* question about enriched category theory
@ 2001-05-11 17:13 Gaunce Lewis
0 siblings, 0 replies; only message in thread
From: Gaunce Lewis @ 2001-05-11 17:13 UTC (permalink / raw)
To: categories
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
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