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Yes, you’re right. Someone should go through the article again and fix the mistakes (I may do so in the near future).
Why is Emily Riehl’s definition of “locally presentable” category in Categories in Context simpler than the nLab definition? Are they equivalent?
The nLab says a categorry $\mathcal{C}$ is locally presentable iff
$\mathcal{C}$ is a locally small category;
$\mathcal{C}$ has all small colimits;
there exists a small set $S \hookrightarrow Obj(\mathcal{C})$ of $\lambda$-small objects that generates $\mathcal{C}$ under $\lambda$-filtered colimits for some regular cardinal $\lambda$.
(meaning that every object of $\mathcal{C}$ may be written as a colimit over a diagram with objects in $S$);
every object in $\mathcal{C}$ is a small object (assuming 3, this is equivalent to the assertion that every object in $S$ is small).
Riehl’s definition is that $\mathcal{C}$ is locally presentable iff it is locally small, cocomplete, and for some regular cardinal $\lambda$ it has a set $S$ of objects such that:
Every object in $\mathcal{C}$ can be written as a colimit of a small diagram whose objects are in $S$;
For each object $s \in S$, the functor preserves $\mathcal{C{\lambda$-filtered colimits.
So, the $n$Lab definition seems to include two extra conditions. First, that every object in $\mathcal{C}$ can be written as a colimit of a $\lambda$-filtered small diagram whose objects are in $S$. Second, condition 4, which seems redundant since it seems to be built into condition 3, at least if $\lambda$-small implies small.
Surely there should be some way to simplify this nLab definition!
The $\lambda$-filtered condition is in Adamek and Rosicky’s book, so either Riehl left it out by accident or somehow she noticed it could be safely dropped - I don’t see how.
Quick reaction: I’m not sure what 4. is doing there either, and I agree that Emily’s 1. needs to be fixed (my taste would be to have her 2. coming before 1., i.e. say what the objects in $S$ are doing before describing other objects in terms of $S$).
Item 4 of the definition of locally presentable category (Def 2.1) was there before Kevin Carlson added “$\lambda$-small” to item 3. So item 4 should be changed to a remark. I’ve just done this.
I also think that “$\lambda$-small” should be changed to “$\lambda$-compact”, to be consistent with the rest of the page and linked pages, unless there is a subtle difference between the two that I’m not aware of. I’ve made this change as well.
I also removed a parenthetical remark in item 3 that was no longer correct and wasn’t adding anything.
Accidentally edited this page instead of creating a new one. I’m rectifying my mistake now.
added (here) statement of sufficient conditions for a Grothendieck construction to be locally presentable (from MO:a/102083)
On closer inspection, I wonder that MO:a/102083 has a misprint:
The condition on the pseudofunctor $\Phi \,\colon\, B^{op} \to Cat$ seems to be to preserve 2-limits, not 2-co-limits — this according to Def. 5.3.1 (4) in Makkai & Paré 1989, who phrase it that $\Phi$ must take colimits in $B$ to pseudo-limits in $Cat$.
added pointer to
for discussion of local presentability in the context of enriched categories
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