Relative symmetric monoidal closed categories I: Autoenrichment and change of base
Abstract
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 twopart 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 2functor valued in an op2fibred 2category of symmetric monoidal closed categories enriched over various bases. For a fixed $V$, we show that this induces a 2functorial passage from symmetric monoidal closed categories $\textit{over}$ $V$ (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 2functor and, consequently, is an adjunction in the 2category of symmetric monoidal closed $V$categories.
 Publication:

arXiv eprints
 Pub Date:
 July 2015
 arXiv:
 arXiv:1507.02220
 Bibcode:
 2015arXiv150702220L
 Keywords:

 Mathematics  Category Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  KTheory and Homology;
 18D15;
 18D10;
 18D20;
 18D25;
 18A40;
 18D05;
 18D30
 EPrint:
 Theory and Applications of Categories, Vol. 31, 2016, No. 6, pp 138174