# Is the category Idem filtered?

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). 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!

• Idem is not equivalent to an $\omega$-small (=finite) simplicial set, and it is certainly not $\kappa$-filtered for $\kappa$ uncountable, so it can't possibly be a counterexample to Prop 5.3.1.15. Apr 4, 2017 at 20:05
• Ah, sorry, I missed your point. You are saying that $Idem$ is in fact $\kappa$-filtered for all $\kappa$. I think I agree with that claim, and I agree that a retraction of $C\to C^\triangleright$ does not imply there is a final object. Apr 5, 2017 at 0:14
• Yep, looks like a mistake to me. Apr 5, 2017 at 8:33
• That answers my question. Thank you very much! Apr 5, 2017 at 16:36
• Jacob has now fixed that section in the copy of the book on his website. Apr 12, 2017 at 2:33