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counit-unit_adjunction [2016/01/03 15:33] nikolaj |
counit-unit_adjunction [2016/01/03 15:52] 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. | ||
$\varepsilon_{FY}\circ F(\eta_Y)=1_{FY}$ | $\varepsilon_{FY}\circ F(\eta_Y)=1_{FY}$ |