Alternative definition of “sheaf”
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Let $(X,tau)$ denote a topological space and $mathcal{O}$ denote a presheaf on this space with codomain $mathbf{Set}$. We can take the category of elements of $mathcal{O}$, which consists of a poset $mathrm{el}(mathcal{O}) = {(U,f) : U in tau, f in mathcal{O}(U)}$ together with a forgetful map $pi : mathrm{el}(mathcal{O}) rightarrow tau$ satisfying certain properties. If $cal O$ happens to be a sheaf, this should be reflected in the structure of $(mathrm{el}(mathcal{O}),pi).$ There should consequently be a definition of sheaf like so:
Let $(X,tau)$ denote a topological space. Then a sheaf on $X$ consists of a poset $P$ togther with a monotone map $pi : P rightarrow tau$ such that the following axioms are satisfied:
(a)
(b)
(c)
(whatever)...
I'm a bit unsure what these conditions should be (even for a presheaf). We want to be able to restrict elements of $P$ to arbitrary opens, which makes me think we should view $P$ as a "$tau$-module", by which I mean that for all opens $U in X$ and all $f in P$, we can form the restriction $U cap f$ which would normally be denoted $f restriction_U$. The usual axioms of an action hold, e.g $$X cap f = f, qquad U cap (V cap f) = (U cap V) cap f.$$
I'm not quite sure whether this module structure should be viewed as extra data, or whether it can be recovered from the map $pi$. Note that we have $pi(U cap f) = U cap pi(f)$, for example.
Ideas, anyone?
Addendum. I just learned that local homeomorphisms into a space $X$ are in bijective correspondence with sheaves on $X$. This doesn't actually answer the question, but it's related.
general-topology category-theory definition order-theory sheaf-theory
asked Aug 6 '18 at 9:12
goblingoblin
37k1159193
$endgroup$
|
show 1 more comment
$begingroup$
Let $(X,tau)$ denote a topological space and $mathcal{O}$ denote a presheaf on this space with codomain $mathbf{Set}$. We can take the category of elements of $mathcal{O}$, which consists of a poset $mathrm{el}(mathcal{O}) = {(U,f) : U in tau, f in mathcal{O}(U)}$ together with a forgetful map $pi : mathrm{el}(mathcal{O}) rightarrow tau$ satisfying certain properties. If $cal O$ happens to be a sheaf, this should be reflected in the structure of $(mathrm{el}(mathcal{O}),pi).$ There should consequently be a definition of sheaf like so:
Let $(X,tau)$ denote a topological space. Then a sheaf on $X$ consists of a poset $P$ togther with a monotone map $pi : P rightarrow tau$ such that the following axioms are satisfied:
(a)
(b)
(c)
(whatever)...
I'm a bit unsure what these conditions should be (even for a presheaf). We want to be able to restrict elements of $P$ to arbitrary opens, which makes me think we should view $P$ as a "$tau$-module", by which I mean that for all opens $U in X$ and all $f in P$, we can form the restriction $U cap f$ which would normally be denoted $f restriction_U$. The usual axioms of an action hold, e.g $$X cap f = f, qquad U cap (V cap f) = (U cap V) cap f.$$
I'm not quite sure whether this module structure should be viewed as extra data, or whether it can be recovered from the map $pi$. Note that we have $pi(U cap f) = U cap pi(f)$, for example.
Ideas, anyone?
Addendum. I just learned that local homeomorphisms into a space $X$ are in bijective correspondence with sheaves on $X$. This doesn't actually answer the question, but it's related.
general-topology category-theory definition order-theory sheaf-theory
asked Aug 6 '18 at 9:12
goblingoblin
37k1159193
$endgroup$
1
$begingroup$
For the "$tau$"-module structure, you just want $pi$ to be a discrete Grothendieck fibration.
$endgroup$
– Pece
Aug 6 '18 at 9:27
1
$begingroup$
Why should $P$ be a poset ? I mean how do you order $(U,f)$ and $(U,g)$ for $f,gin mathcal{O}(U)$ ? Shouldn't $P$ be a category ?
$endgroup$
– Max
Aug 6 '18 at 11:31
1
$begingroup$
@Max, the order relation should be $$(U,f) leq (V,g) iff V supseteq U wedge frestriction_U = g$$ if I'm not mistaken.
$endgroup$
– goblin
Aug 6 '18 at 11:35
1
$begingroup$
@goblin : ah indeed, my bad !
