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What's the point of “created limits�
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Seeing as there's an nLab page about creation of limits, I take it that at least some people think this is an important notion. There's also a discussion here about what the "correct" definition of this notion is. However, and maybe this is just my ignorance speaking, but I haven't actually found a reason to worry about this notion, so I basically just ignore it. I mean, okay, a diagram $D$ that lacks a limit in one category $mathbfC$ might gain a limit in a category with more objects, or arrows, or both. More generally, given a functor $F : mathbfC rightarrow mathbfD$, the composite $F circ D$ might or might not have a limit, quite independently of whether or not $D$ does. But so what? I don't quite get what the notion "creation of limits" really teaches us.
Question. Can someone who finds this to be a helpful notion explain what they find helpful about it? Is there, for example, a theorem about creation of limits that's ubiquitous but impossible to see without this notion? Is there an important definition that can't be elegantly stated unless creation of limits is invoked? What's the point of this idea?
ct.category-theory
asked 4 hours ago
goblin
1,4651127
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up vote
2
down vote
favorite
Seeing as there's an nLab page about creation of limits, I take it that at least some people think this is an important notion. There's also a discussion here about what the "correct" definition of this notion is. However, and maybe this is just my ignorance speaking, but I haven't actually found a reason to worry about this notion, so I basically just ignore it. I mean, okay, a diagram $D$ that lacks a limit in one category $mathbfC$ might gain a limit in a category with more objects, or arrows, or both. More generally, given a functor $F : mathbfC rightarrow mathbfD$, the composite $F circ D$ might or might not have a limit, quite independently of whether or not $D$ does. But so what? I don't quite get what the notion "creation of limits" really teaches us.
Question. Can someone who finds this to be a helpful notion explain what they find helpful about it? Is there, for example, a theorem about creation of limits that's ubiquitous but impossible to see without this notion? Is there an important definition that can't be elegantly stated unless creation of limits is invoked? What's the point of this idea?
ct.category-theory
asked 4 hours ago
goblin
1,4651127
1
The nLab entry mentions monadicity theorems. Does that not satisfy you?
– Yemon Choi
2 hours ago
Isn't it just like creating the limits of $mathbb Q$ by constructing $mathbb R$? Or the definition of completion of a metric space, or of Hilbert space, or of complete local ring? Or various possible compactifications of manifolds/varieties/ moduli spaces? Or the ideal boundary in hyperbolic geometry? There are countless examples in mathematics in which you want to "see" your (uniquely defined or not) limit/degenerate points instead of just a buch of "holes".
– Qfwfq
13 mins ago
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Seeing as there's an nLab page about creation of limits, I take it that at least some people think this is an important notion. There's also a discussion here about what the "correct" definition of this notion is. However, and maybe this is just my ignorance speaking, but I haven't actually found a reason to worry about this notion, so I basically just ignore it. I mean, okay, a diagram $D$ that lacks a limit in one category $mathbfC$ might gain a limit in a category with more objects, or arrows, or both. More generally, given a functor $F : mathbfC rightarrow mathbfD$, the composite $F circ D$ might or might not have a limit, quite independently of whether or not $D$ does. But so what? I don't quite get what the notion "creation of limits" really teaches us.
Question. Can someone who finds this to be a helpful notion explain what they find helpful about it? Is there, for example, a theorem about creation of limits that's ubiquitous but impossible to see without this notion? Is there an important definition that can't be elegantly stated unless creation of limits is invoked? What's the point of this idea?
ct.category-theory
asked 4 hours ago
goblin
1,4651127
Seeing as there's an nLab page about creation of limits, I take it that at least some people think this is an important notion. There's also a discussion here about what the "correct" definition of this notion is. However, and maybe this is just my ignorance speaking, but I haven't actually found a reason to worry about this notion, so I basically just ignore it. I mean, okay, a diagram $D$ that lacks a limit in one category $mathbfC$ might gain a limit in a category with more objects, or arrows, or both. More generally, given a functor $F : mathbfC rightarrow mathbfD$, the composite $F circ D$ might or might not have a limit, quite independently of whether or not $D$ does. But so what? I don't quite get what the notion "creation of limits" really teaches us.
