### A categorical approach to universal algebra [article]

Robert Frank Carslaw Walters, University, The Australian National, University, The Australian National
2017
100 109 Chapter III §3. 1 §3.2 The category of devices Devices as algebras Devices and algebras in Sets1 1 116 116 132 References 144 CHAPTER 0 will be written \$a : A -*■ C . If f is a function and a an element of its domain, f evaluated at a will be written fa or f(a) . The set of all A-morphisms from to A^\ s written ACA^A^) . The set of objects of A we write as obj A , and the opposite category as A°P . The category of functors from B^ to A is written A= . If n is a natural transformation
more » ... l transformation from 0:A-*.B t o H r A -, a n d F:Cj-*A,K:B:-*D, then nF : OF -* HF is the natural trans formation defined by nF(C) = n(FC) , and Kp : KG -* • KH is the natural transformation defined by Kp(A) = K(pA) . A morphism a : A^ -* A^ is a split epimorphism if there exists a morphism 3 : A^ A^ such that aB = 1 : A^ A^ . Dually we have the notion of split monomorphism. §0.4 Universal arrows and adjoint functors. 0.4.1 DEFINITIONS Let U : -> A be a functor and A an object of A . Then a universal arrow from A _t_o U is a pair (a : A -> ■ UB,B) with B an object of I| , such that for any A-morphism : A -+ UB^ , there is a unique B-morphism 3 : B -> • B^ such that = U3.a . We often say in this situation that B is free on A , or that a : A -> UB freely generates B . A universal arrow from U _to A is a pair (B , a : UB -* A) such that for any morphism : UB^ -* A there is a unique morphism PROOF Clearly any monomorphism in A through which ß. U a. factors, J6 J J has the property that each ßa_. factors through it. Conversely, con sider a monomorphism y : C -> B through which each ßa^. factors; say y 6 . = ßa . . Further let a. = U a..e. . Then by Lemma 2.1.7, J J J, j e j 3 J , there exists a map X : U (A.,a.) -* C such that ß. U a. = y.A . J J PROOF Considering the construction 2.2.12 and BS(4) we see that y factors into a map u : Xi -* U (Di.,..rii. factors through a ; say (b..ni, = a.8. J J J J 1 Let 6..ni. = ni*8. (6 . £ A (i.,i)) . Then if 4>. n i J J