#
Relative symmetric monoidal closed categories I: Autoenrichment and
change of base

##
Rory B. B. Lucyshyn-Wright

Symmetric monoidal closed categories may be related to one another not
only by the functors between them but also by enrichment of one in
another, and it was known to G. M. Kelly in the 1960s that there is a very
close connection between these phenomena. In this first part of a
two-part series on this subject, we show that the assignment to each
symmetric monoidal closed category $V$ its associated $V$-enriched
category $underline{V}$ extends to a 2-functor valued in an op-2-fibred
2-category of symmetric monoidal closed categories enriched over various
bases. For a fixed $V$, we show that this induces a 2-functorial passage
from symmetric monoidal closed categories *over* $V$ (i.e., equipped
with a morphism to $V$) to symmetric monoidal closed $V$-categories over
$underline{V}$. As a consequence, we find that the enriched adjunction
determined a symmetric monoidal closed adjunction can be obtained by
applying a 2-functor and, consequently, is an adjunction in the 2-category
of symmetric monoidal closed $V$-categories.

Keywords:
monoidal category; closed category; enriched category; enriched
monoidal category; monoidal functor; monoidal adjunction; 2-category;
2-functor; 2-fibration; pseudomonoid

2010 MSC:
18D15, 18D10, 18D20, 18D25, 18A40, 18D05, 18D30

*Theory and Applications of Categories,*
Vol. 31, 2016,
No. 6, pp 138-174.

Published 2016-01-31.

http://www.tac.mta.ca/tac/volumes/31/6/31-06.pdf

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