From: Gaunce Lewis <lglewis@syr.edu>
To: categories@mta.ca
Subject: question about enriched category theory
Date: Fri, 11 May 2001 13:13:44 -0400 [thread overview]
Message-ID: <5.0.2.1.2.20010511124058.00aca8e0@mailbox.syr.edu> (raw)
Assume that V is a symmetric monoidal closed category and that A and B are
categories enriched over V which have tensors. Let me denote the tensor of
an object v of V and an object a of A by
v \ten a
Assume also that F is a functor from the underlying ordinary category A_0
of A to the underlying category B_0 of B. If F were enriched over V, then
there would be a natural map
f : v \ten Fa --> F(v \ten a)
describing the behavior of F on tensors.
However, assume only that F is a functor on the underlying categories. It
seems to me that, if there is a well-behaved natural map f of the above
form for all v in V and a in A, then F ought to be enriched over V. It is
easy to construct from f the map that ought to be the enrichment for
F. The trick is to decide what properties f must have in order to ensure
that the putative enrichment really works. Is this written up somewhere?
Along the same lines, suppose now that F and G are enriched functors from A
to B with the associated maps
f : v \ten Fa --> F(v \ten a)
and
g : v \ten Ga --> G(v \ten a)
describing their behavior on tensors. Assume also that t is an ordinary
natural tranformation between the ordinary functors F_0 and G_0 underlying
F and G. There is an obvious diagram relating t, f, and g, and it seems
that this diagram ought to commute if t is an enriched natural
transformation. In fact, it seems that the commutativity of this diagram
ought to be equivalent to t being enriched over V. Is this written down
anywhere?
Thanks for any help on this,
Gaunce
reply other threads:[~2001-05-11 17:13 UTC|newest]
Thread overview: [no followups] expand[flat|nested] mbox.gz Atom feed
Reply instructions:
You may reply publicly to this message via plain-text email
using any one of the following methods:
* Save the following mbox file, import it into your mail client,
and reply-to-all from there: mbox
Avoid top-posting and favor interleaved quoting:
https://en.wikipedia.org/wiki/Posting_style#Interleaved_style
* Reply using the --to, --cc, and --in-reply-to
switches of git-send-email(1):
git send-email \
--in-reply-to=5.0.2.1.2.20010511124058.00aca8e0@mailbox.syr.edu \
--to=lglewis@syr.edu \
--cc=categories@mta.ca \
/path/to/YOUR_REPLY
https://kernel.org/pub/software/scm/git/docs/git-send-email.html
* If your mail client supports setting the In-Reply-To header
via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line
before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).