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.


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.


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