Mixing
What does “topological heuristics” refer to here?
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This quote is from the wikipedia page on topoi:
Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic.
What does this article mean by "topological heuristics"? This suggests to me that there are certain principles that are in a sense "deeper" than topology, that have application in broader contexts. What are those principles?
general-topology category-theory topos-theory
asked Jan 30 at 15:08
user56834user56834
3,38621253
$endgroup$
add a comment |
$begingroup$
This quote is from the wikipedia page on topoi:
Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic.
What does this article mean by "topological heuristics"? This suggests to me that there are certain principles that are in a sense "deeper" than topology, that have application in broader contexts. What are those principles?
general-topology category-theory topos-theory
asked Jan 30 at 15:08
user56834user56834
3,38621253
$endgroup$
$begingroup$
"topological heuristics" means that you can draw an intuitive analogy between your situation and a "spatial", topological situation
$endgroup$
– Max
Jan 30 at 15:45
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@Max, Do you by “spatial” mean “geometric”? It seems to me that “topological” has to do with “touching” of things, whereas “geometric” is something more general (e.g. also having to do with distance)
$endgroup$
– user56834
Jan 30 at 15:53
$begingroup$
I don't mean geometric, precisely because usually there isn't this notion of distance or rigidity; it's really topological : related to how things relate to one another, what's local etc.
$endgroup$
– Max
Jan 30 at 16:02
add a comment |
$begingroup$
This quote is from the wikipedia page on topoi:
Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic.
What does this article mean by "topological heuristics"? This suggests to me that there are certain principles that are in a sense "deeper" than topology, that have application in broader contexts. What are those principles?
general-topology category-theory topos-theory
asked Jan 30 at 15:08
user56834user56834
3,38621253
$endgroup$
This quote is from the wikipedia page on topoi:
Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic.
What does this article mean by "topological heuristics"? This suggests to me that there are certain principles that are in a sense "deeper" than topology, that have application in broader contexts. What are those principles?
general-topology category-theory topos-theory
general-topology category-theory topos-theory
asked Jan 30 at 15:08
user56834user56834
3,38621253
asked Jan 30 at 15:08
user56834user56834
3,38621253
asked Jan 30 at 15:08
user56834user56834
3,38621253
asked Jan 30 at 15:08
user56834user56834
3,38621253
asked Jan 30 at 15:08
user56834user56834
3,38621253
3,38621253
$begingroup$
"topological heuristics" means that you can draw an intuitive analogy between your situation and a "spatial", topological situation
$endgroup$
– Max
Jan 30 at 15:45
$begingroup$
@Max, Do you by “spatial” mean “geometric”? It seems to me that “topological” has to do with “touching” of things, whereas “geometric” is something more general (e.g. also having to do with distance)
$endgroup$
– user56834
Jan 30 at 15:53
$begingroup$
I don't mean geometric, precisely because usually there isn't this notion of distance or rigidity; it's really topological : related to how things relate to one another, what's local etc.
$endgroup$
– Max
Jan 30 at 16:02
add a comment |
$begingroup$
"topological heuristics" means that you can draw an intuitive analogy between your situation and a "spatial", topological situation
$endgroup$
– Max
Jan 30 at 15:45
$begingroup$
@Max, Do you by “spatial” mean “geometric”? It seems to me that “topological” has to do with “touching” of things, whereas “geometric” is something more general (e.g. also having to do with distance)
$endgroup$
– user56834
Jan 30 at 15:53
$begingroup$
I don't mean geometric, precisely because usually there isn't this notion of distance or rigidity; it's really topological : related to how things relate to one another, what's local etc.
$endgroup$
– Max
Jan 30 at 16:02
$begingroup$
"topological heuristics" means that you can draw an intuitive analogy between your situation and a "spatial", topological situation
$endgroup$
– Max
Jan 30 at 15:45
$begingroup$
"topological heuristics" means that you can draw an intuitive analogy between your situation and a "spatial", topological situation
$endgroup$
– Max
Jan 30 at 15:45
$begingroup$
@Max, Do you by “spatial” mean “geometric”? It seems to me that “topological” has to do with “touching” of things, whereas “geometric” is something more general (e.g. also having to do with distance)
$endgroup$
– user56834
Jan 30 at 15:53
$begingroup$
@Max, Do you by “spatial” mean “geometric”? It seems to me that “topological” has to do with “touching” of things, whereas “geometric” is something more general (e.g. also having to do with distance)
$endgroup$
– user56834
Jan 30 at 15:53
$begingroup$
I don't mean geometric, precisely because usually there isn't this notion of distance or rigidity; it's really topological : related to how things relate to one another, what's local etc.
$endgroup$
– Max
Jan 30 at 16:02
$begingroup$
I don't mean geometric, precisely because usually there isn't this notion of distance or rigidity; it's really topological : related to how things relate to one another, what's local etc.
$endgroup$
– Max
Jan 30 at 16:02
add a comment |
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$begingroup$
"topological heuristics" means that you can draw an intuitive analogy between your situation and a "spatial", topological situation
$endgroup$
– Max
Jan 30 at 15:45
$begingroup$
@Max, Do you by “spatial” mean “geometric”? It seems to me that “topological” has to do with “touching” of things, whereas “geometric” is something more general (e.g. also having to do with distance)
$endgroup$
– user56834
Jan 30 at 15:53
$begingroup$
I don't mean geometric, precisely because usually there isn't this notion of distance or rigidity; it's really topological : related to how things relate to one another, what's local etc.
$endgroup$
– Max
Jan 30 at 16:02