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counit-unit_adjunction [2016/01/03 15:17] nikolaj |
counit-unit_adjunction [2016/01/03 15:54] nikolaj |
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| $\beta$ in $1_{\bf D}\cong GF$ | $\beta$ in $1_{\bf D}\cong GF$ | ||
| - | In the case of an adjunction, only the unit arrows $\eta_{GX}$ (i.e. the units on the image of $G$) and the $F$-images of $\eta$ (i.e. $F(\eta_Y)$) can be inverted, but the nice thing is that the inverse is already something known, namely the other natural transformation | + | In the case of equivalence, we can go from a category ${\bf D}$ along $F$ (to the image of ${\bf D}$ in ${\bf C}$, call that "image 1") and then back along $G$ (the image of "image 1" in ${\bf D}$, call it "image 2") and find the same (${\bf D}$ and "image 2" are actually isomorphic). This possibility for invertibility means nothing was lost when passing from ${\bf D}$ to "image 1". |
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| + | In the case of an adjunction, not both nats are invertible. However, we need not go two times along a functor to invert! We already know about an left-invertibility relation of $\eta$ (either in the form $F(\eta_Y)$ or $\eta_{GX}$) once we go to the first image. | ||
| $\varepsilon_{FY}\circ F(\eta_Y)=1_{FY}$ | $\varepsilon_{FY}\circ F(\eta_Y)=1_{FY}$ | ||
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| There is also the combined case where you have an equivalence where the natural transformations are related in the sense of above - this is called an adjoint equivalence. | There is also the combined case where you have an equivalence where the natural transformations are related in the sense of above - this is called an adjoint equivalence. | ||
| - | For another perspective relating to universal morphisms, see [[On universal morphisms]] (31.10.2014). | + | == Inducing hom-set adjunctions == |
| + | Say you're given an arrow $f$ from or to the images of one of the functors (in either ${\mathrm{Hom}}(FX,Y)$ or ${\mathrm{Hom}}(X,GY)$). We can now pre- or post-compose with arrows formed from $\eta$ and $\epsilon$, use the functors on arrows and thus algebraically find an image of $f$ in the other category. | ||
| - | ... | + | Of course, each identity morphisms $1_{FX}:{\mathrm{Hom}}(FX,FX)$ in ${\bf C}$ corresponds to a component $\eta_X:{\mathrm{Hom}}(X,GFX)$ of $\eta:1_{\bf D}\xrightarrow{\bullet}GF$. And the claim here is that not only |
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| - | Emphasis: Having an adjoint functor pair really means you also got a nice pair of natural transofmrations (for which functors are only a conditions). Given any functor $G$ (w.l.o.g, say you're in ${\bf D}$ and the functor out of it is $G$), then if there is an $F$ so that $F\dashv G$, you got yourself a [[monad]]. | + | |
| - | + | ||
| - | Each identity morphisms $1_{FX}$ in ${\bf C}$ corresponds to a component $\eta_X$ of $\eta:1_{\bf D}\xrightarrow{\bullet}GF$. And not only | + | |
| $1_{FX}\leftrightarrow \eta_X$ | $1_{FX}\leftrightarrow \eta_X$ | ||
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| ^ ${\mathrm{Hom}}(FX,Y)\cong{\mathrm{Hom}}(X,GY)$ ^ | ^ ${\mathrm{Hom}}(FX,Y)\cong{\mathrm{Hom}}(X,GY)$ ^ | ||
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| + | It's not that hard do the construction in both directions, after you've written down the types of $\eta,\epsilon, F, G$ before you. | ||
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| + | == As universals == | ||
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| + | For another perspective relating to universal morphisms, see [[On universal morphisms]] (31.10.2014). | ||
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| + | == To monads == | ||
| + | |||
| + | Having an adjoint functor pair really means you also got a nice pair of natural transofmrations (for which functors are only a conditions). Given any functor $G$ (w.l.o.g, say you're in ${\bf D}$ and the functor out of it is $G$), then if there is an $F$ so that $F\dashv G$, you got yourself a [[monad]]. | ||
| === Theorems === | === Theorems === | ||
