Functorial definition of continuous map
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My motivation for this question is pure abstract nonsense, but I'd like to define continuous maps or continuity itself in a functorial/categorical way, since every continuous map induces a map (of sets) between the topologies in the opposite direcction.
Therefore, I could say that a continuous map $f:Xto Y$ is just a map of sets, which induces a map $f^*:Top(Y)to Top(X)$. The thing is that given a set, there are lots of topologies on it and given two topological spaces there are in general plenty of continuous maps, so I've thought of taking a subcategory $S$ of Set$times mathbf{Set}^{op}$ whose pairs consist of a set and a topology on it. In this setting, I consider a functor $F:Sto mathbf{Set}$ which is the second projection on objets, and sending each arrow $(x,t(x))to (y,t(y))$ to an arrow $t(y)to t(x)$. In the best case this would give a class of continuous maps $xto y$, but I'm not sure that every map $t(y)to t(x)$ gives rise to a well-defined map $xto y$.
Are there definitions of continuity or other topological concepts following this approach?
general-topology category-theory
asked Dec 18 '18 at 9:31
JaviJavi
2,6842826
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add a comment |
$begingroup$
My motivation for this question is pure abstract nonsense, but I'd like to define continuous maps or continuity itself in a functorial/categorical way, since every continuous map induces a map (of sets) between the topologies in the opposite direcction.
Therefore, I could say that a continuous map $f:Xto Y$ is just a map of sets, which induces a map $f^*:Top(Y)to Top(X)$. The thing is that given a set, there are lots of topologies on it and given two topological spaces there are in general plenty of continuous maps, so I've thought of taking a subcategory $S$ of Set$times mathbf{Set}^{op}$ whose pairs consist of a set and a topology on it. In this setting, I consider a functor $F:Sto mathbf{Set}$ which is the second projection on objets, and sending each arrow $(x,t(x))to (y,t(y))$ to an arrow $t(y)to t(x)$. In the best case this would give a class of continuous maps $xto y$, but I'm not sure that every map $t(y)to t(x)$ gives rise to a well-defined map $xto y$.
Are there definitions of continuity or other topological concepts following this approach?
general-topology category-theory
asked Dec 18 '18 at 9:31
JaviJavi
2,6842826
$endgroup$
2
$begingroup$
I think you might be interested in the notion of frame.
$endgroup$
– Arnaud D.
Dec 18 '18 at 10:22
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Thanks, I just read today that a topological space is a set with a frame of open sets.
$endgroup$
– Javi
Dec 18 '18 at 10:51
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Where did contravariancy disappear? $S$ should rather be a subcategory of ${bf Set}times{bf Set}^{op}$, shouldn't it?
$endgroup$
– Berci
Dec 20 '18 at 23:55
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That's right @Berci
$endgroup$
– Javi
Dec 21 '18 at 15:58
add a comment |
$begingroup$
My motivation for this question is pure abstract nonsense, but I'd like to define continuous maps or continuity itself in a functorial/categorical way, since every continuous map induces a map (of sets) between the topologies in the opposite direcction.
Therefore, I could say that a continuous map $f:Xto Y$ is just a map of sets, which induces a map $f^*:Top(Y)to Top(X)$. The thing is that given a set, there are lots of topologies on it and given two topological spaces there are in general plenty of continuous maps, so I've thought of taking a subcategory $S$ of Set$times mathbf{Set}^{op}$ whose pairs consist of a set and a topology on it. In this setting, I consider a functor $F:Sto mathbf{Set}$ which is the second projection on objets, and sending each arrow $(x,t(x))to (y,t(y))$ to an arrow $t(y)to t(x)$. In the best case this would give a class of continuous maps $xto y$, but I'm not sure that every map $t(y)to t(x)$ gives rise to a well-defined map $xto y$.
Are there definitions of continuity or other topological concepts following this approach?
general-topology category-theory
asked Dec 18 '18 at 9:31
JaviJavi
2,6842826
$endgroup$
My motivation for this question is pure abstract nonsense, but I'd like to define continuous maps or continuity itself in a functorial/categorical way, since every continuous map induces a map (of sets) between the topologies in the opposite direcction.
