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# Relationship between inverse/direct image and co/representable profunctors (@nickhu)
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Given a continuous map $X \xrightarrow{f} Y$, there is an associated direct image (covariant) functor $f_*$ and an associated inverse image (contravariant) functor $f^*$, as described in Laurent's lecture and also here. Moreover, there is an adjointness relationship $f^* \dashv f_*$.

Now suppose we have a functor $\mathcal{C} \xrightarrow{F} \mathcal{D}$. The identity profunctor on $\mathcal{D}$ is given by the Hom-functor $\mathcal{D}(-, =)$. Now $F$ embeds contravariantly as a profunctor from $\mathcal{D} \to \mathcal{C}$, by a left action on this Hom, i.e. it maps $(c, d) \mapsto \mathcal{D}(F c, d)$, and covariantly as a profunctor $\mathcal{C} \to \mathcal{D}$ by a right action similarly. These are denoted $F_*$ and $F^*$ respectively, and also have an adjointness relationship $F^* \dashv F_*$ [1, Proposition 7.9.1]. These profunctors are particularly important, and are called the (co)representable profunctors - morally, every profunctor is the same as some functor precisely when it is representable.

Is there some relationship between these two mappings, or is the connection merely superficial?


F. Borceux, Handbook of categorical algebra. Cambridge [England] ; New York: Cambridge University Press, 1994.

This topic was modified 2 years ago by Nick Hu (@jwrigley)
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This looks a lot like the construction of a geometric morphism $\varphi \colon {\bf Sets}^{\mathcal{C}^{op}} \to {\bf Sets}^{\mathcal{D}^{op}}$ from a functor $\varphi \colon \mathcal{C} \to \mathcal{D}$ given on pages 358-359 of Sheaves in Geometry and Logic by MacLane and Moerdijk.

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