## The group completion theorem via localizations of ring spectra

__Abstract:__In this short expository note we give a proof of the group completion theorem (see [MS76]) using localizations of ring spectra. We also relate the group completion of a topological monoid to the plus construction. Most of the results are well-known but we hope that our treatment can shed some new light on some aspects.## The Balmer spectrum of the equivariant homotopy category of a finite abelian group (with T Barthel, M. Hausmann, N. Naumann, J. Noel and N. Stapleton)

__Abstract:__For a finite abelian group A, we determine the Balmer spectrum of the category of compact objects in genuine A-spectra. This generalizes the case A = Z/p due to Balmer and Sanders, by establishing (a corrected version of) their log_p-conjecture for abelian groups. We work out the consequences for the chromatic type of fixed-points. We also establish a generalization of Kuhn's blue-shift theorem for Tate-constructions.## On topological cyclic homology (with P.Scholze )

Preprint arXiv:1707.01799, 2017.

__Abstract:__Topological cyclic homology is a refinement of Connes' cyclic homology which was introduced by B\"okstedt--Hsiang--Madsen in 1993 as an approximation to algebraic K-theory. There is a trace map from algebraic $K$-theory to topological cyclic homology, and a theorem of Dundas--Goodwillie--McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing K-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and the elaborate notion of a cyclotomic spectrum. The goal of this paper is to revisit this theory using only homotopy-invariant notions. In particular, we give a new construction of topological cyclic homology. This is based on a new definition of the infinity-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum $X$ with $S^1$-action (in the most naive sense) together with $S^1$-equivariant maps $\varphi_p: X\to X^{tC_p}$ for all primes $p$. Here $X^{tC_p}=\cofib(\Nm: X_{hC_p}\to X^{hC_p})$ is the Tate construction. On bounded below spectra, we prove that this agrees with previous definitions. As a consequence, we obtain a new and simple formula for topological cyclic homology.## Localization of Cofibration Categories and Groupoid C^*-algebra (with M.Land and K. Szumilo)

Preprint arXiv:1609.03805, 2016.

__Abstract:__We prove that relative functors out of a cofibration category are essentially the same as relative functors which are only defined on the subcategory of cofibrations. As an application we give a new construction of the functor that assigns to a groupoid its groupoid C^*-algebra and thereby its topological K-theory spectrum.## On the Relation between K- and L-Theory of C^*-algebras (with M.Land )

Preprint arXiv:1608.02903, 2016.

__Abstract:__We prove the existence of a map of spectra $\tau_A: kA \to \ell A$ 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 $KA\adj \xrightarrow{\simeq} LA\adj$ 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.## Stable infinity Operads and the multiplicative Yoneda lemma

Preprint arXiv:1608.02901, 2016.

__Abstract:__We construct for every infinity-operad O with certain finite limits new infinity-operads Sp(O) of spectrum objects and CGrp(O) of commutative group objects in O. We show that these are the universal stable resp. additive infinity-operads obtained from O. We deduce that for a stably (resp. additively) symmetric monoidal infinity-category C the Yoneda embedding factors through the infinity-category of exact, contravariant functors from C to the infinity-category of spectra (resp. connective spectra) and admits a certain multiplicative refinement. As an application we prove that the identity functor Sp to Sp is initial among exact, lax symmetric monoidal endofunctors of the symmetric monoidal infinity-category Sp of spectra with smash product.## The Beilinson regulator is a map of ring spectra (with U. Bunke and G. Tamme)

Preprint arXiv:1509.05667, 2015.

__Abstract:__We prove that the Beilinson regulator, which is a map from K-theory to absolute Hodge cohomology of a smooth variety, admits a refinement to a map of E-infinity-ring spectra in the sense of algebraic topology. To this end we exhibit absolute Hodge cohomology as the cohomology of a commutative differential graded algebra over R. The associated spectrum to this CDGA is the target of the refinement of the regulator and the usual K-theory spectrum is the source. To prove this result we compute the space of maps from the motivic K-theory spectrum to the motivic spectrum that represents absolute Hodge cohomology using the motivic Snaith theorem. We identify those maps which admit an E-infinity-refinement and prove a uniqueness result for these refinements.## Homology of dendroidal sets (with M. Basic)

Preprint arXiv:1509.00702, 2015.

