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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.
@article{TAC_2015_30_a22,
author = {Martin Brandenburg and Alexandru Chirvasitu and Theo Johnson-Freyd},
title = {Reflexivity and dualizability in categorified linear algebra},
journal = {Theory and applications of categories},
pages = {808--835},
publisher = {mathdoc},
volume = {30},
year = {2015},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2015_30_a22/}
}
TY - JOUR AU - Martin Brandenburg AU - Alexandru Chirvasitu AU - Theo Johnson-Freyd TI - Reflexivity and dualizability in categorified linear algebra JO - Theory and applications of categories PY - 2015 SP - 808 EP - 835 VL - 30 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TAC_2015_30_a22/ LA - en ID - TAC_2015_30_a22 ER -
%0 Journal Article %A Martin Brandenburg %A Alexandru Chirvasitu %A Theo Johnson-Freyd %T Reflexivity and dualizability in categorified linear algebra %J Theory and applications of categories %D 2015 %P 808-835 %V 30 %I mathdoc %U http://geodesic.mathdoc.fr/item/TAC_2015_30_a22/ %G en %F TAC_2015_30_a22
Martin Brandenburg; Alexandru Chirvasitu; Theo Johnson-Freyd. Reflexivity and dualizability in categorified linear algebra. Theory and applications of categories, Tome 30 (2015), pp. 808-835. http://geodesic.mathdoc.fr/item/TAC_2015_30_a22/