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This question was already asked on MSE.


In CAT the Yoneda embedding $C^{\text{op}} \stackrel{y_C}{\to} \text{hom}(C, \text{Set})$, induces a map $$\text{hom}(\text{hom}(C, \text{Set}), \text{Set}) \stackrel{y^*_C}{\to} \text{hom}(C^{\text{op}}, \text{Set})$$

Thus, if we call $T = \text{hom}(\_, \text{Set})$. We have a map (which is not even dinatural): $$\_^{\text{op}} \stackrel{\iota}{\to} T$$ and a kind of multiplication,

$$T^2 \stackrel{\mu}{\to} T \circ \_^{\text{op}}. $$

Is there a way to say that $T$ is contravariant monad on CAT?! I looked online for such a notion and I did not find anything.


Note, even if $\iota$ is not dinatural, i.e. $f^*y_Hf \neq y_G$, for a functor $G \stackrel{f}{\to} H$, looking at CAT as a 2-category, there is a $2$-cell $$y_G \Rightarrow f^*y_Hf.$$

What does it mean? If we look at the presheaf construction as a kind of power set the map $y_G \Rightarrow f^*y_Hf$ is witnessing that $\{a\} \subset f^{-1}\{f(a)\}$ for a function $f: A \to B$ between sets and an element $a \in A$.

Another way to say it is that the $2$-cell is $y_G \Rightarrow f^*y_Hf$ is individuated by the universal property of $f^* \cong \text{Lan}_{y_Hf}(y_G)$.

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    $\begingroup$ It seems to me that you are just observing the monadicity of $C \mapsto (T(C^{op}))^{op}$ ? up to size question, this is the "free completion" monad. $\endgroup$ May 31, 2018 at 8:57
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    $\begingroup$ But is monad, even weak $2$-monad the correct word? It is not covariant. $\endgroup$ May 31, 2018 at 9:03
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    $\begingroup$ @IvanDiLiberti: it's both covariant and contravariant, since it takes every functor to an adjunction; see the nlab. The power set functor on $\mathsf{Set}$ behaves in the same way: in comes in a covariant and a contravariant version, for which the reason secretly is that it takes every map of sets to a Galois connection. $\endgroup$ May 31, 2018 at 9:10
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    $\begingroup$ @TobiasFritz I think something different is meant by non-covariance, namely, that it involves "transformations" between functors of different variances $\endgroup$ May 31, 2018 at 9:18
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    $\begingroup$ There are a lot of intertwined issues here, but regarding what kind of transformation your $\iota$ is, you might check out ncatlab.org/nlab/show/2-dinatural+transformation . $\endgroup$ May 31, 2018 at 18:42

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