From: Tony Meman <tonymeman1@googlemail.com>
To: categories <categories@mta.ca>
Subject: Question on "On Closed Categories of Functors"
Date: Sun, 1 Feb 2009 20:11:25 +0100 [thread overview]
Message-ID: <E1LTllR-0001i6-Pf@mailserv.mta.ca> (raw)
Dear category theorists,
I have a question concerning the paper "On Closed Categories of Functors"
from Brian Day (By the way, this is an excellent paper).
Let V be a symmetric monoidal closed category and C a small V-category.
The (ordinary) category [C,V] of V-functors admits the sturcure of a
V-category in a canonical way.
A symmetric monoidal V-category is the enriched analogue of a
symmetric-monoidal structure on an ordinary category, i.e. all the structure
morphisms are V-morphisms and the coherence conditions are fullfilled. The
underlaying category of a symmetric monoidal V-category admits the structure
of an ordinary symmetric monoidal category.
Brian Day constructs a symmetric monoidal closed structure ([C,V],@,E) on
the V-category of V-functors [C,V] for some cases [3.3, 3.6], e.g. if
(C,*,e) is a symmetric monoidal V-category [4.1]. The underlaying *category*
[C,V] of V-functors admits a closed symmetric monoidal structure from the
enriched one by taking the underlaying functor of each V-functor, the
underlaying natural transformation of each V-natural transformation.
Because a closed symmetric monoidal category is canonically enriched over
itself, the category [C,V] gets a [C,V] enrichment in this way.
My question is: What does this [C,V]-enrichment of [C,V] have to do with the
V-enrichment of [C,V]?
Suppose C have a terminal object t. One gets a evaluation functor
Ev_t:[C,V]-CAT-->V-CAT. Is this the connection between the two enrichments?
Thank you in advance for any help.
Tony
next reply other threads:[~2009-02-01 19:11 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2009-02-01 19:11 Tony Meman [this message]
2009-02-02 7:53 street
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