__Abstract:__We define for every dendroidal set X a chain complex and show that this assignment determines a left Quillen functor. Then we define the homology groups Hn(X) as the homology groups of this chain complex. This generalizes the homology of simplicial sets. Our main result is that the homology of X is isomorphic to the homology of the associated spectrum K(X) as discussed in earlier work of the authors. Since these homology groups are sometimes computable we can identify some spectra K(X) which we could not identify before.## Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories (with. S. Sagave)

Preprint arXiv:1506.01475. 2015.

__Abstract:__We prove the theorem stated in the title. More precisely, we show the stronger statement that every symmetric monoidal left adjoint functor between presentably symmetric monoidal infinity-categories is represented by a strong symmetric monoidal left Quillen functor between simplicial, combinatorial and left proper symmetric monoidal model categories.## Lax colimits and free fibrations in infinity-categories (w. D. Gepner and R. Haugseng)

Preprint arXiv:1501.02161, 2015.

__Abstract:__We define and discuss lax and weighted colimits of diagrams in infinity-categories and show that the coCartesian fibration associated to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free Cartesian fibration associated to a a functor of infinity-categories. As an application of these results, we prove that lax representable functors are preserved under exponentiation, and also that the total space of a presentable Cartesian fibration between infinity-categories is presentable, generalizing a theorem of Makkai and Pare to the infinity-categorical setting. Lastly, in the appendix, we observe that pseudofunctors between (2,1)-categories give rise to functors between infinity-categories via the Duskin nerve.## Twisted differential cohomology (with U. Bunke)

Preprint arXiv:1406.3231, 2014.

__Abstract:__The main goal of the present paper is the construction of twisted generalized differential cohomology theories and the comprehensive statement of its basic functorial properties. Technically it combines the homotopy theoretic approach to (untwisted) generalized differential cohomology developed by Hopkins-Singer and later by the first author and D. Gepner with the oo-categorical treatement of twisted cohomology by Ando-Blumberg-Gepner. We introduce the notion of a differential twist for a given generalized cohomology theory and construct twisted differential cohomology groups (resp. spectra). The main technical results of the paper are existence and uniqueness statements for differential twists. These results will be applied in a variety of examples, including K-theory, topological modular forms and other cohomology theories.## Universality of Multiplicative Infinite Loop Space Machines (with D. Gepner and M. Groth)

In: Algebr. Geom. Topol. (to appear) (2015).

__Abstract:__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.## T-duality via gerby geometry and reductions (with U. Bunke)

In: Rev. Math. Phys. 27.5 (2015), pp. 1550013, 46.

__Abstract:__We consider topological T-duality of torus bundles equipped with S^{1}-gerbes. We show how a geometry on the gerbe determines a reduction of its band to the subsheaf of S^{1}-valued functions which are constant along the torus fibres. We observe that such a reduction is exactly the additional datum needed for the construction of a T-dual pair. We illustrate the theory by working out the example of the canonical lifting gerbe on a compact Lie group which is a torus bundles over the associated flag manifold. It was a recent observation of Daenzer and van Erp (arXiv1211.0763) that for certain compact Lie groups and a particular choice of the gerbe, the T-dual torus bundle is given by the Langlands dual group.## Principal oo-bundles: Presentations (with U. Schreiber and D. Stevenson)

In: J. Homotopy Relat. Struct. (2014), pp. 1 - 58.

__Abstract:__We discuss two aspects of the presentation of the theory of principal infinity-bundles in an infinity-topos, introduced in [NSSa], in terms of categories of simplicial (pre)sheaves. First we show that over a cohesive site C and for G a presheaf of simplicial groups which is C-acyclic, G-principal infinity-bundles over any object in the infinity-topos over C are classified by hyper-Cech-cohomology with coefficients in G. Then we show that over a site C with enough points, principal infinity-bundles in the infinity-topos are presented by ordinary simplicial bundles in the sheaf topos that satisfy principality by stalkwise weak equivalences. Finally we discuss explicit details of these presentations for the discrete site (in discrete infinity-groupoids) and the smooth site (in smooth infinity-groupoids, generalizing Lie groupoids and differentiable stacks). In the companion article [NSSc] we use these presentations for constructing classes of examples of (twisted) principal infinity-bundles and for the discussion of various applications.## Principal oo-bundles: General theory (with U. Schreiber and D. Stevenson)

In: J. Homotopy Relat. Struct. (2014), pp. 1 - 53.

