Mixing
Showing an isomorphism between Hom sets
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I am trying to show that if $(F,G,eta,epsilon)$ is the data of an equivalence of categories $mathcal{C}$ and $mathcal{D}$, then $Hom_mathcal{C}(x,y)$ and $Hom_mathcal{C} (GF(x),GF(y))$ are isomorphic.
I have managed to show that it is injective, but I am struggling to show it is surjective. Could anyone provide a hint in the right direction?
category-theory
asked Jan 30 at 22:54
foshofosho
4,7861033
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add a comment |
$begingroup$
I am trying to show that if $(F,G,eta,epsilon)$ is the data of an equivalence of categories $mathcal{C}$ and $mathcal{D}$, then $Hom_mathcal{C}(x,y)$ and $Hom_mathcal{C} (GF(x),GF(y))$ are isomorphic.
I have managed to show that it is injective, but I am struggling to show it is surjective. Could anyone provide a hint in the right direction?
category-theory
asked Jan 30 at 22:54
foshofosho
4,7861033
$endgroup$
$begingroup$
I would probably just explicitly write the pair of inverse maps between those sets, and then verify they are indeed inverses.
$endgroup$
– Daniel Schepler
Jan 30 at 23:00
$begingroup$
And by "isomorphic", do you mean just that the sets are bijective for any particular $x,y$, or that the two functors $mathcal{C}^{op} times mathcal{C} to mathbf{Sets}$ are isomorphic functors?
$endgroup$
– Daniel Schepler
Jan 30 at 23:02
$begingroup$
We already have $hom_{mathcal C}(x,y) cong hom_{mathcal D}(F(x),,F(y))$ since $F$ is an equivalence functor. Now apply it again with $G$
$endgroup$
– Berci
Jan 30 at 23:46
add a comment |
$begingroup$
I am trying to show that if $(F,G,eta,epsilon)$ is the data of an equivalence of categories $mathcal{C}$ and $mathcal{D}$, then $Hom_mathcal{C}(x,y)$ and $Hom_mathcal{C} (GF(x),GF(y))$ are isomorphic.
I have managed to show that it is injective, but I am struggling to show it is surjective. Could anyone provide a hint in the right direction?
category-theory
asked Jan 30 at 22:54
foshofosho
4,7861033
$endgroup$
I am trying to show that if $(F,G,eta,epsilon)$ is the data of an equivalence of categories $mathcal{C}$ and $mathcal{D}$, then $Hom_mathcal{C}(x,y)$ and $Hom_mathcal{C} (GF(x),GF(y))$ are isomorphic.
I have managed to show that it is injective, but I am struggling to show it is surjective. Could anyone provide a hint in the right direction?
category-theory
category-theory
asked Jan 30 at 22:54
foshofosho
4,7861033
asked Jan 30 at 22:54
foshofosho
4,7861033
asked Jan 30 at 22:54
foshofosho
4,7861033
asked Jan 30 at 22:54
foshofosho
4,7861033
asked Jan 30 at 22:54
foshofosho
4,7861033
4,7861033
$begingroup$
I would probably just explicitly write the pair of inverse maps between those sets, and then verify they are indeed inverses.
$endgroup$
– Daniel Schepler
Jan 30 at 23:00
$begingroup$
And by "isomorphic", do you mean just that the sets are bijective for any particular $x,y$, or that the two functors $mathcal{C}^{op} times mathcal{C} to mathbf{Sets}$ are isomorphic functors?
$endgroup$
– Daniel Schepler
Jan 30 at 23:02
$begingroup$
We already have $hom_{mathcal C}(x,y) cong hom_{mathcal D}(F(x),,F(y))$ since $F$ is an equivalence functor. Now apply it again with $G$
$endgroup$
– Berci
Jan 30 at 23:46
add a comment |
$begingroup$
I would probably just explicitly write the pair of inverse maps between those sets, and then verify they are indeed inverses.
$endgroup$
– Daniel Schepler
Jan 30 at 23:00
$begingroup$
And by "isomorphic", do you mean just that the sets are bijective for any particular $x,y$, or that the two functors $mathcal{C}^{op} times mathcal{C} to mathbf{Sets}$ are isomorphic functors?
$endgroup$
– Daniel Schepler
Jan 30 at 23:02
$begingroup$
We already have $hom_{mathcal C}(x,y) cong hom_{mathcal D}(F(x),,F(y))$ since $F$ is an equivalence functor. Now apply it again with $G$
$endgroup$
– Berci
Jan 30 at 23:46
$begingroup$
I would probably just explicitly write the pair of inverse maps between those sets, and then verify they are indeed inverses.
$endgroup$
– Daniel Schepler
Jan 30 at 23:00
$begingroup$
I would probably just explicitly write the pair of inverse maps between those sets, and then verify they are indeed inverses.
$endgroup$
– Daniel Schepler
Jan 30 at 23:00
$begingroup$
And by "isomorphic", do you mean just that the sets are bijective for any particular $x,y$, or that the two functors $mathcal{C}^{op} times mathcal{C} to mathbf{Sets}$ are isomorphic functors?
$endgroup$
– Daniel Schepler
Jan 30 at 23:02
$begingroup$
And by "isomorphic", do you mean just that the sets are bijective for any particular $x,y$, or that the two functors $mathcal{C}^{op} times mathcal{C} to mathbf{Sets}$ are isomorphic functors?
$endgroup$
– Daniel Schepler
Jan 30 at 23:02
$begingroup$
We already have $hom_{mathcal C}(x,y) cong hom_{mathcal D}(F(x),,F(y))$ since $F$ is an equivalence functor. Now apply it again with $G$
$endgroup$
– Berci
Jan 30 at 23:46
$begingroup$
We already have $hom_{mathcal C}(x,y) cong hom_{mathcal D}(F(x),,F(y))$ since $F$ is an equivalence functor. Now apply it again with $G$
$endgroup$
– Berci
Jan 30 at 23:46
add a comment |
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$begingroup$
I would probably just explicitly write the pair of inverse maps between those sets, and then verify they are indeed inverses.
$endgroup$
– Daniel Schepler
Jan 30 at 23:00
$begingroup$
And by "isomorphic", do you mean just that the sets are bijective for any particular $x,y$, or that the two functors $mathcal{C}^{op} times mathcal{C} to mathbf{Sets}$ are isomorphic functors?
$endgroup$
– Daniel Schepler
Jan 30 at 23:02
$begingroup$
We already have $hom_{mathcal C}(x,y) cong hom_{mathcal D}(F(x),,F(y))$ since $F$ is an equivalence functor. Now apply it again with $G$
$endgroup$
– Berci
Jan 30 at 23:46