Mixing
inclusion functor of FinSet into Top
I was reading Codensity and Stone spaces by Andrei Sipos where the author defines an inclusion functor from FinSet to Top without (or perhaps I missed it) specifying the topological structure we would endow on finite sets. Does any one know if there is a canonical choice of topological structure associated with a given finite set?
category-theory
edited Nov 20 '18 at 21:03
Derek Elkins
16.2k11337
asked Nov 20 '18 at 20:48
discretizer
1377
add a comment |
I was reading Codensity and Stone spaces by Andrei Sipos where the author defines an inclusion functor from FinSet to Top without (or perhaps I missed it) specifying the topological structure we would endow on finite sets. Does any one know if there is a canonical choice of topological structure associated with a given finite set?
category-theory
edited Nov 20 '18 at 21:03
Derek Elkins
16.2k11337
asked Nov 20 '18 at 20:48
discretizer
1377
5
The discrete topology is standard. It's the only Hausdorff one, at least.
– anomaly
Nov 20 '18 at 20:52
Probably the discrete topology, but I can't say for sure
– Max
Nov 20 '18 at 20:53
1
@anomaly thanks! The observation that only the discrete topology is helpful! Since the claim is the algebra category associated with the induced codensity monad given by the inclusion functor from FinSet into Top is the category of Stone spaces, the Hausdorff structure is probably necessary.
– discretizer
Nov 20 '18 at 21:01
@Max thanks a lot!
– discretizer
Nov 20 '18 at 21:02
add a comment |
I was reading Codensity and Stone spaces by Andrei Sipos where the author defines an inclusion functor from FinSet to Top without (or perhaps I missed it) specifying the topological structure we would endow on finite sets. Does any one know if there is a canonical choice of topological structure associated with a given finite set?
category-theory
edited Nov 20 '18 at 21:03
Derek Elkins
16.2k11337
asked Nov 20 '18 at 20:48
discretizer
1377
I was reading Codensity and Stone spaces by Andrei Sipos where the author defines an inclusion functor from FinSet to Top without (or perhaps I missed it) specifying the topological structure we would endow on finite sets. Does any one know if there is a canonical choice of topological structure associated with a given finite set?
category-theory
category-theory
edited Nov 20 '18 at 21:03
Derek Elkins
16.2k11337
asked Nov 20 '18 at 20:48
discretizer
1377
edited Nov 20 '18 at 21:03
Derek Elkins
16.2k11337
asked Nov 20 '18 at 20:48
discretizer
1377
edited Nov 20 '18 at 21:03
Derek Elkins
16.2k11337
edited Nov 20 '18 at 21:03
Derek Elkins
16.2k11337
edited Nov 20 '18 at 21:03
Derek Elkins
16.2k11337
16.2k11337
asked Nov 20 '18 at 20:48
discretizer
1377
asked Nov 20 '18 at 20:48
discretizer
1377
asked Nov 20 '18 at 20:48
discretizer
1377
1377
5
The discrete topology is standard. It's the only Hausdorff one, at least.
– anomaly
Nov 20 '18 at 20:52
Probably the discrete topology, but I can't say for sure
– Max
Nov 20 '18 at 20:53
1
@anomaly thanks! The observation that only the discrete topology is helpful! Since the claim is the algebra category associated with the induced codensity monad given by the inclusion functor from FinSet into Top is the category of Stone spaces, the Hausdorff structure is probably necessary.
– discretizer
Nov 20 '18 at 21:01
@Max thanks a lot!
– discretizer
Nov 20 '18 at 21:02
add a comment |
5
The discrete topology is standard. It's the only Hausdorff one, at least.
– anomaly
Nov 20 '18 at 20:52
Probably the discrete topology, but I can't say for sure
– Max
Nov 20 '18 at 20:53
1
@anomaly thanks! The observation that only the discrete topology is helpful! Since the claim is the algebra category associated with the induced codensity monad given by the inclusion functor from FinSet into Top is the category of Stone spaces, the Hausdorff structure is probably necessary.
– discretizer
Nov 20 '18 at 21:01
@Max thanks a lot!
– discretizer
Nov 20 '18 at 21:02
5
5
The discrete topology is standard. It's the only Hausdorff one, at least.
– anomaly
Nov 20 '18 at 20:52
The discrete topology is standard. It's the only Hausdorff one, at least.
– anomaly
Nov 20 '18 at 20:52
Probably the discrete topology, but I can't say for sure
– Max
Nov 20 '18 at 20:53
Probably the discrete topology, but I can't say for sure
– Max
Nov 20 '18 at 20:53
1
1
@anomaly thanks! The observation that only the discrete topology is helpful! Since the claim is the algebra category associated with the induced codensity monad given by the inclusion functor from FinSet into Top is the category of Stone spaces, the Hausdorff structure is probably necessary.
– discretizer
Nov 20 '18 at 21:01
@anomaly thanks! The observation that only the discrete topology is helpful! Since the claim is the algebra category associated with the induced codensity monad given by the inclusion functor from FinSet into Top is the category of Stone spaces, the Hausdorff structure is probably necessary.
– discretizer
Nov 20 '18 at 21:01
@Max thanks a lot!
– discretizer
Nov 20 '18 at 21:02
@Max thanks a lot!
– discretizer
Nov 20 '18 at 21:02
add a comment |
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5
The discrete topology is standard. It's the only Hausdorff one, at least.
– anomaly
Nov 20 '18 at 20:52
Probably the discrete topology, but I can't say for sure
– Max
Nov 20 '18 at 20:53
1
@anomaly thanks! The observation that only the discrete topology is helpful! Since the claim is the algebra category associated with the induced codensity monad given by the inclusion functor from FinSet into Top is the category of Stone spaces, the Hausdorff structure is probably necessary.
– discretizer
Nov 20 '18 at 21:01
@Max thanks a lot!
– discretizer
Nov 20 '18 at 21:02