solutions to exercises of lesson 1
Solutions lesson 1
Ex 1. The identity morphism has itself as inverse morphism.
Ex 2. If f and g are composable and f’ and g’ are, the inverse of f and g , respectively, then
f’◦g’ in the inverse of g◦f.
In fact, if we consider (g◦f )◦(f’◦g’), by associativity it follows
(g◦f )◦ (f’◦g’) = g◦ (f◦ f’) ◦ g’ = g◦1B◦g’ = g◦g’= 1C
by the identity property.
Similarly
(f’◦g’ )◦(g ◦f) = f’◦ (g’◦g ) ◦ f =f’◦1B ◦f= f’◦f = 1A
If you draw a diagram with points and arrows everything is clearer.
Ex 3. Let f: A ® B be a morphism with two inverse morphisms g: B ® A and h: B ® A,
then h=g.
if we consider the composite g◦f◦h, by associativity we get
g◦(f◦h) = (g◦f) ◦h
since g◦(f◦h) = g◦1B = g and (g◦f) ◦ h= 1A◦ h = h
hence g= h
Remark: the properties that we proved are following directly from the 2 axioms of morphisms in a category, i. e. associativity and identity.
Ex.4 The isomorphisms in Set are exactly the bijections. This statement is not quite a logical triviality. It amounts to the assertion that a function has a two-sided inverse if and only if it is injective and surjective.
The isomorphisms in Grp are exactly the isomorphisms of groups. Again,
this is not quite trivial, at least if you were taught that the definition of
group isomorphism is ‘bijective homomorphism’. In order to show that this is
equivalent to being an isomorphism in Grp, you have to prove that the inverse function
of a bijective homomorphism is also a homomorphism.
Similarly, the categorical isomorphisms in Ring are exactly the classical isomorphisms of rings.
The isomorphisms in Top are exactly the homeomorphisms. Note that, in contrast to the situation in Grp and Ring, a bijective morphism in Top is not necessarily an isomorphism. A classic example is the identity function 1X: (X, t1) ® (X, t2) where t1 and t2 are topologies on the same set and t1 is strictly finer than t2.
1X: (X, t1) ® (X,t2) is then continuos but its inverse function is not continuous, so it is not an isomorphism in Top.
In an order S (=preorder plus the antisymmetry property), considered as a category, the only isomorphisms are the identities arrows.
Infact f: x ® y means x £ y; if there exists g: y ® x it means y £ x hence x=y