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Hochschild homology, lax codescent, and duplicial structure. (English) Zbl 1387.18027

In this brilliant paper, duplicial objects in the sense of [W. G. Dwyer and D. M. Kan, Comment. Math. Helv. 60, 582–600 (1985; Zbl 0593.18010)] are studied in a 2-categorical framework. The main achievements can be grouped as follows.
First, a construction of duplicial objects in [G. Böhm and D. Ştefan, Commun. Math. Phys. 282, No. 1, 239–286 (2008; Zbl 1153.18004)] is interpreted as the action of a cleverly designed functor built on a 2-categorical generalization of Hochschild homology. This gives a moral analogue of (and relies on) the construction of simplicial objects via bar resolution with respect to a comonad. As a byproduct, duplicial structures on simplicial objects are described as compatible natural transformations between the right and left decalage comonads.
Second, it is analyzed what additional structure is needed on a bicategory (so by instance on a monoidal category or a mere category) in order for the simplicial nerve to inherit a duplicial structure. For example, paracyclic structures (i.e. duplicial structures with invertible structure morphisms) on the nerve of a monoidal category \(\mathsf M\) are proven to correspond to \(\ast\)-autonomous structures; and cyclic structures (i.e. paracyclic structures where the structure morphism has order \(n+1\) at each grade \(n\)) to \(\ast\)-autonomous structures with cyclic dualizing object.

MSC:

18G30 Simplicial sets; simplicial objects in a category (MSC2010)
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
16T05 Hopf algebras and their applications
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