EXERCISES IN CATEGORY THEORY 3
Try either (2) or (2*) depending upon whether you prefer monoids or vector spaces.
(1) Let PX = {[/:[/ C 1} be the power set. Extend this construction to a functor P : Set —>• Set. Show that the functions r\x ■ X —>• PX : x i->- {x} give a natural transformation
(2) The free monoid FX on a set X has elements given by words (lists) [xiX2 ■ ■ -xn]. Multiplication in FX is given by joining words together (eg. [xy] o [z] — [xyz]), and the identity element is the empty word [—]. Show that F is part of a functor F : Set ->• Mon.
Let U : Mon —>• Set be the forgetful functor and consider the composite W — UF : Set —>• Set. Describe a natural transformation 1 =^> W. (2*) The free vector space FX on a set X is the vector space with basis {e^ : i G X}: thus elements of are finite sum Eir^ei where each e M. Show that F extends to a functor F : Set —>• Veci. Letting U : yect —>• Set denote the forgetful functor, describe a natural transformation 1 —> UF.
(3) Let N — (N, +, 0) be the monoid of natural numbers with addition. Describe a natural transformation
Mon(N-)
Mon
and show that it is a natural isomorphism. (4) Let X be a category with products and X E X. Given / : B unique map
X x f :X xA^ X x B
making the diagram
C there exists a
commute. Use the universal property of products to show that this gives a functor X x — : X —> X. (More generally product gives a functor — x — : X x X —> X.)
Date: October 2, 2014.
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2 EXERCISES IN CATEGORY THEORY 3
(5) Show that the product projections qa : X x A ^ A form the components of a natural transformation
Ix-
X'.