Is this a 'relative' colimit or some other categorical construction?












1












$begingroup$


Let $mathcal{A}$ be a closed covering of some space $X$, then let $Sigma[mathcal{A}]$ be the category of intersections of subsets of $mathcal{A}$, further let $F: Sigma[mathcal{A}] to mathsf{Top}$ be the a $Sigma[mathcal{A}]$-diagram of spaces with all corresponding maps nice enough (cofibrations and cofibrant images, etc.). Then for some subcategory $J hookrightarrow Sigma[mathcal{A}]$, let



$$ S = bigcup_{i in J} left[F[i] setminus bigcup_{j in left(i downarrow Sigma[mathcal{A}]right) } F[j] right] $$



where $(i downarrow Sigma[mathcal{A}])$ is the undercategory of $Sigma[mathcal{A}]$ at $i$. Can we frame this at all as a "relative" colimit or some other categorical construction?



Further we might be able to pull the internal union (the of the setminus) out of the construction by realizing something about the combination of all of these undercategories.



I eventually wish relate $S$ to some sort of relative homotopy colimit, and eventually a combinatorial model in terms of $J$ and $Sigma[mathcal{A}]$ so any help would be greatly appreciated.






algebraic-topology category-theory homotopy-theory simplicial-stuff







share|cite|improve this question









$endgroup$








  • 2




    $begingroup$
    These sorts of things are typically coends.
    $endgroup$
    – Randall
    Jan 16 at 20:22










  • $begingroup$
    Could you elaborate? I am not familiar with coends, to the degree that I could properly understand $S$ in that language in any reasonable amount of time. @Randall
    $endgroup$
    – MadcowD
    Jan 16 at 22:29










  • $begingroup$
    In a short comment? No. This is an entire chapter in MacLane.
    $endgroup$
    – Randall
    Jan 17 at 15:27
















1












$begingroup$


Let $mathcal{A}$ be a closed covering of some space $X$, then let $Sigma[mathcal{A}]$ be the category of intersections of subsets of $mathcal{A}$, further let $F: Sigma[mathcal{A}] to mathsf{Top}$ be the a $Sigma[mathcal{A}]$-diagram of spaces with all corresponding maps nice enough (cofibrations and cofibrant images, etc.). Then for some subcategory $J hookrightarrow Sigma[mathcal{A}]$, let



$$ S = bigcup_{i in J} left[F[i] setminus bigcup_{j in left(i downarrow Sigma[mathcal{A}]right) } F[j] right] $$



where $(i downarrow Sigma[mathcal{A}])$ is the undercategory of $Sigma[mathcal{A}]$ at $i$. Can we frame this at all as a "relative" colimit or some other categorical construction?



Further we might be able to pull the internal union (the of the setminus) out of the construction by realizing something about the combination of all of these undercategories.



I eventually wish relate $S$ to some sort of relative homotopy colimit, and eventually a combinatorial model in terms of $J$ and $Sigma[mathcal{A}]$ so any help would be greatly appreciated.






algebraic-topology category-theory homotopy-theory simplicial-stuff







share|cite|improve this question









$endgroup$








  • 2




    $begingroup$
    These sorts of things are typically coends.
    $endgroup$
    – Randall
    Jan 16 at 20:22










  • $begingroup$
    Could you elaborate? I am not familiar with coends, to the degree that I could properly understand $S$ in that language in any reasonable amount of time. @Randall
    $endgroup$
    – MadcowD
    Jan 16 at 22:29










  • $begingroup$
    In a short comment? No. This is an entire chapter in MacLane.
    $endgroup$
    – Randall
    Jan 17 at 15:27














1












1








1


1



$begingroup$


Let $mathcal{A}$ be a closed covering of some space $X$, then let $Sigma[mathcal{A}]$ be the category of intersections of subsets of $mathcal{A}$, further let $F: Sigma[mathcal{A}] to mathsf{Top}$ be the a $Sigma[mathcal{A}]$-diagram of spaces with all corresponding maps nice enough (cofibrations and cofibrant images, etc.). Then for some subcategory $J hookrightarrow Sigma[mathcal{A}]$, let



$$ S = bigcup_{i in J} left[F[i] setminus bigcup_{j in left(i downarrow Sigma[mathcal{A}]right) } F[j] right] $$



where $(i downarrow Sigma[mathcal{A}])$ is the undercategory of $Sigma[mathcal{A}]$ at $i$. Can we frame this at all as a "relative" colimit or some other categorical construction?



