Category theory MOC

Pushforward and pullback

A pushforward 𝑓∗ is a function associated with a morphism 𝑓 :𝑋 →𝑌 and an object 𝑍 representing postcomposition so that 𝑓∗ :𝖢(𝑍,𝑋) →𝖢(𝑍,𝑌) :𝑔 →𝑓 ∘𝑔. #m/def/cat Likewise, a pullback 𝑓∗ is defined by precomposition so that 𝑓∗ :𝖢(𝑌,𝑍),𝖢(𝑋,𝑍) :𝑔 ↦𝑔 ∘𝑓.¹ #m/def/cat

There is a sense in which pushforwards and pullbacks are natural transformations. See Yoneda embedding.

Properties

• Pushforward and pullback of an isomorphism: 𝑓 is iso iff 𝑓⋆ is bijective iff 𝑓⋆ is bijective.

#state/develop | #lang/en | #SemBr