Abstract: A complex of stable infinity-categories is a categorification of a chain complex, meaning a sequence of stable infinity-categories together with a differential that squares to the zero functor. We refer to such categorified complexes as categorical complexes. We give a categorification of the totalization construction, which associates a categorical complex with a categorical multi-complex. Special cases include the totalizations of commutative squares or higher cubes of stable infinity categories. This can be used to construct interesting examples of categorical complexes, for instance coming from normal crossing divisors.
The study of categorical complexes can be seen as part of the conjectural/emerging subject of categorified homological algebra. We will also indicate a partial formalisation of this, based on the notion of a lax additive (infinity,2)-category, categorifying the notion of an additive 1-category.
This talk is based on joint work with T. Dyckerhoff and T. Walde, see https://arxiv.org/abs/2301.02606.