universal actions of pseudo-monoids in 'biclosed' monoidal 2-categories

universal actions of pseudo-monoids in 'biclosed' monoidal 2-categories

[mgroth]

Dear category theorists,

   I have a question in the context of 'biclosed' monoidal 2-categories. To
give an analogy, let us begin with one dimension less, i.e., with a closed
monoidal category C. Given an arbitrary object X \in C it is then easy to
show that the internal hom-object END(X)=HOM(X,X) is canonically a monoid
and, in fact, the terminal example of a monoid acting on $X$. One way to
make the latter result precise is as follows. The category Mod(C) of
modules in C, where objects are pairs (M,Y) consisting of a monoid M and a
left M-module Y is endowed with a forgetful functor U:Mod(C)-->C. This
functor U can also be obtained by first observing that the monoidal unit S
can be endowed with the structure of a monoid making it the initial monoid
and such that the category S-Mod is isomorphic to C. U is then induced by
restriction of scalars along the unique monoid morphisms S-->M and the
isomorphism S-Mod\cong C. The statement that the canonical action of
END(X) on X is the terminal example of a monoid acting on X is now
precisely the statement that it gives us a terminal object of U^{-1}(X).

   I would now love to have corresponding results in the following
2-categorical situation where there are two 'degenerations': the
closedness of the monoidal structure is not expressed by a 2-adjunction
but only by a 'biadjunction' and we consider pseudo-monoids instead of
monoids. Thus, let us consider a symmetric monoidal 2-category C which is
'biclosed' in the sense that for every X there is a right biadjoint
2-functor to -\otimes X, i.e., we have an internal hom 2-functor HOM(X,-)
and natural equivalences of categories

Hom(W\otimes X,Y)-->Hom(W,HOM(X,Y))

(where Hom is the enriched hom of C). I would now love to have the
following results:

i) For an arbitrary object X \in C, the internal hom END(X)=HOM(X,X) can
be canonically endowed with the structure of a pseudo-monoid.

ii) There is a canonical action of the pseudo-monoid END(X) on X induced
by the 'biadjunction counit'.

iii) This action is 'the' 'biterminal' example of such an action: using a
2-categorical version of the Grothendieck construction one can form the
2-category PsMod(C) of pseudo-modules in C which is again endowed with a
projection functor U:PsMod(C)-->C. Given an object X \in C the canonical
action of ii) is then a bi-terminal object of U^{-1}(X) in the sense that
all hom-categories of morphisms into that object are equivalent to the
terminal category.

It would be of great help if someone could give me a reference to the
literature where such issues are discussed. Just in case that someone of
you has too much time I would also like to get something close to

iv) The monoidal S unit is 'the' initial example a of pseudo-monoid and we
have an equivalence of categories S-PsMod \simeq C

but i)-iii) are more important to me. Thanks a lot in advance!

Best, Moritz Groth.


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