Is there a "non-trivial" characterization of the concrete categories admitting and adjoint pair of functors $F \dashv G$ were $G$ is defined on the category sBan of separable Banach spaces and bounded linear maps? (Besdies checking if the adjoint functor theorems hold)
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2$\begingroup$ What do you mean by "admitting a forgetful functor"? Do you mean admitting a functor that has a left adjoint, or something more specific? $\endgroup$– Yemon ChoiCommented Sep 5, 2019 at 18:29
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$\begingroup$ Just that there is a free functor from sBan to $\mathfrak{D}$. $\endgroup$– ABIMCommented Sep 5, 2019 at 19:03
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2$\begingroup$ terms like free and forgetful do not make sense in isolation. What is your definition of "free functor"? Do you just mean "functor which has a right adjoint" $\endgroup$– Yemon ChoiCommented Sep 5, 2019 at 19:21
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4$\begingroup$ Do you have any examples in mind? Your category of Banach spaces isn't very well-behaved, I think; much better behaved is the category of Banach spaces and short maps (maps of norm $\le 1$), which e.g. is complete and cocomplete, and has the property that two Banach spaces are isomorphic iff they're isometric. $\endgroup$– Qiaochu YuanCommented Sep 5, 2019 at 23:19
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5$\begingroup$ @QiaochuYuan as you probably know, for the category with short maps, there's a functor to Set given by "take the closed unit ball" which has as its left adjoint "form the $\ell^1$-space" $\endgroup$– Yemon ChoiCommented Sep 6, 2019 at 1:16
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