## Enrichment Through Variation

**R. Gordon and A. J. Power**
*Abstract:* We show that, for a closed bicategory *W*,
the 2-category of tensored *W*-categories and all
*W*-functors between them is equivalent to the 2-category of
closed *W*- representations and maps of such, which in turn is
isomorphic to a full sub-2-category of Lax(*W*, Cat). We
further show that, if *w* is a locally dense subbicategory of
*W* and *W* is biclosed, then the 2-category of
*W*-categories having tensors with 1-cells of *w* embeds
fully into the 2-category of *w*-representations. This allows
us to generalize Gabriel-Ulmer duality to *W*-categories and
to prove, for *W*-categories, that for locally finitely
presentable *A* and for *B* admitting finite tensors and
filtered colimits, the category of *W*-functors from *Af*
to *B* is equivalent to that of finitary *W*-functors
from *A* to *B*.

*LFCS report ECS-LFCS-93-254*

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