$endgroup$
– Max
Aug 6 '18 at 11:45
$begingroup$
It seems to me that with this ordering there may be a way to recover some stuff with $pi$ and the notions of lower bounds : $s,tin P$ are compatible if and only if they have a lower bound $r$ such that $pi(r) = pi(s)cap pi(t)$; and so you can express the gluing axiom (it seems)
$endgroup$
– Max
Aug 6 '18 at 11:52
|
show 1 more comment
$begingroup$
Let $(X,tau)$ denote a topological space and $mathcal{O}$ denote a presheaf on this space with codomain $mathbf{Set}$. We can take the category of elements of $mathcal{O}$, which consists of a poset $mathrm{el}(mathcal{O}) = {(U,f) : U in tau, f in mathcal{O}(U)}$ together with a forgetful map $pi : mathrm{el}(mathcal{O}) rightarrow tau$ satisfying certain properties. If $cal O$ happens to be a sheaf, this should be reflected in the structure of $(mathrm{el}(mathcal{O}),pi).$ There should consequently be a definition of sheaf like so:
Let $(X,tau)$ denote a topological space. Then a sheaf on $X$ consists of a poset $P$ togther with a monotone map $pi : P rightarrow tau$ such that the following axioms are satisfied:
(a)
(b)
(c)
(whatever)...
I'm a bit unsure what these conditions should be (even for a presheaf). We want to be able to restrict elements of $P$ to arbitrary opens, which makes me think we should view $P$ as a "$tau$-module", by which I mean that for all opens $U in X$ and all $f in P$, we can form the restriction $U cap f$ which would normally be denoted $f restriction_U$. The usual axioms of an action hold, e.g $$X cap f = f, qquad U cap (V cap f) = (U cap V) cap f.$$
I'm not quite sure whether this module structure should be viewed as extra data, or whether it can be recovered from the map $pi$. Note that we have $pi(U cap f) = U cap pi(f)$, for example.
Ideas, anyone?
Addendum. I just learned that local homeomorphisms into a space $X$ are in bijective correspondence with sheaves on $X$. This doesn't actually answer the question, but it's related.
general-topology category-theory definition order-theory sheaf-theory
asked Aug 6 '18 at 9:12
goblingoblin
37k1159193
$endgroup$
Let $(X,tau)$ denote a topological space and $mathcal{O}$ denote a presheaf on this space with codomain $mathbf{Set}$. We can take the category of elements of $mathcal{O}$, which consists of a poset $mathrm{el}(mathcal{O}) = {(U,f) : U in tau, f in mathcal{O}(U)}$ together with a forgetful map $pi : mathrm{el}(mathcal{O}) rightarrow tau$ satisfying certain properties. If $cal O$ happens to be a sheaf, this should be reflected in the structure of $(mathrm{el}(mathcal{O}),pi).$ There should consequently be a definition of sheaf like so:
Let $(X,tau)$ denote a topological space. Then a sheaf on $X$ consists of a poset $P$ togther with a monotone map $pi : P rightarrow tau$ such that the following axioms are satisfied:
(a)
(b)
(c)
(whatever)...
I'm a bit unsure what these conditions should be (even for a presheaf). We want to be able to restrict elements of $P$ to arbitrary opens, which makes me think we should view $P$ as a "$tau$-module", by which I mean that for all opens $U in X$ and all $f in P$, we can form the restriction $U cap f$ which would normally be denoted $f restriction_U$. The usual axioms of an action hold, e.g $$X cap f = f, qquad U cap (V cap f) = (U cap V) cap f.$$
I'm not quite sure whether this module structure should be viewed as extra data, or whether it can be recovered from the map $pi$. Note that we have $pi(U cap f) = U cap pi(f)$, for example.
Ideas, anyone?
Addendum. I just learned that local homeomorphisms into a space $X$ are in bijective correspondence with sheaves on $X$. This doesn't actually answer the question, but it's related.
general-topology category-theory definition order-theory sheaf-theory
general-topology category-theory definition order-theory sheaf-theory
asked Aug 6 '18 at 9:12
goblingoblin
37k1159193
asked Aug 6 '18 at 9:12
goblingoblin
37k1159193
edited Dec 15 '18 at 6:06
goblin
asked Aug 6 '18 at 9:12
goblingoblin
37k1159193
asked Aug 6 '18 at 9:12
goblingoblin
37k1159193
asked Aug 6 '18 at 9:12
goblingoblin
37k1159193
37k1159193
1
$begingroup$
For the "$tau$"-module structure, you just want $pi$ to be a discrete Grothendieck fibration.
$endgroup$
– Pece
Aug 6 '18 at 9:27
1
$begingroup$
Why should $P$ be a poset ? I mean how do you order $(U,f)$ and $(U,g)$ for $f,gin mathcal{O}(U)$ ? Shouldn't $P$ be a category ?
$endgroup$
– Max
Aug 6 '18 at 11:31
1
$begingroup$
@Max, the order relation should be $$(U,f) leq (V,g) iff V supseteq U wedge frestriction_U = g$$ if I'm not mistaken.
$endgroup$
– goblin
Aug 6 '18 at 11:35
1
$begingroup$
@goblin : ah indeed, my bad !
