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<zeta_0> hello guys, i am currently learning category theory, i cannot find a good definition on what `covariant` and `contravariant` mean in terms of `category theory`, could you guys give my literal translations for `co`, `contra`, and `variant` and how they are used in category theory?
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<pyan> zeta_0: “covariant” means “preserves the direction of the arrows”, and “contravariant” means “reverses the direction of the arrows”
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<ski> "covariant" here means "varies in the same direction", while "contravariant" means "varies in the opposite direction"
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<ski> say `C',`D' are categories, `A',`B' are objects of `C', and `f : A >---> B' is a morphism of `C'
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<ski> if `F' is a (covariant) functor from `C' to `D', then `F f : F A >---> F B'
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<ski> if `F' is a contravariant functor from `C' to `D', then `F f : F B >---> F A', or, if you prefer, `F f : F A <---< F B'
<ski> so, if `F' is a covariant functor, then applying it to a morphism preserves the direction, in the sense that if the morphism (in `C') goes from `A' to `B', then applying the functor to the morphism will be a morphism (in `D') that goes from `F A' to `F B'
<ski> while if `F' is a contravariant functor, then applying it to a morphism reverses the direction, in the sense that if the morphism (in `C') goes from `A' to `B', then applying the functor to the morphism will be a morphism (in `D') that goes *to* `F A' *from* `F B'
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<ski> (btw, note that this use of "co-" is not directly related to the "co-" that is used in "coproduct","coequalizer","colimit","cocone","comonad","comonoid",&c. .. there is no "cofunctor")
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