I have recently been reading Lurie's "Higher Topos Theory", and come upon what I believe to be an erronous claim. However, the author goes to some pain as a result of that claim, and the error seems to affect multiple other statements. Also, I am relatively unfamiliar with arguments about sizes of categories, which seem to be relevant here. As a result, I am hesitant to trust my understanding.


Specifically, the author defines the category $Idem$ in definition 4.4.5.2 (it is also defined in [nlab](https://ncatlab.org/nlab/show/idempotent+complete+(infinity,1)-category)). To my understanding, it is the nerve of the ordinary category with one object and two morphisms, $id$, $e$ with $e\circ e=e$.


Above proposition 4.4.5.15, the author claims that this category is not filtered. This does not seem true to me: for any simplicial set $K$ and a map $K\rightarrow Idem$, it is possible to extend it to a cone point under $K$ via $e$.


This issue resurfaces again later. Indeed, in proposition 5.3.1.15, it is claimed that a sufficiently small filtered category has a final object. To my understanding, $Idem$ *must* be a counterexample to that theorem. Otherwise, consider a very small $\infty$-category $\mathcal{C}$, and a much larger cardinal $\kappa$. The ind-category $Ind\_\kappa(\mathcal{C})$ should be the idempotent completion of $\mathcal{C}$. However, it is also given by the $\kappa$-right-exact functors on $\mathcal{C}$, which are the $\kappa$-filtered right fibrations over $\mathcal{C}$. By choosing $\kappa$ sufficiently large (and perhaps enlarging the universe), we may assume that such fibrations are $\kappa$-small, and thus have a final object and are representable. This would imply that $Ind\_\kappa(\mathcal{C})=\mathcal{C}$, a contradiction. This suggests that $Idem$ should, in a sense, be the only counterexample.


In fact, the above argument could be made even more concrete: let $\mathcal{C}=Idem$. Its idempotent completion $Idem^+$ contains an extra object $x$, which defines a right fibration over $Idem$ via $Idem\times\_{Idem^+}Idem^+\_{/x}$. This is easily seen to be equivalent to the trivial right fibration $Idem\rightarrow Idem$. Since this fibration belongs to $Ind\_\kappa(Idem)$ for all $\kappa$, it should be $\kappa$-filtered.


Finally, I believe that I also know where the error is in the proof of 5.3.1.15. The author claims that the existence of a retract $\mathcal{C}^\triangleright\rightarrow\mathcal{C}$ implies that $\mathcal{C}$ has a final object. This is false, again as can be demonstrated by $Idem$.


I am uncertain how to reconcile any of this with the author's claims, and would gladly appreciate any help!



Is the category Idem filtered?
As noted in the comments, this was a mistake in the book, and is now corrected in the online version.