$endgroup$
– Max
Aug 6 '18 at 11:45
$begingroup$
It seems to me that with this ordering there may be a way to recover some stuff with $pi$ and the notions of lower bounds : $s,tin P$ are compatible if and only if they have a lower bound $r$ such that $pi(r) = pi(s)cap pi(t)$; and so you can express the gluing axiom (it seems)
$endgroup$
– Max
Aug 6 '18 at 11:52
|
show 1 more comment
1
$begingroup$
For the "$tau$"-module structure, you just want $pi$ to be a discrete Grothendieck fibration.
$endgroup$
– Pece
Aug 6 '18 at 9:27
1
$begingroup$
Why should $P$ be a poset ? I mean how do you order $(U,f)$ and $(U,g)$ for $f,gin mathcal{O}(U)$ ? Shouldn't $P$ be a category ?
$endgroup$
– Max
Aug 6 '18 at 11:31
1
$begingroup$
@Max, the order relation should be $$(U,f) leq (V,g) iff V supseteq U wedge frestriction_U = g$$ if I'm not mistaken.
$endgroup$
– goblin
Aug 6 '18 at 11:35
1
$begingroup$
@goblin : ah indeed, my bad !
$endgroup$
– Max
Aug 6 '18 at 11:45
$begingroup$
It seems to me that with this ordering there may be a way to recover some stuff with $pi$ and the notions of lower bounds : $s,tin P$ are compatible if and only if they have a lower bound $r$ such that $pi(r) = pi(s)cap pi(t)$; and so you can express the gluing axiom (it seems)
$endgroup$
– Max
Aug 6 '18 at 11:52
1
1
$begingroup$
For the "$tau$"-module structure, you just want $pi$ to be a discrete Grothendieck fibration.
$endgroup$
– Pece
Aug 6 '18 at 9:27
$begingroup$
For the "$tau$"-module structure, you just want $pi$ to be a discrete Grothendieck fibration.
$endgroup$
– Pece
Aug 6 '18 at 9:27
1
1
$begingroup$
Why should $P$ be a poset ? I mean how do you order $(U,f)$ and $(U,g)$ for $f,gin mathcal{O}(U)$ ? Shouldn't $P$ be a category ?
$endgroup$
– Max
Aug 6 '18 at 11:31
$begingroup$
Why should $P$ be a poset ? I mean how do you order $(U,f)$ and $(U,g)$ for $f,gin mathcal{O}(U)$ ? Shouldn't $P$ be a category ?
$endgroup$
– Max
Aug 6 '18 at 11:31
1
1
$begingroup$
@Max, the order relation should be $$(U,f) leq (V,g) iff V supseteq U wedge frestriction_U = g$$ if I'm not mistaken.
$endgroup$
– goblin
Aug 6 '18 at 11:35
$begingroup$
@Max, the order relation should be $$(U,f) leq (V,g) iff V supseteq U wedge frestriction_U = g$$ if I'm not mistaken.
$endgroup$
– goblin
Aug 6 '18 at 11:35
1
1
$begingroup$
@goblin : ah indeed, my bad !
$endgroup$
– Max
Aug 6 '18 at 11:45
$begingroup$
@goblin : ah indeed, my bad !
$endgroup$
– Max
Aug 6 '18 at 11:45
$begingroup$
It seems to me that with this ordering there may be a way to recover some stuff with $pi$ and the notions of lower bounds : $s,tin P$ are compatible if and only if they have a lower bound $r$ such that $pi(r) = pi(s)cap pi(t)$; and so you can express the gluing axiom (it seems)
$endgroup$
– Max
Aug 6 '18 at 11:52
$begingroup$
It seems to me that with this ordering there may be a way to recover some stuff with $pi$ and the notions of lower bounds : $s,tin P$ are compatible if and only if they have a lower bound $r$ such that $pi(r) = pi(s)cap pi(t)$; and so you can express the gluing axiom (it seems)
$endgroup$
– Max
Aug 6 '18 at 11:52
|
show 1 more comment
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$begingroup$
For the "$tau$"-module structure, you just want $pi$ to be a discrete Grothendieck fibration.
$endgroup$
– Pece
Aug 6 '18 at 9:27
1
$begingroup$
Why should $P$ be a poset ? I mean how do you order $(U,f)$ and $(U,g)$ for $f,gin mathcal{O}(U)$ ? Shouldn't $P$ be a category ?
$endgroup$
– Max
Aug 6 '18 at 11:31
1
$begingroup$
@Max, the order relation should be $$(U,f) leq (V,g) iff V supseteq U wedge frestriction_U = g$$ if I'm not mistaken.
$endgroup$
– goblin
Aug 6 '18 at 11:35
1
$begingroup$
@goblin : ah indeed, my bad !
$endgroup$
– Max
Aug 6 '18 at 11:45
$begingroup$
It seems to me that with this ordering there may be a way to recover some stuff with $pi$ and the notions of lower bounds : $s,tin P$ are compatible if and only if they have a lower bound $r$ such that $pi(r) = pi(s)cap pi(t)$; and so you can express the gluing axiom (it seems)
$endgroup$
– Max
Aug 6 '18 at 11:52