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

]]>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.

]]>Added a corollary: locally presentable categories are complete.

]]>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 $S$).

]]>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.

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!

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

]]>Added well-poweredness and well-copoweredness to properties

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

]]>Am I right? I don't (yet) feel confident enough with my category theory to change this single-handedly. ]]>