Therefore, I could say that a continuous map $f:Xto Y$ is just a map of sets, which induces a map $f^*:Top(Y)to Top(X)$. The thing is that given a set, there are lots of topologies on it and given two topological spaces there are in general plenty of continuous maps, so I've thought of taking a subcategory $S$ of Set$times mathbf{Set}^{op}$ whose pairs consist of a set and a topology on it. In this setting, I consider a functor $F:Sto mathbf{Set}$ which is the second projection on objets, and sending each arrow $(x,t(x))to (y,t(y))$ to an arrow $t(y)to t(x)$. In the best case this would give a class of continuous maps $xto y$, but I'm not sure that every map $t(y)to t(x)$ gives rise to a well-defined map $xto y$.
Are there definitions of continuity or other topological concepts following this approach?
general-topology category-theory
general-topology category-theory
asked Dec 18 '18 at 9:31
JaviJavi
2,6842826
asked Dec 18 '18 at 9:31
JaviJavi
2,6842826
edited Dec 21 '18 at 15:59
Javi
asked Dec 18 '18 at 9:31
JaviJavi
2,6842826
asked Dec 18 '18 at 9:31
JaviJavi
2,6842826
asked Dec 18 '18 at 9:31
JaviJavi
2,6842826
2,6842826
2
$begingroup$
I think you might be interested in the notion of frame.
$endgroup$
– Arnaud D.
Dec 18 '18 at 10:22
$begingroup$
Thanks, I just read today that a topological space is a set with a frame of open sets.
$endgroup$
– Javi
Dec 18 '18 at 10:51
$begingroup$
Where did contravariancy disappear? $S$ should rather be a subcategory of ${bf Set}times{bf Set}^{op}$, shouldn't it?
$endgroup$
– Berci
Dec 20 '18 at 23:55
$begingroup$
That's right @Berci
$endgroup$
– Javi
Dec 21 '18 at 15:58
add a comment |
2
$begingroup$
I think you might be interested in the notion of frame.
$endgroup$
– Arnaud D.
Dec 18 '18 at 10:22
$begingroup$
Thanks, I just read today that a topological space is a set with a frame of open sets.
$endgroup$
– Javi
Dec 18 '18 at 10:51
$begingroup$
Where did contravariancy disappear? $S$ should rather be a subcategory of ${bf Set}times{bf Set}^{op}$, shouldn't it?
$endgroup$
– Berci
Dec 20 '18 at 23:55
$begingroup$
That's right @Berci
$endgroup$
– Javi
Dec 21 '18 at 15:58
2
2
$begingroup$
I think you might be interested in the notion of frame.
$endgroup$
– Arnaud D.
Dec 18 '18 at 10:22
$begingroup$
I think you might be interested in the notion of frame.
$endgroup$
– Arnaud D.
Dec 18 '18 at 10:22
$begingroup$
Thanks, I just read today that a topological space is a set with a frame of open sets.
$endgroup$
– Javi
Dec 18 '18 at 10:51
$begingroup$
Thanks, I just read today that a topological space is a set with a frame of open sets.
$endgroup$
– Javi
Dec 18 '18 at 10:51
$begingroup$
Where did contravariancy disappear? $S$ should rather be a subcategory of ${bf Set}times{bf Set}^{op}$, shouldn't it?
$endgroup$
– Berci
Dec 20 '18 at 23:55
$begingroup$
Where did contravariancy disappear? $S$ should rather be a subcategory of ${bf Set}times{bf Set}^{op}$, shouldn't it?
$endgroup$
– Berci
Dec 20 '18 at 23:55
$begingroup$
That's right @Berci
$endgroup$
– Javi
Dec 21 '18 at 15:58
$begingroup$
That's right @Berci
$endgroup$
– Javi
Dec 21 '18 at 15:58
add a comment |
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2
$begingroup$
I think you might be interested in the notion of frame.
$endgroup$
– Arnaud D.
Dec 18 '18 at 10:22
$begingroup$
Thanks, I just read today that a topological space is a set with a frame of open sets.
$endgroup$
– Javi
Dec 18 '18 at 10:51
$begingroup$
Where did contravariancy disappear? $S$ should rather be a subcategory of ${bf Set}times{bf Set}^{op}$, shouldn't it?
$endgroup$
– Berci
Dec 20 '18 at 23:55
$begingroup$
That's right @Berci
$endgroup$
– Javi
Dec 21 '18 at 15:58