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Abstract Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $$\mathsf V$$, as well as for small $$\mathsf V$$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and $$C_n$$-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic $$K$$-theory to twisted topological Hochschild homology.
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This content will become publicly available on April 1, 2026
Symmetries of the cyclic nerve
We undertake a systematic study of the Hochschild homology, i.e. (the geometric realization of) the cyclic nerve, of -categories (and more generally of category-objects in an ∞-category), as a version of factorization homology. In order to do this, we codify -categories in terms of quiver representations in them. By examining a universal instance of such Hochschild homology, we explicitly identify its natural symmetries, and construct a non-stable version of the cyclotomic trace map. Along the way we give a unified account of the cyclic, paracyclic, and epicyclic categories. We also prove that this gives a combinatorial description of the case of factorization homology as presented in [4], which parametrizes -categories by solidly 1-framed stratified spaces.
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- Award ID(s):
- 1945639
- PAR ID:
- 10627576
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Advances in Mathematics
- Volume:
- 466
- Issue:
- C
- ISSN:
- 0001-8708
- Page Range / eLocation ID:
- 110170
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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