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# Relationship between inverse/direct image and co/representable profunctors

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(@nickhu)
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Joined: 2 years ago
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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?

[1]
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)
Eminent Member
Joined: 3 years ago
Posts: 20

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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