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Reflexivity and dualizability in categorified linear algebra

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Martin Brandenburg, Alexandru Chirvasitu, and Theo Johnson-Freyd

The "linear dual" of a cocomplete linear category $C$ is the category of
all cocontinuous linear functors $C \to Vect$. We study the questions of
when a cocomplete linear category is reflexive (equivalent to its double dual)
or dualizable (the pairing with its dual comes with a corresponding copairing).
Our main results are that the category of comodules for a countable-dimensional
coassociative coalgebra is always reflexive, but (without any dimension
hypothesis) dualizable if and only if it has enough projectives, which rarely
happens. Along the way, we prove that the category $QCoh(X)$ of
quasi-coherent sheaves on a stack $X$ is not dualizable if $X$ is the
classifying stack of a semisimple algebraic group in positive characteristic or
if $X$ is a scheme containing a closed projective subscheme of positive
dimension, but is dualizable if $X$ is the quotient of an affine scheme by a
virtually linearly reductive group. Finally we prove tensoriality (a type of
Tannakian duality) for affine ind-schemes with countable indexing poset.

Keywords:
locally presentable, dualizable, cocomplete, cocontinuous

2010 MSC:
18A30, 18A35, 18A40, 14A15, 14R20

*Theory and Applications of Categories,*
Vol. 30, 2015,
No. 23, pp 808-835.

Published 2015-06-22.

http://www.tac.mta.ca/tac/volumes/30/23/30-23.pdf

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