Diagonal functor

In category theory, a branch of mathematics, the diagonal functor \mathcal{C} \rightarrow \mathcal{C} \times \mathcal{C} is given by \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 \mathcal{C}: 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 small index category \mathcal{J}, one may construct the functor category \mathcal{C}^\mathcal{J}, the objects of which are called diagrams. For each object a in \mathcal{C}, there is a constant diagram \Delta_a : \mathcal{J} \to \mathcal{C} that maps every object in \mathcal{J} to a and every morphism in \mathcal{J} to 1_a. The diagonal functor \Delta : \mathcal{C} \rightarrow \mathcal{C}^\mathcal{J} assigns to each object a of \mathcal{C} the diagram \Delta_a, and to each morphism f: a \rightarrow b in \mathcal{C} the natural transformation \eta in \mathcal{C}^\mathcal{J} (given for every object j of \mathcal{J} by \eta_j = f). Thus, for example, in the case that \mathcal{J} is a discrete category with two objects, the diagonal functor \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 \mathcal{F} : \mathcal{J} \rightarrow \mathcal{C}, a natural transformation \Delta_a \to \mathcal{F} (for some object a of \mathcal{C}) is called a cone for \mathcal{F}. These cones and their factorizations correspond precisely to the objects and morphisms of the comma category (\Delta\downarrow\mathcal{F}), and a limit of \mathcal{F} is a terminal object in (\Delta\downarrow\mathcal{F}), i.e., a universal arrow \Delta \rightarrow \mathcal{F}. Dually, a colimit of \mathcal{F} is an initial object in the comma category (\mathcal{F}\downarrow\Delta), i.e., a universal arrow \mathcal{F} \rightarrow \Delta.

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

See also

References

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  • {{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}}

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Category:Category theory

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