We define a commutative monoid structure on the poset of left-exact localizations of a higher topos, that we call the acyclic product. Our approach is anchored in a structural analogy between the poset of left-exact localizations of a topos and the poset of ideals of a commutative ring. The acyclic product is analogous to the product of ideals. The sequence of powers of a given left-exact localization defines a tower of localizations. We show how this recovers the towers of Goodwillie calculus in the unstable homotopical setting. We use this to describe the topoi of n-excisive functors as classifying n-nilpotent objects.

We put Goodwillie's calculus of functors and Weiss' orthogonal calculus in a unified framework. We do so in two ways. On the one hand, the relevant categories are all symmetric monoidal and controlled by their compact objects. We introduce the notion of tidy map as a means to generate symmetric monoidal localizations in this setting. These localizations are always left exact. Then we show that both the Goodwillie and Weiss towers are generated by such maps. On the other hand, the relevant categories are also topoi, for which there is a general theory of completion towers of left exact localizations. We had shown in a previous work that the Goodwillie tower is an instance a such a tower. We show here that the Weiss tower is a completion tower as well, and therefore that the general theory applies to orthogonal calculus.

We revisit the work of Toen-Vezzosi and Lurie on Grothendieck topologies, using the new tools of acyclic classes and congruences. We introduce a notion of extended Grothendieck topology on any 8-topos, and prove that the poset of extended Grothendieck topologies is isomorphic to that of topological localizations, hypercomplete localizations, Lawvere-Tierney topologies, and covering topologies (a variation on the notion of pretopology). It follows that these posets are small and have the structure of a frame. We revisit also the topological-cotopological factorization by introducing the notion of a cotopological morphism. And we revisit the notions of hypercompletion, hyperdescent, hypercoverings and hypersheaves associated to an extended Grothendieck topology. We also introduce the notion of forcing, which is a tool to compute with localizations of 8-topoi. We use this in particular to show that the topological part of a left-exact localization of an 8-topos is universally forcing the generators of this localization to be 8-connected instead of inverting them.

We are developing tools for working with arbitrary left-exact localizations of ∞-topoi. We introduce a notion of higher sheaf with respect to an arbitrary set of maps Σ in an ∞-topos E. We show that the full subcategory of higher sheaves Sh(E,Σ) is an ∞-topos, and that the sheaf reflection E→Sh(E,Σ) is the left-exact localization generated by Σ. The proof depends on the notion of congruence, which is a substitute for the notion of Grothendieck topology in 1-topos theory.

We propose a definition of higher sheaf with respect to an arbitrary set of maps $\Sigma$ in an $\infty$-topos $\mathcal{E}$. We then show that the associated reflection $\mathcal{E} \to {\rm Sh}(\mathcal{E},\Sigma)$ is left-exact so that the subcategory of sheaves with respect to $\Sigma$ is itself an $\infty$-topos. Furthermore, we show that the reflection $\mathcal{E} \to {\rm Sh}(\mathcal{E},\Sigma)$ may be characterized as the left-exact localization generated by $\Sigma$. In the course of the proof, we study the interaction of various types of factorization systems, and make essential use of the notion of a \emph{modality}, that is, a factorization system whose left class is stable by base change.

We prove a generalization of the classical connectivity theorem of Blakers-Massey, valid in an arbitrary higher topos and with respect to an arbitrary modality, that is, a factorization system (L,R) in which the left class is stable by base change. We explain how to rederive the classical result, as well as the recent generalization of Chacholski, Scherer and Werndli (Ann. Inst. Fourier 66 (2016) 2641-2665). Our proof is inspired by the one given in homotopy-type theory in Favonia et al. (2016).

We develop an approach to Goodwillie's calculus of functors using the techniques of higher topos theory. Central to our method is the introduction of the notion of fiberwise orthogonality, a strengthening of ordinary orthogonality which allows us to give a number of useful characterizations of the class of n-excisive maps. We use these results to show that the pushout product of a P_n-equivalence with a P_m-equivalence is a P_m+n+1-equivalence. Then, building on our previous work, we prove a Blakers-Massey type theorem for the Goodwillie tower. We show how to use the resulting techniques to rederive some foundational theorems in the subject, such as delooping of homogeneous functors.

To every homotopy n-nilpotent group, defined in earlier work by Dwyer and the author, we associate an endofunctor of pointed spaces and prove that it is looped and n-excisive. As a tool we prove that Ω P_n( id) commutes with sifted colimits of connected spaces.

