### Abstract

The “linear dual” of a cocomplete linear category C is the category of all cocontinuous linear functors C → Vect. We study the questions of when a cocomplete linear category is re exive (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 re exive, 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.

Original language | English (US) |
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Pages (from-to) | 808-835 |

Number of pages | 28 |

Journal | Theory and Applications of Categories |

Volume | 30 |

State | Published - Jun 22 2015 |

### Keywords

- Cocomplete
- Cocontinuous
- Dualizable
- Locally presentable

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

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## Cite this

*Theory and Applications of Categories*,

*30*, 808-835.