Pushforward and pullback of morphisms

Pushforward and pullback of an isomorphism

Let be a morphism in an arbitrary category , and Then the following conditions are equivalent cat

• is an isomorphism
• is a bijection
• is a bijection

Proof

Suppose is an isomorphism. Then there exists an inverse . For any , there exist pushforwards and . Let , then clearly . Likewise for , clearly . Hence is the inverse of Similarly for any , there exist pullbacks and . Proceeding as before, is the inverse of . Therefore, if is an isomorphism, so are and bijections.

Next, assume for any the pushforward is a bijection. If we let , from surjectivity it follows there exists such that . If we let , it follows that , and hence from injectivity . Therefore is the inverse of , whence is an isomorphism.

Finally, assume for any the pullback is a bijection. If we let , from surjectivity it follows there exists such that . If we let , it follows that , and hence from injectivity . Therefore is the inverse of , whence is an isomorphism.

In summary, if you understand all the morphisms , you know up to isomorphism. In summary, if you understand all the morphisms , you know up to isomorphism.¹

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Footnotes

1. 2020, Topology: A categorical approach, p. 9 ↩