Monoidal categories and categorification

In this series of talks, I will give an introduction to some of the ideas of “categorification” which have created a new point of view in representation theory centered around various monoidal categories of a diagrammatic nature. I will likely start by discussing classical examples such as the Temperley-Lieb and HOMFLY-PT skein categories, before focussing on the Kac-Moody 2-category of Khovanov, Lauda and Rouquier. Many of the categories appearing in classical representation theory, especially of symmetric and general linear groups, admit additional structure making them into module categories (“2-representations”) over the Kac-Moody 2-category. This has consequences both at a combinatorial level (related to crystals and labelling sets of irreducible modules) and at a categorical level (related to the construction of Morita and derived equivalences between blocks).

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