1
$\begingroup$

The following is at Emily Riehl’s Category theory in context, page 90.

Firstly the Definition of creation of limits:

Definition 3.3.1. : For any class of diagrams $K : J → C$ valued in $C$, a functor $F : C → D$ creates those limits if whenever $FK : J → D$ has a limit in $D$, there is some limit cone over $FK$ that can be lifted to a limit cone over $K$, and moreover $F$ reflects the limits in the class of diagrams.

But later the author writes the following at the same page, which I think is a weaker hypothesis than that in Definition 3.3.1, because it seems not to require the limit cone existing in $C$ to be a lift of a limit cone in $D$.

Remark 3.3.2. To ground intuition for these terms, the following metaphor may prove helpful: a functor $F : C → D $ maps from “upstairs” (the category $C$) to “downstairs” (the category $D$). An upstairs diagram (a functor $K : J → C$) sits above a downstairs diagram (by composing with $F$).

$F$ creates limits if the mere presence of a downstairs limit cone implies the existence of an upstairs limit cone, and if any cone sitting above a limit cone is a limit cone.

May I miss something? Thanks for any hint.

$\endgroup$
0
$\begingroup$

This is indeed a weaker hypothesis. The formal definition supposed that $F$ lifts and reflects limits, while the intuitive one suppose only that $F$ reflects limits and the existence of limits. As long as $\mathcal C$ has limits, the formal definition implies that $F$ preserves them, but the intuitive one does not, so is missing a piece of the concept.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.