Fwd: Terminology problem with monoidal adjunctions

Fwd: Terminology problem with monoidal adjunctions

[Tony Meman]

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---------- Forwarded message ----------
From: Tony Meman <tonymeman1@googlemail.com>
Date: 2009/10/3
Subject: Terminology problem with monoidal adjunctions
To: categories <categories@mta.ca>


Dear mathematicians,

I have a terminology problem concerning monoidal adjunctions and would
therefore like to ask some experts.

Let V and W be two symmetric monoidal categories and
L: V <--> W :R
an adjunction of (lax) symmetric monoidal functors, .i.e. the unit and the
counit are monoidal natural transformations. In a previous post, it was
pointed out to me that L have to be automatically a strong symmetric
monoidal functor then (I do not remember if the monoidal structures have to
be closed for this implication). I have read the term 'strong symmetric
monoidal adjunction' and it seems to me that this is just a monoidal
adjunction with a strong monoidal L. Why is this definition not redundant? I
have also read about a 'strict symmetric mnonoidal adjunction'. This
confuses me totally, since I have the impression that there is sometimes
inconsistency in the use of the term 'strict' and 'strong'.

Thank you for any help.
Tony

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