# A locally $\kappa$-presentable category is also locally $\lambda$-presentable for $\lambda>\kappa$? (Typo?)

In Riehl's Category Theory in Context, Sect. 4.6, we find the following:

Definition 4.6.16. Let $$\kappa$$ be a regular cardinal.¹ A locally small category $$\mathsf{C}$$ is locally $$\kappa$$-presentable if it is cocomplete and if it has a set of objects $$S$$ so that:

1. Every object in $$\mathsf{C}$$ can be written as a colimit of a diagram valued in the subcategory spanned by the objects in $$S$$.
2. For each object $$s \in S$$, the functor $$\mathsf{C}(s,-): \mathsf{C} \rightarrow\mathsf{Set}$$ preserves $$\kappa$$-filtered colimits.

A functor between locally $$\kappa$$-presentable categories is accessible if it preserves $$\kappa$$-filtered colimits.

For example, a large variety of categories whose objects are sets equipped with some sort of "algebraic" structure are locally finitely presentable (meaning locally $$\omega$$-presentable); see Definition 5.5.5. A locally $$\kappa$$-presentable category is also locally $$\lambda$$-presentable for any $$\lambda>\kappa$$.

¹ A regular cardinal is an infinite cardinal $$\kappa$$ with the property that every union of fewer than $$\kappa$$ sets each of cardinality less than $$\kappa$$ has cardinality less than $$\kappa$$.

First of all: I ignore the definition of "cardinal." I don't know any set theory, the only intuition of "cardinal" that I have is "isomorphism class in $$\mathsf{Set}$$."

My only question is: in Riehl's text, is "$$\lambda>\kappa$$" a typo? Shouldn't it be $$\lambda<\kappa$$ instead? It is what my intuition tells me, but again, I have no formal background to support this (since I ignore the relevant set-theoretic definitions). The definition of "$$\kappa$$-filtered colimit" I know is the one from Wikipedia. (I should also say, the naive definition of the order in cardinals I know is: $$\kappa<\lambda$$ if there is an injection $$K\to\Lambda$$ for some sets $$K$$, $$\Lambda$$ with cardinal $$\kappa$$, $$\lambda$$, respectively.)

• (Small remark: there being an injection $K \to \Lambda$ is the definition of $\kappa \le \lambda$. The strict inequality $\kappa < \lambda$ means that $\kappa \le \lambda$ and $\kappa \ne \lambda$ - or equivalently, "there is an injection $K \to \Lambda$ but no injection/no bijection $\Lambda \to K$ for some/any sets $K$ and $\Lambda$ of cardinalities $\kappa, \lambda$".) Dec 25, 2023 at 1:12

It is not a typo*. When a category is locally $$\lambda$$-presentable, you can think of its objects being "smaller than $$\lambda$$" (in a suitable sense, don't take this literally!). When $$\lambda \geq \kappa$$, every object that is smaller than $$\kappa$$ is of course also smaller than $$\lambda$$, and hence locally $$\kappa$$-presentable implies locally $$\lambda$$-presentable. This is, of course, not a formal proof. But I hope that it corrects your intution you had which had it vice versa. For example, every locally $$\omega$$-presentable category is also locally $$\lambda$$-presentable (since $$\omega \leq \lambda$$).

Formal proof: It is enough to show that every functor preserving $$\kappa$$-filtered colimits also preserves $$\lambda$$-filtered colimits**. For this, it is enough to prove that every $$\lambda$$-filtered poset is also $$\kappa$$-filtered. Well, if $$T$$ is a subset of the poset with $$< \kappa$$ elements, it also has $$< \lambda$$ elements, hence has an upper bound.

*Funny enough, I also thought this is a typo when reading the book by Adamek-Rosicky for the first time many years ago.

**By definition, a $$\lambda$$-filtered colimit is a colimit indexed by a $$\lambda$$-filtered poset, meaning that every subset with $$< \lambda$$ elements has an upper bound.

• Thank you so much! To sum it all up: If $\mathsf{J}$ is a $\kappa$-filtered category, then the bigger $\kappa$ is, the "smaller" $\mathsf{J}$ will be, right? Dec 24, 2023 at 12:39
• I don't agree with that summary. Also, notice that $\kappa$ is always $\kappa$-filtered (since $\kappa$ is regular). Dec 24, 2023 at 12:40
• Thanks again. Then I guess the relevant phenomenon at play may be more subtle. Dec 24, 2023 at 12:43
• Is anything unclear? Dec 24, 2023 at 15:13
• No, it's just that I've never worked with these ideas : ) Dec 24, 2023 at 15:31