Functors encode invariants of isomorphism classes
Since functors take compositions to compositions and identities to identities, they also take isomorphisms to isomorphisms, thereby preserving isomorphism classes. cat
Proof
Let
and πΉ : π’ β π£ be an isomorphism. Then there exists π β π’ ( π , π ) such that π β 1 β π’ ( π , π ) and π β 1 π = i d π . It follows that π π β 1 = i d π and ( πΉ π β 1 ) ( πΉ π ) = πΉ i d π = i d πΉ π . Therefore ( πΉ π ) ( πΉ π β 1 ) = ( πΉ i d π ) = i d πΉ π has inverse πΉ π β π£ ( πΉ π , πΉ π ) , whence πΉ π β 1 β π£ ( πΉ π , πΉ π ) is an isomorphism. πΉ π
This is a fundamental idea that captures the very essence of what makes category theory useful.
For example, in Topology MOC, the value an arbitrary functor
Fully faithful
If a functor
Footnotes
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2020, Topology: A categorical approach, p. 11 β©