thus, for example, in the case that $\backslash mathcal$ 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 $\backslash mathcal\; \backslash rightarrow\; \backslash mathcal\; \backslash times\; \backslash mathcal$ is recovered.
Diagonal functors provide a way to define limits
Limit or Limits may refer to:
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* "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 ...

$\backslash mathcal\; :\; \backslash mathcal\; \backslash rightarrow\; \backslash mathcal$, a natural transformation $\backslash Delta\_a\; \backslash to\; \backslash mathcal$ (for some object $a$ of $\backslash mathcal$) 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 $\backslash mathcal$. 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 ...

$(\backslash Delta\backslash downarrow\backslash mathcal)$, and a limit of $\backslash mathcal$ is a terminal object in $(\backslash Delta\backslash downarrow\backslash mathcal)$, i.e., a universal arrow $\backslash Delta\; \backslash rightarrow\; \backslash mathcal$. Dually, a colimit of $\backslash mathcal$ is an initial object in the comma category $(\backslash mathcal\backslash downarrow\backslash Delta)$, i.e., a universal arrow $\backslash mathcal\; \backslash rightarrow\; \backslash Delta$.
If every functor from $\backslash mathcal$ to $\backslash mathcal$ has a limit (which will be the case if $\backslash mathcal$ is complete), then the operation of taking limits is itself a functor from $\backslash mathcal^\backslash mathcal$ to $\backslash mathcal$. 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 $\backslash mathcal\; \backslash rightarrow\; \backslash mathcal\; \backslash times\; \backslash mathcal$ 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.
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)
* Diagonal morphism
References

* * Category theory {{cattheory-stub