Further we might be able to pull the internal union (the of the setminus) out of the construction by realizing something about the combination of all of these undercategories.



I eventually wish relate $S$ to some sort of relative homotopy colimit, and eventually a combinatorial model in terms of $J$ and $Sigma[mathcal{A}]$ so any help would be greatly appreciated.






algebraic-topology category-theory homotopy-theory simplicial-stuff







share|cite|improve this question









$endgroup$




Let $mathcal{A}$ be a closed covering of some space $X$, then let $Sigma[mathcal{A}]$ be the category of intersections of subsets of $mathcal{A}$, further let $F: Sigma[mathcal{A}] to mathsf{Top}$ be the a $Sigma[mathcal{A}]$-diagram of spaces with all corresponding maps nice enough (cofibrations and cofibrant images, etc.). Then for some subcategory $J hookrightarrow Sigma[mathcal{A}]$, let



$$ S = bigcup_{i in J} left[F[i] setminus bigcup_{j in left(i downarrow Sigma[mathcal{A}]right) } F[j] right] $$



where $(i downarrow Sigma[mathcal{A}])$ is the undercategory of $Sigma[mathcal{A}]$ at $i$. Can we frame this at all as a "relative" colimit or some other categorical construction?



Further we might be able to pull the internal union (the of the setminus) out of the construction by realizing something about the combination of all of these undercategories.



I eventually wish relate $S$ to some sort of relative homotopy colimit, and eventually a combinatorial model in terms of $J$ and $Sigma[mathcal{A}]$ so any help would be greatly appreciated.






algebraic-topology category-theory homotopy-theory simplicial-stuff




algebraic-topology category-theory homotopy-theory simplicial-stuff






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 16 at 20:21









MadcowDMadcowD

326114




326114








  • 2




    $begingroup$
    These sorts of things are typically coends.
    $endgroup$
    – Randall
    Jan 16 at 20:22










  • $begingroup$
    Could you elaborate? I am not familiar with coends, to the degree that I could properly understand $S$ in that language in any reasonable amount of time. @Randall
    $endgroup$
    – MadcowD
    Jan 16 at 22:29










  • $begingroup$
    In a short comment? No. This is an entire chapter in MacLane.
    $endgroup$
    – Randall
    Jan 17 at 15:27














  • 2




    $begingroup$
    These sorts of things are typically coends.
    $endgroup$
    – Randall
    Jan 16 at 20:22










  • $begingroup$
    Could you elaborate? I am not familiar with coends, to the degree that I could properly understand $S$ in that language in any reasonable amount of time. @Randall
    $endgroup$
    – MadcowD
    Jan 16 at 22:29










  • $begingroup$
    In a short comment? No. This is an entire chapter in MacLane.
    $endgroup$
    – Randall
    Jan 17 at 15:27








2




2




$begingroup$
These sorts of things are typically coends.
$endgroup$
– Randall
Jan 16 at 20:22




$begingroup$
These sorts of things are typically coends.
$endgroup$
– Randall
Jan 16 at 20:22












$begingroup$
Could you elaborate? I am not familiar with coends, to the degree that I could properly understand $S$ in that language in any reasonable amount of time. @Randall
$endgroup$
– MadcowD
Jan 16 at 22:29




$begingroup$
Could you elaborate? I am not familiar with coends, to the degree that I could properly understand $S$ in that language in any reasonable amount of time. @Randall
$endgroup$
– MadcowD
Jan 16 at 22:29












$begingroup$
In a short comment? No. This is an entire chapter in MacLane.
$endgroup$
– Randall
Jan 17 at 15:27




$begingroup$
In a short comment? No. This is an entire chapter in MacLane.
$endgroup$
– Randall
Jan 17 at 15:27










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