Diagonal functor

In the mathematical subfield of category theory of Diagonalfunktor is a functor that allows a category in the category of functors for any non-empty (small ) category embed. The name comes from the fact that for a discrete two-element of Diagonalfunktor is just the picture.

Definition and functoriality

Be a category and a small category. Then the Diagonalfunktor is defined as a map that assigns to each morphism is a natural transformation, thereby is given that it assigns to each object, and thus to each morphism in the morphism. For an object is obviously a functor. To view now, that actually is a functor, consider for morphisms of the category and the chain of natural transformations and, by definition, this yields for each in the following commutative diagram:

This is nothing more than:

This corresponds to the natural transformation, is thus proved that. For non-empty is obviously injective, ie embeds into the corresponding functor. Under a certain condition is also full: Be natural transformation, ie that for each in the chart

Commutes (for and ). This means nothing less than that whenever a morphism exists between and. If the category regarded as a graph is weakly connected, so is constant and thus in the image of, which is full. This is true, for example, an arrow or a general category for with start or end object, but not for a product for discrete with at least two elements.

Connection with Limites

A cone with respect to a functor is nothing more than an object in after provided with a natural transformation. A limit of is a special cone, namely a - kouniverselle solution. Dually, a colimit of a special Kokegel, namely a - universal solution for. Has a rechtsadjungierten functor, so is complete with respect to Limites, the converse is also true. This adjoint functor is just the Limesfunktor. According to the Kolimesfunktor ( if it exists ) is linksadjungiert to Diagonalfunktor.

The Diagonalfunktor is continuous, that is, it receives all Limites that exist in. Likewise, it receives all colimits.