__Abstract:__The theory of principal bundles makes sense in any infinity-topos, such as that of topological, of smooth, or of otherwise geometric infinity-groupoids/infinity-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 infinity-group G these G-principal infinity-bundles reproduce the theories of ordinary principal bundles, of bundle gerbes/principal 2-bundles and of bundle 2-gerbes and generalize these to their further higher and equivariant analogs. The induced associated infinity-bundles subsume the notions of gerbes and higher gerbes in the literature. We discuss here this general theory of principal infinity-bundles, intimately related to the axioms of Giraud, Toen-Vezzosi, Rezk and Lurie that characterize infinity-toposes. We show a natural equivalence between principal infinity-bundles and intrinsic nonabelian cocycles, implying the classification of principal infinity-bundles by nonabelian sheaf hyper-cohomology. We observe that the theory of geometric fiber infinity-bundles associated to principal infinity-bundles subsumes a theory of infinity-gerbes and of twisted infinity-bundles, with twists deriving from local coefficient infinity-bundles, which we define, relate to extensions of principal infinity-bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice infinity-topos. In a companion article [NSSb] we discuss explicit presentations of this theory in categories of simplicial (pre)sheaves by hyper-Cech cohomology and by simplicial weakly-principal bundles; and in [NSSc] we discuss various examples and applications of the theory.## Algebraic K-Theory of infinity-Operads

In: J. K-Theory 14.3 (2014), pp. 614 - 641

__Abstract:__The theory of dendroidal sets has been developed to serve as a combinatorial model for homotopy coherent operads by Moerdijk and Weiss. An infinity-operad is a dendroidal set D satisfying certain lifting conditions. In this paper we give a definition of K-groups K_n(D) for a dendroidal set D. These groups generalize the K-theory of symmetric monoidal (resp. permutative) categories and algebraic K-theory of rings. We establish some useful properties like invariance under the appropriate equivalences and long exact sequences which allow us to compute these groups in some examples. We show that the K-theory groups of D can be realized as homotopy groups of a K-theory spectrum K(D).## Differential cohomology theories as sheaves of spectra (with U. Bunke and M. Völkl)

In: J. Homotopy Relat. Struct. (to appear) (2014)

__Abstract:__We show that every sheaf on the site of smooth manifolds with values in a stable (infinity,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.## Dendroidal sets as models for connective spectra (with M. Basic)

In:J. K-Theory 14.3 (2014), pp. 387 - 421.

__Abstract:__Dendroidal sets have been introduced as a combinatorial model for homotopy coherent operads. We introduce the notion of fully Kan dendroidal sets and show that there is a model structure on the category of dendroidal sets with fibrant objects given by fully Kan dendroidal sets. Moreover we show that the resulting homotopy theory is equivalent to the homotopy theory of connective spectra.## Lifting Problems and Transgression for Non-Abelian Gerbes (with K. Waldorf)

In: Adv. Math. 242 (2013), pp. 50 - 79.

__Abstract:__We discuss various lifting and reduction problems for bundles and gerbes in the context of a strict Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen's long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in non-abelian cohomology. This transgression map relates the degree one non-abelian cohomology of a smooth manifold (represented by non-abelian gerbes) with the degree zero non-abelian cohomology of the free loop space (represented by principal bundles). We prove several properties for this transgression map. For instance, it reduces - in case of a Lie 2-group with a single object - to the ordinary transgression in ordinary cohomology. We describe applications of our results to string manifolds: first, we obtain a new comparison theorem for different notions of string structures. Second, our transgression map establishes a direct relation between string structures and spin structure on the loop space.## Four Equivalent Versions of Non-Abelian Gerbes (with K. Waldorf)

In: Pacific J. Math. 264.2 (2013), pp. 355 - 419.

