Notation in Category Theory, related to total categories, colimit$_{xin C, z in P(x)} x$
$begingroup$
A definition says:
Call a category C total if the Yoneda embedding has a left adjoint
$F:$ PShv(C) $ to $ C.
If $P in$ PShv(C), then $F(P) cong$ colimit$_{xin C, z in P(x)} x$.
Moreover, a category C is total if this colimit exists.
What is this symbol colimit$_{xin C, z in P(x)} x$? Why is there this isomorphism, and why is C total if this colimit exists?
category-theory notation
asked Dec 28 '18 at 18:17
MariahMariah
2,1471718
$endgroup$
add a comment |
$begingroup$
A definition says:
Call a category C total if the Yoneda embedding has a left adjoint
$F:$ PShv(C) $ to $ C.
If $P in$ PShv(C), then $F(P) cong$ colimit$_{xin C, z in P(x)} x$.
Moreover, a category C is total if this colimit exists.
What is this symbol colimit$_{xin C, z in P(x)} x$? Why is there this isomorphism, and why is C total if this colimit exists?
category-theory notation
asked Dec 28 '18 at 18:17
MariahMariah
2,1471718
$endgroup$
add a comment |
$begingroup$
A definition says:
Call a category C total if the Yoneda embedding has a left adjoint
$F:$ PShv(C) $ to $ C.
If $P in$ PShv(C), then $F(P) cong$ colimit$_{xin C, z in P(x)} x$.
Moreover, a category C is total if this colimit exists.
What is this symbol colimit$_{xin C, z in P(x)} x$? Why is there this isomorphism, and why is C total if this colimit exists?
category-theory notation
asked Dec 28 '18 at 18:17
MariahMariah
2,1471718
$endgroup$
A definition says:
Call a category C total if the Yoneda embedding has a left adjoint
$F:$ PShv(C) $ to $ C.
If $P in$ PShv(C), then $F(P) cong$ colimit$_{xin C, z in P(x)} x$.
Moreover, a category C is total if this colimit exists.
What is this symbol colimit$_{xin C, z in P(x)} x$? Why is there this isomorphism, and why is C total if this colimit exists?
category-theory notation
category-theory notation
asked Dec 28 '18 at 18:17
MariahMariah
2,1471718
asked Dec 28 '18 at 18:17
MariahMariah
2,1471718
edited Dec 28 '18 at 19:48
Mariah
asked Dec 28 '18 at 18:17
MariahMariah
2,1471718
asked Dec 28 '18 at 18:17
MariahMariah
2,1471718
asked Dec 28 '18 at 18:17
MariahMariah
2,1471718
2,1471718
add a comment |
add a comment |
1 Answer
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The colimit is really indexed by the category of elements $int_{mathcal{C}} P$, whose objects are pairs $(x,z)$ with $x in mathrm{ob}(mathcal{C})$ and $z in P(x)$, and where a morphism $f : (x,z) to (x', z')$ is a morphism $f : x to x'$ in $mathcal{C}$ such that $P(f)(z') = z$.
Every presheaf $P$ is a colimit of representables indexed by $int_{mathcal{C}} P$, namely
$$P cong mathrm{colim}_{(x,z) in int P} ~ mathsf{y}(x)$$
in $mathrm{Psh}(mathcal{C})$, where $mathsf{y} : mathcal{C} to mathrm{Psh}(mathcal{C})$ is the Yoneda embedding.
