Lift uniqueness for connected colimits of the projection functor $Pi : c/C to C$
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I'm currently working through proposition 3.3.8 of Emily Riehl's Category Theory in Context, which proves that the projection functor $Pi: c/C to C$ strictly creates limits and connected colimits (i.e. if the image of a diagram is a (co) limit in $C$, there is a unique lift to a (co) limit on $c/C$). The first remark notes that a diagram $(K,kappa) : J to c/C$ in $c/C$ is a functor $K: J to C$ together with a cone $kappa: c Rightarrow K$, whose image via $Pi$ is the diagram $K$. Hence the idea is to prove that if $K$ is a limit/connected colimit then there is a unique (co) limit cone over $(K,kappa)$ whose image is the (co) limit cone over $K$.
I have understood the case for limits, but there is a subtlety in the connected colimits case which I am failing to understand. The author takes a colimit cone $mu : K Rightarrow p$, and defines an arrow $c xrightarrow{zeta} p in operatorname{obj} c/C$ via $zeta := mu_jkappa_j$ for some $j in operatorname{obj} J$. Immediately after, it is claimed that $zeta$ is independent of the choice of $j$ since $J$ is assumed to be connected. Hence $mu$ together with $zeta$ give a colimit cone over $(K,kappa)$, proving that $K$ has a colimit lift, and moreover it is unique since $zeta$ is determined by $mu$ and $kappa$.
I get the outline of the argument, but I am not yet convinced of why $J$ being connected implies that $zeta = mu_jkappa_j$ for all $j$ objects of $J$, which seems a central step in the proof (both for uniqueness and to define a lift cone in the slice category to begin with).
Any help would be greatly appreciated!
category-theory proof-explanation
asked Dec 28 '18 at 9:00
Guido A.Guido A.
8,2651730
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I'm currently working through proposition 3.3.8 of Emily Riehl's Category Theory in Context, which proves that the projection functor $Pi: c/C to C$ strictly creates limits and connected colimits (i.e. if the image of a diagram is a (co) limit in $C$, there is a unique lift to a (co) limit on $c/C$). The first remark notes that a diagram $(K,kappa) : J to c/C$ in $c/C$ is a functor $K: J to C$ together with a cone $kappa: c Rightarrow K$, whose image via $Pi$ is the diagram $K$. Hence the idea is to prove that if $K$ is a limit/connected colimit then there is a unique (co) limit cone over $(K,kappa)$ whose image is the (co) limit cone over $K$.
I have understood the case for limits, but there is a subtlety in the connected colimits case which I am failing to understand. The author takes a colimit cone $mu : K Rightarrow p$, and defines an arrow $c xrightarrow{zeta} p in operatorname{obj} c/C$ via $zeta := mu_jkappa_j$ for some $j in operatorname{obj} J$. Immediately after, it is claimed that $zeta$ is independent of the choice of $j$ since $J$ is assumed to be connected. Hence $mu$ together with $zeta$ give a colimit cone over $(K,kappa)$, proving that $K$ has a colimit lift, and moreover it is unique since $zeta$ is determined by $mu$ and $kappa$.
I get the outline of the argument, but I am not yet convinced of why $J$ being connected implies that $zeta = mu_jkappa_j$ for all $j$ objects of $J$, which seems a central step in the proof (both for uniqueness and to define a lift cone in the slice category to begin with).
Any help would be greatly appreciated!
category-theory proof-explanation
asked Dec 28 '18 at 9:00
Guido A.Guido A.
8,2651730
$endgroup$
add a comment |
$begingroup$
I'm currently working through proposition 3.3.8 of Emily Riehl's Category Theory in Context, which proves that the projection functor $Pi: c/C to C$ strictly creates limits and connected colimits (i.e. if the image of a diagram is a (co) limit in $C$, there is a unique lift to a (co) limit on $c/C$). The first remark notes that a diagram $(K,kappa) : J to c/C$ in $c/C$ is a functor $K: J to C$ together with a cone $kappa: c Rightarrow K$, whose image via $Pi$ is the diagram $K$. Hence the idea is to prove that if $K$ is a limit/connected colimit then there is a unique (co) limit cone over $(K,kappa)$ whose image is the (co) limit cone over $K$.
I have understood the case for limits, but there is a subtlety in the connected colimits case which I am failing to understand. The author takes a colimit cone $mu : K Rightarrow p$, and defines an arrow $c xrightarrow{zeta} p in operatorname{obj} c/C$ via $zeta := mu_jkappa_j$ for some $j in operatorname{obj} J$. Immediately after, it is claimed that $zeta$ is independent of the choice of $j$ since $J$ is assumed to be connected. Hence $mu$ together with $zeta$ give a colimit cone over $(K,kappa)$, proving that $K$ has a colimit lift, and moreover it is unique since $zeta$ is determined by $mu$ and $kappa$.
I get the outline of the argument, but I am not yet convinced of why $J$ being connected implies that $zeta = mu_jkappa_j$ for all $j$ objects of $J$, which seems a central step in the proof (both for uniqueness and to define a lift cone in the slice category to begin with).
Any help would be greatly appreciated!
category-theory proof-explanation
asked Dec 28 '18 at 9:00
Guido A.Guido A.
8,2651730
$endgroup$
I'm currently working through proposition 3.3.8 of Emily Riehl's Category Theory in Context, which proves that the projection functor $Pi: c/C to C$ strictly creates limits and connected colimits (i.e. if the image of a diagram is a (co) limit in $C$, there is a unique lift to a (co) limit on $c/C$). The first remark notes that a diagram $(K,kappa) : J to c/C$ in $c/C$ is a functor $K: J to C$ together with a cone $kappa: c Rightarrow K$, whose image via $Pi$ is the diagram $K$. Hence the idea is to prove that if $K$ is a limit/connected colimit then there is a unique (co) limit cone over $(K,kappa)$ whose image is the (co) limit cone over $K$.