__Abstract:__We recall and partially improve four versions of smooth, non-abelian gerbes: Cech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We prove that all these four versions are equivalent, and so establish new relations between interesting recent developments. Prominent partial results we prove are a bijection between continuous and smooth non-abelian cohomology, and an explicit equivalence between bundle gerbes and principal 2-bundles as 2-stacks.## Bicategories in field theories - an invitation (with C. Schweigert)

In: Strings, gauge fields, and the geometry behind. World Sci. Publ., Hackensack, NJ, 2013, pp. 119 - 132.

__Abstract:__We explain some applications of bicategories in both classical and quantum field theory. This includes a modern perspective on some pioneering work of Max Kreuzer and Bert Schellekens on rational conformal field theory.## A Smooth Model for the String Group (with C. Sachse and C. Wockel)

In: Int. Math. Res. Not. IMRN 16 (2013), pp. 3678 - 3721.

__Abstract:__We construct a model for the string group as an infinite-dimensional Lie group. In a second step we extend this model by a contractible Lie group to a Lie 2-group model. To this end we need to establish some facts on the homotopy theory of Lie 2-groups. Moreover, we provide an explicit comparison of string structures for the two models and a uniqueness result for Lie 2-group models.## Strictification of weakly equivariant Hopf algebras (with J. Maier and C. Schweigert)

In: Bull. Belg. Math. Soc. Simon Stevin 20.2 (2013), pp. 269 - 285.

__Abstract:__A weakly equivariant Hopf algebra is a Hopf algebra A with an action of a finite group G up to inner automorphisms. We show that each weakly equivariant Hopf algebra can be replaced by a Morita equivalent algebra B with a strict action of G and with a coalgebra structure that leads to a tensor equivalent representation category. However, the coproduct of this strictification cannot, in general, be chosen to be unital, so that a strictification of the G-action can only be found on a weak Hopf algebra B.## Equivariant Modular Categories via Dijkgraaf-Witten Theory (with J. Maier and C. Schweigert)

In: Adv. Theor. Math. Phys. 16.1 (2012), pp. 289 - 358.

__Abstract:__Based on a weak action of a finite group J on a finite group G, we present a geometric construction of J-equivariant Dijkgraaf-Witten theory as an extended topological field theory. The construction yields an explicitly accessible class of equivariant modular tensor categories. For the action of a group J on a group G, the category is described as the representation category of a J-ribbon algebra that generalizes the Drinfel'd double of the finite group G.## Equivariance In Higher Geometry (with C. Schweigert)

In: Adv. Math. 226.4 (2011), pp. 3367 - 3408

__Abstract:__We study (pre-)sheaves in bicategories on geometric categories: smooth manifolds, manifolds with a Lie group action and Lie groupoids. We present three main results: we describe equivariant descent, we generalize the plus construction to our setting and show that the plus construction yields a 2-stackification for 2-prestacks. Finally we show that, for a 2-stack, the pullback functor along a Morita-equivalence of Lie groupoids is an equivalence of bicategories. Our results have direct applications to gerbes and 2-vector bundles. For instance, they allow to construct equivariant gerbes from local data and can be used to simplify the description of the local data. We illustrate the usefulness of our results in a systematic discussion of holonomies for unoriented surfaces.## Algebraic models for higher categories

In: Indag. Math. (N.S.) 21.1- 2 (2011), pp. 52 - 75.

__Abstract:__We introduce the notion of algebraic fibrant objects in a general model category and establish a (combinatorial) model category structure on algebraic fibrant objects. Based on this construction we propose algebraic Kan complexes as an algebraic model for oo-groupoids and algebraic quasi-categories as an algebraic model for (oo,1)-categories. We furthermore give an explicit proof of the homotopy hypothesis.## Bundle gerbes and surface holonomy (with J. Fuchs, C. Schweigert, and K. Waldorf)

In: European Congress of Mathematics. Ed. by A. Ran, H. te Riele, and J. Wiegerinck. EMS Publishing House, 2008, pp. 167-197.

__Abstract:__Hermitian bundle gerbes with connection are geometric objects for which a notion of surface holonomy can be defined for closed oriented surfaces. We systematically introduce bundle gerbes by closing the pre-stack of trivial bundle gerbes under descent. Inspired by structures arising in a representation theoretic approach to rational conformal field theories, we introduce geometric structure that is appropriate to define surface holonomy in more general situations: Jandl gerbes for unoriented surfaces, D-branes for surfaces with boundaries, and bi-branes for surfaces with defect lines.