By Jeff Strom
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Example text
30 (a) Consider the category whose objects are the integers 1, 2, 3, . . and with arrows given by divisibility. Give number-theoretical descriptions of pushouts in this category. (b) Repeat (a) but using the category whose objects are real numbers and whose morphisms correspond to inequalities x ≤ y. 31 Determine the pushout of the diagram {1} o B f /A in the category of groups and homomorphisms. 24; what should the object you constructed here be called? 32 Define the dual notion of coequalizer, and compare coequalizers with pushouts.
The unreduced suspension of a space X is the space Σ0 X which is obtained from X × I by collapsing the X × {0} to a single point [0] and also collapsing X × {1} to a single point [1]. 2 Show that Σ0 S n ∼ = S n+1 for each n. Hint Draw a picture with both S n × I and S n+1 for the cases n = 0, 1. Let X = S n − {N }, where N = (0, 0, . . , 0, 1) ∈ S n is the north pole of the sphere. We can define a function σ : X → Rn by the following rule: 1. if x ∈ X, then x = N , and there is a unique line joining x and N ; 2.
There is an inclusion of categories T◦ → T(2) given by X → (X, ∅). Corollary 27 The space map((X, A), (Y, B)) is in T◦ . 20 Prove Corollary 27 by showing that map((X, A), (Y, B)) / map(X, Y ) map(A, B) / map(A, Y ) is a pullback square. 60 3. 7 Thus, we consider space of maps map((X, A), (Y, B)) as the pair map((X, A), (Y, B)), map(X, B) . big space subspace The product of two pairs (X, A) and (Y, B) in T is also a pair in T : (X, A) × (Y, B) = X × Y , A × Y ∪ X × B . big space subspace With these preliminaries, you can generalize some of the results of Theorem 25 to maps of pairs.