I have understood the case for limits, but there is a subtlety in the connected colimits case which I am failing to understand. The author takes a colimit cone $mu : K Rightarrow p$, and defines an arrow $c xrightarrow{zeta} p in operatorname{obj} c/C$ via $zeta := mu_jkappa_j$ for some $j in operatorname{obj} J$. Immediately after, it is claimed that $zeta$ is independent of the choice of $j$ since $J$ is assumed to be connected. Hence $mu$ together with $zeta$ give a colimit cone over $(K,kappa)$, proving that $K$ has a colimit lift, and moreover it is unique since $zeta$ is determined by $mu$ and $kappa$.
I get the outline of the argument, but I am not yet convinced of why $J$ being connected implies that $zeta = mu_jkappa_j$ for all $j$ objects of $J$, which seems a central step in the proof (both for uniqueness and to define a lift cone in the slice category to begin with).
Any help would be greatly appreciated!
category-theory proof-explanation
category-theory proof-explanation
asked Dec 28 '18 at 9:00
Guido A.Guido A.
8,2651730
asked Dec 28 '18 at 9:00
Guido A.Guido A.
8,2651730
edited Dec 28 '18 at 11:08
Guido A.
asked Dec 28 '18 at 9:00
Guido A.Guido A.
8,2651730
asked Dec 28 '18 at 9:00
Guido A.Guido A.
8,2651730
asked Dec 28 '18 at 9:00
Guido A.Guido A.
8,2651730
8,2651730
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1 Answer
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Pick any projection of the cone $kappa$, i.e. $kappa_j : c to Kj$. Then for any other projection $kappa_i : c to Ki$, we have either $kappa_j=Kfcirckappa_i$ or $kappa_i=Kgcirckappa_j$ by connectedness. For the colimiting cocone, we have the opposite: given the coprojection $mu_j : Kjto p$ we have $mu_i=mu_jcirc Kf$ or $mu_j=mu_icirc Kg$ respectively. In the first case, we have $mu_jcirckappa_j = mu_icirc Kfcirckappa_i = mu_icirckappa_i$ and similarly for the second case.
answered Dec 28 '18 at 19:12
Derek ElkinsDerek Elkins
17.7k11437
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$begingroup$
Pick any projection of the cone $kappa$, i.e. $kappa_j : c to Kj$. Then for any other projection $kappa_i : c to Ki$, we have either $kappa_j=Kfcirckappa_i$ or $kappa_i=Kgcirckappa_j$ by connectedness. For the colimiting cocone, we have the opposite: given the coprojection $mu_j : Kjto p$ we have $mu_i=mu_jcirc Kf$ or $mu_j=mu_icirc Kg$ respectively. In the first case, we have $mu_jcirckappa_j = mu_icirc Kfcirckappa_i = mu_icirckappa_i$ and similarly for the second case.
answered Dec 28 '18 at 19:12
Derek ElkinsDerek Elkins
17.7k11437
$endgroup$
add a comment |
$begingroup$
Pick any projection of the cone $kappa$, i.e. $kappa_j : c to Kj$. Then for any other projection $kappa_i : c to Ki$, we have either $kappa_j=Kfcirckappa_i$ or $kappa_i=Kgcirckappa_j$ by connectedness. For the colimiting cocone, we have the opposite: given the coprojection $mu_j : Kjto p$ we have $mu_i=mu_jcirc Kf$ or $mu_j=mu_icirc Kg$ respectively. In the first case, we have $mu_jcirckappa_j = mu_icirc Kfcirckappa_i = mu_icirckappa_i$ and similarly for the second case.
answered Dec 28 '18 at 19:12
Derek ElkinsDerek Elkins
17.7k11437
$endgroup$
add a comment |
$begingroup$
Pick any projection of the cone $kappa$, i.e. $kappa_j : c to Kj$. Then for any other projection $kappa_i : c to Ki$, we have either $kappa_j=Kfcirckappa_i$ or $kappa_i=Kgcirckappa_j$ by connectedness. For the colimiting cocone, we have the opposite: given the coprojection $mu_j : Kjto p$ we have $mu_i=mu_jcirc Kf$ or $mu_j=mu_icirc Kg$ respectively. In the first case, we have $mu_jcirckappa_j = mu_icirc Kfcirckappa_i = mu_icirckappa_i$ and similarly for the second case.
answered Dec 28 '18 at 19:12
Derek ElkinsDerek Elkins
17.7k11437
$endgroup$
Pick any projection of the cone $kappa$, i.e. $kappa_j : c to Kj$. Then for any other projection $kappa_i : c to Ki$, we have either $kappa_j=Kfcirckappa_i$ or $kappa_i=Kgcirckappa_j$ by connectedness. For the colimiting cocone, we have the opposite: given the coprojection $mu_j : Kjto p$ we have $mu_i=mu_jcirc Kf$ or $mu_j=mu_icirc Kg$ respectively. In the first case, we have $mu_jcirckappa_j = mu_icirc Kfcirckappa_i = mu_icirckappa_i$ and similarly for the second case.
answered Dec 28 '18 at 19:12
Derek ElkinsDerek Elkins
17.7k11437
answered Dec 28 '18 at 19:12
Derek ElkinsDerek Elkins
17.7k11437
answered Dec 28 '18 at 19:12
Derek ElkinsDerek Elkins
17.7k11437
answered Dec 28 '18 at 19:12
Derek ElkinsDerek Elkins
17.7k11437
17.7k11437
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