6
$\begingroup$

For every pseudomonad $T$ on the 2-category of (locally small) categories $\mathbf{Cat}$, we can consider the 2-category of $T$-pseudoalgebras and pseudomorphisms $T\text{-PsAlg}_p$, which is equipped with a forgetful 2-functor $U_T : T\text{-PsAlg}_p \to \mathbf{Cat}$.

Is there an example of such a pseudomonad $T$ for which there does not exist a 2-monad $T'$ for which the 2-category of $T'$-pseudoalgebras and pseudomorphisms $T'\text{-PsAlg}_p$ is biequivalent to $T\text{-PsAlg}_p$, concretely over $\mathbf{Cat}$, in the sense that the following triangle commutes up to pseudomonad equivalence?

The obvious triangle

Intuitively, I am looking for an example of a pseudomonad on $\mathbf{Cat}$ that cannot be strictified into a 2-monad, or a proof that no such example exists. Note that it is certainly the case that a large class of pseudomonads may be strictified (e.g. as in the answer to this question).

(I have previously asked a much more general question, but it seems plausible that this special case admits a simpler answer.)

Improve this question
$\endgroup$
4
  • $\begingroup$ Maybe symmetric or braided monoidal categories? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Feb 27 at 17:10
  • $\begingroup$ @მამუკაჯიბლაძე: these are the pseudoalgebras for the free strict symmetric/braided monoidal category 2-monads on $\mathbf{Cat}$. $\endgroup$
    – varkor
    Commented Feb 27 at 17:50
  • 1
    $\begingroup$ What exactly do you mean by strict? For me it would mean strictly commutative, i. e. the symmetry isomorphisms being just identities. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Feb 27 at 19:20
  • $\begingroup$ @მამუკაჯიბლაძე: sorry, I meant to write "symmetric/braided strict monoidal category", i.e. the braiding is not strict. $\endgroup$
    – varkor
    Commented Feb 27 at 19:26

0

Reset to default

You must log in to answer this question.