To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

# Diagonal functor

In category theory, a branch of mathematics, the diagonal functor ${\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}$ is given by ${\displaystyle \Delta (a)=\langle a,a\rangle }$, which maps objects as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects within the category ${\displaystyle {\mathcal {C}}}$: a product ${\displaystyle a\times b}$ is a universal arrow from ${\displaystyle \Delta }$ to ${\displaystyle \langle a,b\rangle }$. The arrow comprises the projection maps.

More generally, given a small index category ${\displaystyle {\mathcal {J}}}$, one may construct the functor category ${\displaystyle {\mathcal {C}}^{\mathcal {J}}}$, the objects of which are called diagrams. For each object ${\displaystyle a}$ in ${\displaystyle {\mathcal {C}}}$, there is a constant diagram ${\displaystyle \Delta _{a}:{\mathcal {J}}\to {\mathcal {C}}}$ that maps every object in ${\displaystyle {\mathcal {J}}}$ to ${\displaystyle a}$ and every morphism in ${\displaystyle {\mathcal {J}}}$ to ${\displaystyle 1_{a}}$. The diagonal functor ${\displaystyle \Delta :{\mathcal {C}}\rightarrow {\mathcal {C}}^{\mathcal {J}}}$ assigns to each object ${\displaystyle a}$ of ${\displaystyle {\mathcal {C}}}$ the diagram ${\displaystyle \Delta _{a}}$, and to each morphism ${\displaystyle f:a\rightarrow b}$ in ${\displaystyle {\mathcal {C}}}$ the natural transformation ${\displaystyle \eta }$ in ${\displaystyle {\mathcal {C}}^{\mathcal {J}}}$ (given for every object ${\displaystyle j}$ of ${\displaystyle {\mathcal {J}}}$ by ${\displaystyle \eta _{j}=f}$). Thus, for example, in the case that ${\displaystyle {\mathcal {J}}}$ is a discrete category with two objects, the diagonal functor ${\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}$ is recovered.

Diagonal functors provide a way to define limits and colimits of diagrams. Given a diagram ${\displaystyle {\mathcal {F}}:{\mathcal {J}}\rightarrow {\mathcal {C}}}$, a natural transformation ${\displaystyle \Delta _{a}\to {\mathcal {F}}}$ (for some object ${\displaystyle a}$ of ${\displaystyle {\mathcal {C}}}$) is called a cone for ${\displaystyle {\mathcal {F}}}$. These cones and their factorizations correspond precisely to the objects and morphisms of the comma category ${\displaystyle (\Delta \downarrow {\mathcal {F}})}$, and a limit of ${\displaystyle {\mathcal {F}}}$ is a terminal object in ${\displaystyle (\Delta \downarrow {\mathcal {F}})}$, i.e., a universal arrow ${\displaystyle \Delta \rightarrow {\mathcal {F}}}$. Dually, a colimit of ${\displaystyle {\mathcal {F}}}$ is an initial object in the comma category ${\displaystyle ({\mathcal {F}}\downarrow \Delta )}$, i.e., a universal arrow ${\displaystyle {\mathcal {F}}\rightarrow \Delta }$.

If every functor from ${\displaystyle {\mathcal {J}}}$ to ${\displaystyle {\mathcal {C}}}$ has a limit (which will be the case if ${\displaystyle {\mathcal {C}}}$ is complete), then the operation of taking limits is itself a functor from ${\displaystyle {\mathcal {C}}^{\mathcal {J}}}$ to ${\displaystyle {\mathcal {C}}}$. 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 ${\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}$ 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, which is the limit of the empty category.