- #1
Euge
Gold Member
MHB
POTW Director
- 2,073
- 242
Here's this week's problem!
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Problem. Let $L : \mathcal{C} \to \mathcal{C}$ be a left-adjoint functor from category $\mathcal{C}$ to category $\mathcal{C'}$. Show that if $F : \mathcal{D} \to \mathcal{C}$ is a functor such that $\operatorname{colim} F$ is an object of $\mathcal{C}$, then $L(\operatorname{colim} F)$ is a colimit of $L \circ F : \mathcal{D} \to \mathcal{C'}$.
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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
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Problem. Let $L : \mathcal{C} \to \mathcal{C}$ be a left-adjoint functor from category $\mathcal{C}$ to category $\mathcal{C'}$. Show that if $F : \mathcal{D} \to \mathcal{C}$ is a functor such that $\operatorname{colim} F$ is an object of $\mathcal{C}$, then $L(\operatorname{colim} F)$ is a colimit of $L \circ F : \mathcal{D} \to \mathcal{C'}$.
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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!