Diagonal functor
In category theory, a branch of mathematics, the diagonal functor is given by , 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 : a product is a universal arrow from to . The arrow comprises the projection maps.
More generally, given a small index category , one may construct the functor category , the objects of which are called diagrams. For each object in , there is a constant diagram that maps every object in to and every morphism in to . The diagonal functor assigns to each object of the diagram , and to each morphism in the natural transformation in (given for every object of by ). Thus, for example, in the case that is a discrete category with two objects, the diagonal functor is recovered.
Diagonal functors provide a way to define limits and colimits of diagrams. Given a diagram , a natural transformation (for some object of ) is called a cone for . These cones and their factorizations correspond precisely to the objects and morphisms of the comma category , and a limit of is a terminal object in , i.e., a universal arrow . Dually, a colimit of is an initial object in the comma category , i.e., a universal arrow .
If every functor from to has a limit (which will be the case if is complete), then the operation of taking limits is itself a functor from to . 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 described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor.
See also
References
{{reflist}}
{{refbegin}}
- {{cite book |doi=10.1093/acprof:oso/9780198568612.003.0007 |chapter=Functors and Naturality |title=Category Theory |date=2006 |last1=Awodey |first1=Steve |pages=125–158 |isbn=978-0-19-856861-2 }}
- {{cite book|title=Sheaves in geometry and logic a first introduction to topos theory|last=Mac Lane|first=Saunders|last2=Moerdijk|first2=Ieke|publisher=Springer-Verlag|year=1992|isbn=9780387977102|location=New York|pages=20–23}}
- {{cite book|title=A Concise Course in Algebraic Topology|last=May|first=J. P.|publisher=University of Chicago Press|year=1999|isbn=0-226-51183-9|pages=16|url=https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf}}
{{refend}}
{{Functors}}
{{DEFAULTSORT:Diagonal Functor}}
{{cattheory-stub}}