Which simplicial sets are filtered colimits of standard simplices?
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The question is all in the title : every simplicial set is a colimit of the standard simplices $Delta^n$, but I'm wondering which ones are filtered or directed colimits of these, if there's a nice characterization of these simplicial sets.
The motivation is the following : we can see by hand that $|Delta^ptimes Delta^q|simeq |Delta^p|times|Delta^q$, and since geometric relization commutes with colimits as a left adjoint, if $X,Y$ are filtered colimits of standard simplices, since filtered colimits commute with products in $mathbf{Set, Top}$, we get $|Xtimes Y|simeq |X|times |Y|$, but here the $times$ would be the usual product topology, not the compactly generated product topology.
If the subcategory of filtered colimits of standard simplices is big enough, this would give an easy proof of this last result for a large chunk of simplicial sets and so would be interesting.
(As I'm writing this I'm realizing that I'm not sure that filtered colimits commute with products in $mathbf{Top}$; is it the case ? )
general-topology category-theory simplicial-stuff limits-colimits
asked Dec 6 '18 at 9:47
MaxMax
13.2k11040
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add a comment |
$begingroup$
The question is all in the title : every simplicial set is a colimit of the standard simplices $Delta^n$, but I'm wondering which ones are filtered or directed colimits of these, if there's a nice characterization of these simplicial sets.
The motivation is the following : we can see by hand that $|Delta^ptimes Delta^q|simeq |Delta^p|times|Delta^q$, and since geometric relization commutes with colimits as a left adjoint, if $X,Y$ are filtered colimits of standard simplices, since filtered colimits commute with products in $mathbf{Set, Top}$, we get $|Xtimes Y|simeq |X|times |Y|$, but here the $times$ would be the usual product topology, not the compactly generated product topology.
If the subcategory of filtered colimits of standard simplices is big enough, this would give an easy proof of this last result for a large chunk of simplicial sets and so would be interesting.
(As I'm writing this I'm realizing that I'm not sure that filtered colimits commute with products in $mathbf{Top}$; is it the case ? )
general-topology category-theory simplicial-stuff limits-colimits
asked Dec 6 '18 at 9:47
MaxMax
13.2k11040
$endgroup$
add a comment |
$begingroup$
The question is all in the title : every simplicial set is a colimit of the standard simplices $Delta^n$, but I'm wondering which ones are filtered or directed colimits of these, if there's a nice characterization of these simplicial sets.
The motivation is the following : we can see by hand that $|Delta^ptimes Delta^q|simeq |Delta^p|times|Delta^q$, and since geometric relization commutes with colimits as a left adjoint, if $X,Y$ are filtered colimits of standard simplices, since filtered colimits commute with products in $mathbf{Set, Top}$, we get $|Xtimes Y|simeq |X|times |Y|$, but here the $times$ would be the usual product topology, not the compactly generated product topology.
If the subcategory of filtered colimits of standard simplices is big enough, this would give an easy proof of this last result for a large chunk of simplicial sets and so would be interesting.
(As I'm writing this I'm realizing that I'm not sure that filtered colimits commute with products in $mathbf{Top}$; is it the case ? )
general-topology category-theory simplicial-stuff limits-colimits
asked Dec 6 '18 at 9:47
MaxMax
13.2k11040
$endgroup$
The question is all in the title : every simplicial set is a colimit of the standard simplices $Delta^n$, but I'm wondering which ones are filtered or directed colimits of these, if there's a nice characterization of these simplicial sets.
The motivation is the following : we can see by hand that $|Delta^ptimes Delta^q|simeq |Delta^p|times|Delta^q$, and since geometric relization commutes with colimits as a left adjoint, if $X,Y$ are filtered colimits of standard simplices, since filtered colimits commute with products in $mathbf{Set, Top}$, we get $|Xtimes Y|simeq |X|times |Y|$, but here the $times$ would be the usual product topology, not the compactly generated product topology.
If the subcategory of filtered colimits of standard simplices is big enough, this would give an easy proof of this last result for a large chunk of simplicial sets and so would be interesting.
(As I'm writing this I'm realizing that I'm not sure that filtered colimits commute with products in $mathbf{Top}$; is it the case ? )
general-topology category-theory simplicial-stuff limits-colimits
general-topology category-theory simplicial-stuff limits-colimits
asked Dec 6 '18 at 9:47
MaxMax
13.2k11040
asked Dec 6 '18 at 9:47
MaxMax
13.2k11040
asked Dec 6 '18 at 9:47
MaxMax
13.2k11040
asked Dec 6 '18 at 9:47
MaxMax
13.2k11040
asked Dec 6 '18 at 9:47
MaxMax
13.2k11040
13.2k11040
add a comment |
add a comment |
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