Question. Can someone who finds this to be a helpful notion explain what they find helpful about it? Is there, for example, a theorem about creation of limits that's ubiquitous but impossible to see without this notion? Is there an important definition that can't be elegantly stated unless creation of limits is invoked? What's the point of this idea?
ct.category-theory
ct.category-theory
asked 4 hours ago
goblin
1,4651127
asked 4 hours ago
goblin
1,4651127
asked 4 hours ago
goblin
1,4651127
asked 4 hours ago
goblin
1,4651127
asked 4 hours ago
goblin
1,4651127
1,4651127
1
The nLab entry mentions monadicity theorems. Does that not satisfy you?
– Yemon Choi
2 hours ago
Isn't it just like creating the limits of $mathbb Q$ by constructing $mathbb R$? Or the definition of completion of a metric space, or of Hilbert space, or of complete local ring? Or various possible compactifications of manifolds/varieties/ moduli spaces? Or the ideal boundary in hyperbolic geometry? There are countless examples in mathematics in which you want to "see" your (uniquely defined or not) limit/degenerate points instead of just a buch of "holes".
– Qfwfq
13 mins ago
add a comment |Â
1
The nLab entry mentions monadicity theorems. Does that not satisfy you?
– Yemon Choi
2 hours ago
Isn't it just like creating the limits of $mathbb Q$ by constructing $mathbb R$? Or the definition of completion of a metric space, or of Hilbert space, or of complete local ring? Or various possible compactifications of manifolds/varieties/ moduli spaces? Or the ideal boundary in hyperbolic geometry? There are countless examples in mathematics in which you want to "see" your (uniquely defined or not) limit/degenerate points instead of just a buch of "holes".
– Qfwfq
13 mins ago
1
1
The nLab entry mentions monadicity theorems. Does that not satisfy you?
– Yemon Choi
2 hours ago
The nLab entry mentions monadicity theorems. Does that not satisfy you?
– Yemon Choi
2 hours ago
Isn't it just like creating the limits of $mathbb Q$ by constructing $mathbb R$? Or the definition of completion of a metric space, or of Hilbert space, or of complete local ring? Or various possible compactifications of manifolds/varieties/ moduli spaces? Or the ideal boundary in hyperbolic geometry? There are countless examples in mathematics in which you want to "see" your (uniquely defined or not) limit/degenerate points instead of just a buch of "holes".
– Qfwfq
13 mins ago
Isn't it just like creating the limits of $mathbb Q$ by constructing $mathbb R$? Or the definition of completion of a metric space, or of Hilbert space, or of complete local ring? Or various possible compactifications of manifolds/varieties/ moduli spaces? Or the ideal boundary in hyperbolic geometry? There are countless examples in mathematics in which you want to "see" your (uniquely defined or not) limit/degenerate points instead of just a buch of "holes".
– Qfwfq
13 mins ago
add a comment |Â
1 Answer
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This is not what "creates limits" means, although I also find this terminology slightly confusing (in the sense that if I had to guess the definition just from the word "creates" my guess would probably have been the same as you, but it's just not that).
The definition does require the limit of $D$ to exists in $mathbf C$ in the first place, but what it says is basically that you can compute it in $mathbf D$, ie roughly we have both:
- $F$ of a limit of $D$, is a limit of $Fcirc D$
- somewhat conversely (and imprecisely), if some $F(x)$ happens to be a limit of $Fcirc D$, then $x$ was already a limit of $D$.
Now as far as I know this definition is most often used for some shape of diagrams rather than for a single one, and as pointed out in the nlab is used in Beck's monadicity theorem. The main example of that phenomena (the colimit version) is that a direct sum of modules over some algebra is just the direct sum of the underlying vector spaces with the "obvious" module structure.
answered 2 hours ago
Adrien
4,25621635
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
This is not what "creates limits" means, although I also find this terminology slightly confusing (in the sense that if I had to guess the definition just from the word "creates" my guess would probably have been the same as you, but it's just not that).
