204t. nikolaus and p. scholze t. nikolaus and p. scholze the analogy between TC and syntomic cohomology has been known, and pursued for example by Kaledin,[57],[58]; see also at the end of the introduction of [13] for an early suggestion of such a relation. As explained to us by Kaledin, our main theorem is closely related to his results relating cyclotomic complexes and filtered Dieudonné modules.By Theorem 1.4, the information stored in the genuine cyclotomic spectrum (THH(A), (Φ p ) p ) can be characterized explicitly. In order for this to be useful, however, we need to give a direct construction of this information. In other words, for A∈Alg E1 (Sp), we have to define directly a T/C p ∼ =T-equivariant Frobenius mapWe will give two discussions of this, first for associative algebras, and then indicate a much more direct construction for E ∞ -algebras.Let us discuss the associative case for simplicity for p=2. Note that, by definition, the source THH(A) is the realization of the cyclic spectrum

The theory of principal bundles makes sense in any ∞-topos, such as the ∞-topos of topological, of smooth, or of otherwise geometric ∞-groupoids/∞-stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure ∞-group G these G-principal ∞-bundles reproduce various higher structures that have been considered in the literature and further generalize these to a full geometric model for twisted higher nonabelian sheaf cohomology. We discuss here this general abstract theory of principal ∞-bundles, observing that it is intimately related to the axioms that characterize ∞-toposes. A central result is a natural equivalence between principal ∞-bundles and intrinsic nonabelian cocycles, implying the classification of principal ∞-bundles by nonabelian sheaf hyper-cohomology. We observe that the theory of geometric fiber ∞-bundles associated to principal ∞-bundles subsumes a theory of ∞-gerbes and of twisted ∞-bundles, with twists deriving from local coefficient ∞-bundles, which we define, relate to extensions of principal ∞-bundles and show to be classified by Communicated by Antonio Cegarra. T. Nikolaus 123 T. Nikolaus et al. a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice ∞-topos.

We establish a canonical and unique tensor product for commutative monoids and groups in an infinity-category C which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that E_n-(semi)ring objects in C give rise to E_n-ring spectrum objects in C. In the case that C is the infinity-category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic K-theory of rings and ring spectra. The main tool we use to establish these results is the theory of smashing localizations of presentable infinity-categories. In particular, we identify preadditive and additive infinity-categories as the local objects for certain smashing localizations. A central theme is the stability of algebraic structures under basechange; for example, we show Ring(D \otimes C) = Ring(D) \otimes C. Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in infinity-categories.Comment: 27 page

We prove the existence of a map of spectra τ A : kA → ℓA between connective topological K-theory and connective algebraic L-theory of a complex C *-algebra A which is natural in A and compatible with multiplicative structures. We determine its effect on homotopy groups and as a consequence obtain a natural equivalence KA[ 1 2 ] ≃ − → LA[ 1 2 ] of periodic K-and L-theory spectra after inverting 2. We show that this equivalence extends to K-and L-theory of real C *-algebras. Using this we give a comparison between the real Baum-Connes conjecture and the L-theoretic Farrell-Jones conjecture. We conclude that these conjectures are equivalent after inverting 2 if and only if a certain completion conjecture in L-theory is true.

We show that every sheaf on the site of smooth manifolds with values in a stable (∞, 1)-category (like spectra or chain complexes) gives rise to a "differential cohomology diagram" and a homotopy formula, which are common features of all classical examples of differential cohomology theories. These structures are naturally derived from a canonical decomposition of a sheaf into a homotopy invariant part and a piece which has a trivial evaluation on a point. In the classical examples the latter is the contribution of differential forms. This decomposition suggest a natural scheme to analyse new sheaves by determining these pieces and the gluing data. We perform this analysis for a variety of classical and not so classical examples. *