Since $F$ is a left adjoint, it preserves colimits, meaning that
$$F(P) cong mathrm{colim}_{(x,z) in int P} ~ Fmathsf{y}(x)$$
Since the Yoneda embedding is full and faithful, the counit $varepsilon : F circ mathsf{y} to mathrm{id}_{mathcal{C}}$ of the adjunction $F dashv mathsf{y}$ is a natural isomorphism, and so we obtain
$$F(P) cong mathrm{colim}_{(x,z) in int P} ~ Fmathsf{y}(x) cong mathrm{colim}_{(x,z) in int P} ~ x$$
You now need to verify that if this colimit exists for all presheaves $P$, then the assignment $P mapsto mathrm{colim}_{(x,z) in int P} ~ x$ determines a functor $mathrm{Psh}(mathcal{C}) to mathcal{C}$ which is left adjoint to the Yoneda embedding.
answered Dec 28 '18 at 19:59
Clive NewsteadClive Newstead
52.1k474137
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add a comment |
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1 Answer
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$begingroup$
The colimit is really indexed by the category of elements $int_{mathcal{C}} P$, whose objects are pairs $(x,z)$ with $x in mathrm{ob}(mathcal{C})$ and $z in P(x)$, and where a morphism $f : (x,z) to (x', z')$ is a morphism $f : x to x'$ in $mathcal{C}$ such that $P(f)(z') = z$.
Every presheaf $P$ is a colimit of representables indexed by $int_{mathcal{C}} P$, namely
$$P cong mathrm{colim}_{(x,z) in int P} ~ mathsf{y}(x)$$
in $mathrm{Psh}(mathcal{C})$, where $mathsf{y} : mathcal{C} to mathrm{Psh}(mathcal{C})$ is the Yoneda embedding.
Since $F$ is a left adjoint, it preserves colimits, meaning that
$$F(P) cong mathrm{colim}_{(x,z) in int P} ~ Fmathsf{y}(x)$$
Since the Yoneda embedding is full and faithful, the counit $varepsilon : F circ mathsf{y} to mathrm{id}_{mathcal{C}}$ of the adjunction $F dashv mathsf{y}$ is a natural isomorphism, and so we obtain
$$F(P) cong mathrm{colim}_{(x,z) in int P} ~ Fmathsf{y}(x) cong mathrm{colim}_{(x,z) in int P} ~ x$$
You now need to verify that if this colimit exists for all presheaves $P$, then the assignment $P mapsto mathrm{colim}_{(x,z) in int P} ~ x$ determines a functor $mathrm{Psh}(mathcal{C}) to mathcal{C}$ which is left adjoint to the Yoneda embedding.
answered Dec 28 '18 at 19:59
Clive NewsteadClive Newstead
52.1k474137
$endgroup$
add a comment |
$begingroup$
The colimit is really indexed by the category of elements $int_{mathcal{C}} P$, whose objects are pairs $(x,z)$ with $x in mathrm{ob}(mathcal{C})$ and $z in P(x)$, and where a morphism $f : (x,z) to (x', z')$ is a morphism $f : x to x'$ in $mathcal{C}$ such that $P(f)(z') = z$.
Every presheaf $P$ is a colimit of representables indexed by $int_{mathcal{C}} P$, namely
$$P cong mathrm{colim}_{(x,z) in int P} ~ mathsf{y}(x)$$
in $mathrm{Psh}(mathcal{C})$, where $mathsf{y} : mathcal{C} to mathrm{Psh}(mathcal{C})$ is the Yoneda embedding.
Since $F$ is a left adjoint, it preserves colimits, meaning that
$$F(P) cong mathrm{colim}_{(x,z) in int P} ~ Fmathsf{y}(x)$$
Since the Yoneda embedding is full and faithful, the counit $varepsilon : F circ mathsf{y} to mathrm{id}_{mathcal{C}}$ of the adjunction $F dashv mathsf{y}$ is a natural isomorphism, and so we obtain
$$F(P) cong mathrm{colim}_{(x,z) in int P} ~ Fmathsf{y}(x) cong mathrm{colim}_{(x,z) in int P} ~ x$$
You now need to verify that if this colimit exists for all presheaves $P$, then the assignment $P mapsto mathrm{colim}_{(x,z) in int P} ~ x$ determines a functor $mathrm{Psh}(mathcal{C}) to mathcal{C}$ which is left adjoint to the Yoneda embedding.
answered Dec 28 '18 at 19:59
Clive NewsteadClive Newstead
52.1k474137
$endgroup$
add a comment |
$begingroup$
The colimit is really indexed by the category of elements $int_{mathcal{C}} P$, whose objects are pairs $(x,z)$ with $x in mathrm{ob}(mathcal{C})$ and $z in P(x)$, and where a morphism $f : (x,z) to (x', z')$ is a morphism $f : x to x'$ in $mathcal{C}$ such that $P(f)(z') = z$.
