5
$\begingroup$

I'm curious about how Yoneda structures on a 2-category $\mathcal{K}$ play with various 2-categorical constructions. For example, if I have a 2-(co)monad on $\mathcal{K}$, are there any conditions guaranteeing the existence of a Yoneda structure on the 2-category of (co)algebras?

$\endgroup$
4
  • 1
    $\begingroup$ I have the impression that for a lex-comonad T on Cat, coAlg(T) has a Yoneda structure. The same for a lex-reflective subcategory of a 2-topos. $\endgroup$ May 19, 2020 at 20:02
  • $\begingroup$ I can believe that, thanks! I actually found Charles Walker's paper and it had the sort of results I was looking for. $\endgroup$ May 19, 2020 at 21:16
  • 1
    $\begingroup$ Proposition 4.4. in Weber's paper citeseerx.ist.psu.edu/viewdoc/… says that you can find a discrete opfibration classifier for each fully faithful subobject of a given one. The entire paper describes a procedure to generate a YS on $\cal K$ out of a dopfib classifier $\Omega$. $\endgroup$
    – fosco
    May 19, 2020 at 21:28
  • $\begingroup$ Day convolution lifts the yoneda embedding for a monoidal category to a yoneda embedding in the 2-category of monoidal categories, and this formalises to liftings of yoneda embeddings of (co)lax algebras for well-behaved 2-monads T. Section 8 of link describes the case in which T extends to a double monad on an (augmented virtual) ``proarrow equipment''. For such T lifts of yoneda embeddings for colax T-algebras exist whenever T preserves composition of proarrows and both its unit and associativity cells satisfy certain Beck-Chevalley conditions. $\endgroup$ May 20, 2020 at 20:46

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.