By Milgram R. (ed.)
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Additional info for Algebraic and Geometric Topology, Part 2
Prove that the class of morphisms of a category is an abstract category, and conversely that the elements of any abstract category are the morphisms of a category if which is determined uniquely up to an isomorphism of categories. 8. Functors Let's' and 2 be categories. A covariant /unctor 7' on 'f; to -0 is a pair of functions (both denoted by the same letter T): An "object function" which assigns to each object CE's an object T(C)E2, and a "mapping 8. Functors 29 function" which assigns to each morphism y: C>C' of ' a morphism T(y): T(C)->T(C') of 2.
The corresponding differential group C(d) is the free abelian group on six generators (0), (1), (2), (01), (12), (02), with the differen- tial given by d(0)=d(1)=d(2)=0, d(01)=(1)-(0),d(02)=(2)--(0), d (12) = (2) - (1) ; in other words, the boundary of each edge is the difference of its two end vertices. One finds H(C(d)) =Z (cls (0)) ED Z (cls ((12) - (02) + (01)]). This group is indeed isomorphic to that found for the circle in Example 1; both are free abelian with two generators. An isomorphism may be given by first specifying a homomorphism / of the differential group C(S') into C(A); we set, say, / (p) _ (0), / (q) _ (1), / (a) _ (01), and / (b) _ (12) - (02).
Conversely, exactness shows that c' gives the isomorphism A1-Kern", so (i) implies (iii) by Prop. 2. Similarly, (ii) implies (iii). Now consider pairs of coterminal homomorphisms al, a,, as in the diagram D: Al °;B:-A,. 4) Such a diagram is said to be universal with ends Al and A. B'+-A, with the same ends there exists a unique 4. Direct Sums 17 homomorphism of D to D' which is the identity on each A.. 5) A1 B'*'-' A,, with D as first row and end maps the identities, there is a unique way of inserting the middle dotted arrow so that the whole diagram becomes commutative (fia1=a;,#a2=a2 ).
Algebraic and Geometric Topology, Part 2 by Milgram R. (ed.)