Every presheaf $P$ is a colimit of representables indexed by $int_{mathcal{C}} P$, namely
$$P cong mathrm{colim}_{(x,z) in int P} ~ mathsf{y}(x)$$
in $mathrm{Psh}(mathcal{C})$, where $mathsf{y} : mathcal{C} to mathrm{Psh}(mathcal{C})$ is the Yoneda embedding.
Since $F$ is a left adjoint, it preserves colimits, meaning that
$$F(P) cong mathrm{colim}_{(x,z) in int P} ~ Fmathsf{y}(x)$$
Since the Yoneda embedding is full and faithful, the counit $varepsilon : F circ mathsf{y} to mathrm{id}_{mathcal{C}}$ of the adjunction $F dashv mathsf{y}$ is a natural isomorphism, and so we obtain
$$F(P) cong mathrm{colim}_{(x,z) in int P} ~ Fmathsf{y}(x) cong mathrm{colim}_{(x,z) in int P} ~ x$$
You now need to verify that if this colimit exists for all presheaves $P$, then the assignment $P mapsto mathrm{colim}_{(x,z) in int P} ~ x$ determines a functor $mathrm{Psh}(mathcal{C}) to mathcal{C}$ which is left adjoint to the Yoneda embedding.
answered Dec 28 '18 at 19:59
Clive NewsteadClive Newstead
52.1k474137
$endgroup$
The colimit is really indexed by the category of elements $int_{mathcal{C}} P$, whose objects are pairs $(x,z)$ with $x in mathrm{ob}(mathcal{C})$ and $z in P(x)$, and where a morphism $f : (x,z) to (x', z')$ is a morphism $f : x to x'$ in $mathcal{C}$ such that $P(f)(z') = z$.
Every presheaf $P$ is a colimit of representables indexed by $int_{mathcal{C}} P$, namely
$$P cong mathrm{colim}_{(x,z) in int P} ~ mathsf{y}(x)$$
in $mathrm{Psh}(mathcal{C})$, where $mathsf{y} : mathcal{C} to mathrm{Psh}(mathcal{C})$ is the Yoneda embedding.
Since $F$ is a left adjoint, it preserves colimits, meaning that
$$F(P) cong mathrm{colim}_{(x,z) in int P} ~ Fmathsf{y}(x)$$
Since the Yoneda embedding is full and faithful, the counit $varepsilon : F circ mathsf{y} to mathrm{id}_{mathcal{C}}$ of the adjunction $F dashv mathsf{y}$ is a natural isomorphism, and so we obtain
$$F(P) cong mathrm{colim}_{(x,z) in int P} ~ Fmathsf{y}(x) cong mathrm{colim}_{(x,z) in int P} ~ x$$
You now need to verify that if this colimit exists for all presheaves $P$, then the assignment $P mapsto mathrm{colim}_{(x,z) in int P} ~ x$ determines a functor $mathrm{Psh}(mathcal{C}) to mathcal{C}$ which is left adjoint to the Yoneda embedding.
answered Dec 28 '18 at 19:59
Clive NewsteadClive Newstead
52.1k474137
answered Dec 28 '18 at 19:59
Clive NewsteadClive Newstead
52.1k474137
answered Dec 28 '18 at 19:59
Clive NewsteadClive Newstead
52.1k474137
answered Dec 28 '18 at 19:59
Clive NewsteadClive Newstead
52.1k474137
52.1k474137
add a comment |
add a comment |
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