The definition does require the limit of $D$ to exists in $mathbf C$ in the first place, but what it says is basically that you can compute it in $mathbf D$, ie roughly we have both:
- $F$ of a limit of $D$, is a limit of $Fcirc D$
- somewhat conversely (and imprecisely), if some $F(x)$ happens to be a limit of $Fcirc D$, then $x$ was already a limit of $D$.
Now as far as I know this definition is most often used for some shape of diagrams rather than for a single one, and as pointed out in the nlab is used in Beck's monadicity theorem. The main example of that phenomena (the colimit version) is that a direct sum of modules over some algebra is just the direct sum of the underlying vector spaces with the "obvious" module structure.
answered 2 hours ago
Adrien
4,25621635
add a comment |Â
up vote
3
down vote
This is not what "creates limits" means, although I also find this terminology slightly confusing (in the sense that if I had to guess the definition just from the word "creates" my guess would probably have been the same as you, but it's just not that).
The definition does require the limit of $D$ to exists in $mathbf C$ in the first place, but what it says is basically that you can compute it in $mathbf D$, ie roughly we have both:
- $F$ of a limit of $D$, is a limit of $Fcirc D$
- somewhat conversely (and imprecisely), if some $F(x)$ happens to be a limit of $Fcirc D$, then $x$ was already a limit of $D$.
Now as far as I know this definition is most often used for some shape of diagrams rather than for a single one, and as pointed out in the nlab is used in Beck's monadicity theorem. The main example of that phenomena (the colimit version) is that a direct sum of modules over some algebra is just the direct sum of the underlying vector spaces with the "obvious" module structure.
answered 2 hours ago
Adrien
4,25621635
add a comment |Â
up vote
3
down vote
up vote
3
down vote
This is not what "creates limits" means, although I also find this terminology slightly confusing (in the sense that if I had to guess the definition just from the word "creates" my guess would probably have been the same as you, but it's just not that).
The definition does require the limit of $D$ to exists in $mathbf C$ in the first place, but what it says is basically that you can compute it in $mathbf D$, ie roughly we have both:
- $F$ of a limit of $D$, is a limit of $Fcirc D$
- somewhat conversely (and imprecisely), if some $F(x)$ happens to be a limit of $Fcirc D$, then $x$ was already a limit of $D$.
Now as far as I know this definition is most often used for some shape of diagrams rather than for a single one, and as pointed out in the nlab is used in Beck's monadicity theorem. The main example of that phenomena (the colimit version) is that a direct sum of modules over some algebra is just the direct sum of the underlying vector spaces with the "obvious" module structure.
answered 2 hours ago
Adrien
4,25621635
This is not what "creates limits" means, although I also find this terminology slightly confusing (in the sense that if I had to guess the definition just from the word "creates" my guess would probably have been the same as you, but it's just not that).
The definition does require the limit of $D$ to exists in $mathbf C$ in the first place, but what it says is basically that you can compute it in $mathbf D$, ie roughly we have both:
- $F$ of a limit of $D$, is a limit of $Fcirc D$
- somewhat conversely (and imprecisely), if some $F(x)$ happens to be a limit of $Fcirc D$, then $x$ was already a limit of $D$.
Now as far as I know this definition is most often used for some shape of diagrams rather than for a single one, and as pointed out in the nlab is used in Beck's monadicity theorem. The main example of that phenomena (the colimit version) is that a direct sum of modules over some algebra is just the direct sum of the underlying vector spaces with the "obvious" module structure.
answered 2 hours ago
Adrien
4,25621635
edited 1 hour ago
answered 2 hours ago
Adrien
4,25621635
answered 2 hours ago
Adrien
4,25621635
answered 2 hours ago
Adrien
4,25621635
4,25621635
add a comment |Â
add a comment |Â
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1
The nLab entry mentions monadicity theorems. Does that not satisfy you?
– Yemon Choi
2 hours ago
Isn't it just like creating the limits of $mathbb Q$ by constructing $mathbb R$? Or the definition of completion of a metric space, or of Hilbert space, or of complete local ring? Or various possible compactifications of manifolds/varieties/ moduli spaces? Or the ideal boundary in hyperbolic geometry? There are countless examples in mathematics in which you want to "see" your (uniquely defined or not) limit/degenerate points instead of just a buch of "holes".
– Qfwfq
13 mins ago