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This is a series of two talks about an emerging relation between homological algebra and low-dimensional topology. While 3D quantum topology is based on semisimple representation theory of quantum groups, we expect nonsemisimple representations of Frobenius algebras and derived categories to be in charge of the 4D quantum topology. We will supplement general ideas by exibiting a new invariant of tangles, which takes values in the chain homology category of complexes of bimodules over certain algebras. This invariant has the same relation to the Kauffman bracket of tangles as homology to the Euler characteristic.