Unstable coalgebras over the Steenrod algebra form a natural target category for singular homology with prime field coefficients. The realization problem asks whether an unstable coalgebra is isomorphic to the homology of a topological space. We study the moduli space of such realizations and give a description of this in terms of cohomological invariants of the unstable coalgebra. This is accomplished by a thorough comparative study of the homotopy theories of cosimplicial unstable coalgebras and of cosimplicial spaces.

The homotopy theory. In this paper, we develop the homotopy theory of small functors from spectra to spectra, and study its interplay with Spanier-Whitehead duality and enriched representability in the dual category of spectra.We note that the Spanier-Whitehead duality functor D: Sp -> Sp(op) factors through the category of small functors from spectra to spectra, and construct a new model structure on the category of small functors, which is Quillen equivalent to Sp(op). In this new framework for the Spanier-Whitehead duality, Sp and Sp(op) are full subcategories of the category of small functors and dualization becomes just a fibrant replacement in our new model structure.

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the $n$-homogeneous model structure, the $n$-th derivative is a Quillen functor to the category of spectra with $\Sigma_n$-action. After taking into account only finitary functors -- which may be done in two different ways -- the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T. G. Goodwillie.

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define homotopy n-nilpotent groups as homotopy algebras over certain simplicial algebraic theories. This notion interpolates between infinite loop spaces and loop spaces, but backwards. We study the relation to ordinary nilpotent groups. We prove that n-excisive functors of the form Omega F factor over the category of homotopy n-nilpotent groups.

An n-truncated model structure on simplicial (pre-)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine's intermediate model structures we construct such an n-type model structure via Bousfield-Friedlander localization and exhibit useful generating sets of trivial cofibrations. Injectively fibrant objects in these categories are called n-hyperstacks. The whole setup can consequently be viewed as a description of the homotopy theory of higher hyperstacks. More importantly, we construct analogous n-truncations on simplicial groupoids and prove a Quillen equivalence between these settings. We achieve a classification of n-types of simplicial presheaves in terms of (n-1)-types of presheaves of simplicial groupoids. Our classification holds for general n. Therefore this can also be viewed as the homotopy theory of (pre-)sheaves of (weak) higher groupoids.

Using the dual of Bousfield–Friedlander localization, we colocalize resolution model structures on cosimplicial objects over a left proper model category to get truncated resolution model structures. These are useful for studying realization and moduli problems in algebraic topology.

We generalize and greatly simplify the approach of Lydakis and Dundas-R\ondigs-{\O}stv{\ae}r to construct an L-stable model structure for small functors from a closed symmetric monoidal model category V to a V-model category M, where L is a small cofibrant object of V. For the special case V=M=S_* pointed simplicial sets and L=S^1 this is the classical case of linear functors and has been described as the first stage of the Goodwillie tower of a homotopy functor. We show, that our various model structures are compatible with a closed symmetric monoidal product on small functors. We compare them with other L-stabilizations described by Hovey, Jardine and others. This gives a particularly easy construction of the classical and the motivic stable homotopy category with the correct smash product. We establish the monoid axiom under certain conditions.

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions, we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by using truncated versions of resolution model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the Adams–Atiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory. As an application, we establish an isomorphism between certain E(n)-local Picard groups and some Ext-groups.

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by using truncated versions of resolution model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the Adams-Atiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory. As application we establish an isomorphism between certain E(n)-local Picard groups and some Ext-groups.

Generalizing F-nilpotent completion for a ring spectrum F we first define the notion of completion with respect to a thick subcategory in a monogenic stable homotopy category. Specializing this to the thick subcategory generated by F-injectives gives an injective completion functor. This is the completion functor adapted to the modified Adams spectral sequence, which uses absolute instead of relative injective resolutions. Finally we show, that both constructions coincide for suitable ring spectra.

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTEstimation of medium effect on dissociation constant of ammonium ion and formation constants of silver(I)-ammine complexes in aqueous solutionMasunobu Maeda, Genkichi Nakagawa, and Georg BiedermannCite this: J. Phys. Chem. 1983, 87, 1, 121–125Publication Date (Print):January 1, 1983Publication History Published online1 May 2002Published inissue 1 January 1983https://pubs.acs.org/doi/10.1021/j100224a028https://doi.org/10.1021/j100224a028research-articleACS PublicationsRequest reuse permissionsArticle Views178Altmetric-Citations22LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access options Get e-Alerts

Durch EMK‐Mcssungen an Ketten mit Hgkedoxelcktrodcn wird das Standard‐Potential Eu(III)/ELI(II) bei 25°C in 1m LiClO 4 zu 37′) i l mV bestimmt.