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• CommentRowNumber1.
• CommentAuthorTobias Fritz
• CommentTimeNov 9th 2012
• (edited Nov 9th 2012)
In the definition, the article states "every object in C is a small object (which follows from 2 and 3)". The bracketed remark doesn't seem quite right to me, since neither 2 nor 3 talk about smallness of objects. Presumably this should better be phrased as in A.1.1 of HTT, "assuming 3, this is equivalent to the assertion that every object in S is small".

Am I right? I don't (yet) feel confident enough with my category theory to change this single-handedly.
• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeNov 9th 2012

Yes, you’re right. Someone should go through the article again and fix the mistakes (I may do so in the near future).

• CommentRowNumber3.
• CommentAuthorTobias Fritz
• CommentTimeNov 10th 2012
Thanks, Todd, I fixed this. (I feel a bit stupid for having my name there on the bottom although I've barely done anything...)
• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeFeb 18th 2019

Added well-poweredness and well-copoweredness to properties

• CommentRowNumber5.
• CommentAuthorJohn Baez
• CommentTimeFeb 8th 2020

Changed “presentable” to “locally presentable” in first paragraph, to reduce the chance that people think a second distinct notion is being introduced.

• CommentRowNumber6.
• CommentAuthorJohn Baez
• CommentTimeFeb 8th 2020
• (edited Feb 8th 2020)

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

1. $\mathcal{C}$ is a locally small category;

2. $\mathcal{C}$ has all small colimits;

3. 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 $S$);

4. 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:

1. Every object in $\mathcal{C}$ can be written as a colimit of a small diagram whose objects are in $S$;

2. 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!

• CommentRowNumber7.
• CommentAuthorJohn Baez
• CommentTimeFeb 8th 2020

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.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeFeb 9th 2020

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 $S$).

• CommentRowNumber9.
• CommentAuthorJohn Baez
• CommentTimeNov 3rd 2020

Added a corollary: locally presentable categories are complete.

• CommentRowNumber10.
• CommentAuthorjdc
• CommentTimeNov 19th 2021

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.

• CommentRowNumber11.
• CommentAuthorvarkor
• CommentTimeDec 17th 2021

Clarified remark about “locally presentable category” versus “presentable category”.