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Nuclei of categories with tensor products. (English) Zbl 1132.18004

The well-know categorification of the centre of an associative algebra leads the author to consider the construction of nucleus for non-associative algebras (following the analogy between algebras (monoids) and monoidal categories). In both cases the categorifications are categories of pairs. The first component of a pair is an object of the original category when the second should be seen as a specialisation of the appropriate constraint (braiding in the case of the monoidal centre, associativity in the case of nucleus). Coherence is then used to define tensor product of pairs and to check the desired properties of resulting monoidal categories. This could give an impression that coherence laws of monoidal categories guarantee that categorifications of algebraic constructions are nicely coherent themselves. Unfortunately it is not always true, and the paper emphasizes this.
Nuclei of categories of modules are considered as an example.

MSC:

18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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