# Is $\mathit{Topos}^\text{op} \to \mathit{Pr}^L$ monadic?


Question 1: Is the inclusion $$\Logos \to \Pr^L$$ monadic?

Question 2: If so, is the induced monad lax-idempotent?

I believe this functor preserves limits and filtered colimits. It doesn't preserve coproducts. I'm not sure if it actually has a left adjoint.

If the answer is "yes, up to size issues", that would be interesting too.

I think this might be one of those questions which is cleaner to consider in the $$\infty$$-categorical context than in the 1-categorical context, but I could be wrong. I'd be interested to hear about the 1-categorical case as well (where I suppose one would consider the $$(2,1)$$-categories of 1-logoi and locally presentable 1-categories).

Regarding monadicity (rather than comonadicity), the (2-categorical variant of the) question is answered in Bunge–Carboni's The symmetric topos. In their paper, $$\mathbf A$$ denotes the 2-category of locally presentable categories and cocontinuous functors (i.e. left adjoint functors), and $$\mathbf R$$ denotes the 2-category of logoi. There is a 2-adjunction $$\Sigma : \mathbf A \rightleftarrows \mathbf R : U$$ (Theorem 3.1) and the induced 2-monad is lax idempotent (Theorem 4.1 and the following discussion).
Presumably everything works out similarly in the $$(\infty, 2)$$-categorical setting.