Kan spectra, group spectra and twisting structures

Spectra and symmetric spectra in general model categories

The work of Hopkins and Smith [HS] has shown that the stable homotopy category has layered periodic behavior. On the (p-local) sphere, the only non-nilpotent selfmaps are multiplication by a power of p. But if we kill such a power to form the Moore space M(p), then we get a new family of non-nilpotent self maps, called the v1-self maps. Similarly, if we kill one of those, we get v2-self maps, and this behavior continues. One of the great advantages of the Brown-Peterson spectrum BP is that the periodicities are not layered, but they all appear as homotopy classes vn ∈ π2(pn−1)BP . Another great advantage of BP is that it is comparatively simple algebraically. Its coefficient ring is polynomial, and it is possible to calculate in the Adams-Novikov spectral sequence based on the operations in BP -homology. In fact, most of the spectra used by algebraic topologists are complex oriented, in that they admit maps from BP . But there is one crucial example that does not, namely, real K-theory KO. Hopkins and Miller [HMi] have recently shown that KO is the tip of an iceberg of non-complex oriented theories which have interesting torsion. It would be nice to have a bordism spectrum that did admit maps to the HopkinsMiller theories EOn. Recall that MO〈k〉 is the Thom spectrum arising from the k − 1-connected Postnikov cover BO〈k〉 of BO, and similarly for MU〈k〉. Note that MO〈4〉 = MSpin and MU〈4〉 = MSU both admit orientations to KO. I hope that this also is the beginning of a general phenomenon, and that the MO〈k〉 and MU〈k〉 will admit orientations to EOn when k is sufficiently large. Such an orientation may have some analytic meaning. Witten interprets a (conjectural) orientation from MO〈8〉 to elliptic cohomology, which should be EO2, as the index of an S-equivariant Dirac operator on the free loop space of a manifold with MO〈8〉 structure. However, it will be hard to get at the algebraic meaning of such an orientation because we know so little about the MO〈k〉. This paper is an attempt to get some qualititative understanding of the MO〈k〉 and MU〈k〉. We try to find analogues of vn in the homotopy of MO〈k〉 and MU〈k〉. It turns out to be easier to study the existence and properties of such analogues in

In this paper we initiate the study of enriched $\infty$-operads. We introduce several models for these objects, including enriched versions of Barwick's Segal operads and the dendroidal Segal spaces of Cisinski and Moerdijk, and show these are equivalent. Our main results are a version of Rezk's completion theorem for enriched $\infty$-operads: localization at the fully faithful and essentially surjective morphisms is given by the full subcategory of complete objects, and a rectification theorem: the homotopy theory of $\infty$-operads enriched in the $\infty$-category arising from a nice symmetric monoidal model category is equivalent to the homotopy theory of strictly enriched operads.

The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D = sets), the syntactic semirings of Polak (D = semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is a commutative variety of algebras, we prove that the syntactic D-monoid of a language L can be constructed as a quotient of a free D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the transition D-monoid of the minimal automaton for L in D. Furthermore, in the case where the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids.

Monoidal indeterminates and categories of possible worlds

We introduce the notion of <em>strong concatenable process</em> for Petri nets as the least refinement of non-sequential (concatenable) processes which can be expressed abstractly by means of a <em> functor</em> Q[_] from the category of Petri nets to an appropriate category of symmetric strict monoidal categories with free non-commutative monoids of objects, in the precise sense that, for each net N, the strong concatenable processes of N are isomorphic to the arrows of Q[N]. This yields an axiomatization of the causal behaviour of Petri nets in terms of symmetric strict monoidal categories.<br /> <br />In addition, we identify a <em>coreflection</em> right adjoint to Q[_] and we characterize its replete image in the category of symmetric monoidal categories, thus yielding an abstract description of the category of net computations.

We establish a highly flexible condition that guarantees that all colored symmetric operads in a symmetric monoidal model category are admissible, i.e., the category of algebras over any operad admits a model structure transferred from the original model category. We also give a necessary and sufficient criterion that ensures that a given weak equivalence of admissible operads admits rectification, i.e., the corresponding Quillen adjunction between the categories of algebras is a Quillen equivalence. In addition, we show that Quillen equivalences of underlying symmetric monoidal model categories yield Quillen equivalences of model categories of algebras over operads. Applications of these results include enriched categories, colored operads, prefactorization algebras, and commutative symmetric ring spectra.

We show that non-Archimedean analytic geometry can be viewed as relative algebraic geometry in the sense of Toën–Vaquié–Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we define a topology on certain subcategories of the category of (relative) affine schemes. In the case that the monoidal category is the category of abelian groups, the topology reduces to the ordinary Zariski topology. By examining this topology in the case that the monoidal category is the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in rigid or Berkovich analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry. We demonstrate that the category of Berkovich analytic spaces (and also rigid analytic spaces) embeds fully faithfully into the category of (relative) schemes in our version of relative algebraic geometry. We define a notion of quasi-coherent sheaf on analytic spaces which we use to characterize surjectivity of covers. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

Luminous Glass: A Study on the Optics Governing Luminescent Solar Concentrators and Optimization of Luminescent Materials through Combinatorial Gradient Sputter Deposition

We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

K-theory and the bridge from motives to noncommutative motives

One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor L:A→X, a `structured cospan' is a diagram in X of the form L(a)→x←L(b). We give a new proof that if A and X have finite colimits and L preserves them, there is a symmetric monoidal double category whose objects are those of A and whose horizontal 1-cells are structured cospans. Second, given a pseudofunctor F:A→Cat, a `decorated cospan' is a diagram in A of the form a→m←b together with an object of F(m). Generalizing the work of Fong, we show that if A has finite colimits and F:(A,+)→(Cat,×) is symmetric lax monoidal, there is a symmetric monoidal double category whose objects are those of A and whose horizontal 1-cells are decorated cospans. We prove that under certain conditions, these two constructions become isomorphic when we take X=∫F to be the Grothendieck category of F. We illustrate these ideas with applications to electrical circuits, Petri nets, dynamical systems and epidemiological modeling.

This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad. In Part 4 we define general monoidal categories equipped with an action operad equivariant structure, and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras. Many illustrations and examples are included. Assuming only basic category theory, this monograph is intended for graduate students and researchers. In addition to being a coherent reference for the topics covered, this book is also suitable for a graduate student seminar and a reading course.

In this chapter, we introduce the notions of 1-Segal and 2-Segal objects in a combinatorial model category C. If further C admits the structure of a left proper, tractable, symmetric monoidal model category, then we introduce model structures for 1-Segal and 2-Segal objects which arise as enriched Bousfield localizations of the injective model structure on CΔ. For \( {\mathbf C} = {\mathbb S}\), the model structure for 1-Segal objects in \({\mathbb S}\) recovers the Rezk model structure for 1-Segal spaces introduced in Rezk (Trans Am Math Soc 353(3):973–1007, 2001).

ObJective To study the hemodynamic characteristics of ophthalmic artery and central retinal artery in patients with thyroid-associated ophthalmopathy (TAO),and to analyze the relevance between the blood flow parameters and the clinical characteristics.At last,discuss the different influence factors on retinal blood supply of patients with TAO.Methods Use the U.S.company GE's Logic7 Color Doppler Ultrasound diagnostic apparatus,to detect the ophthalmic artery and central retinal artery on 49 TAO cases (90 eyes) whose thyroid function was normal or stable over six months and 24 normal controls (48 eyes),simultaneously,record its spectrum and measure the peak systolic velocity (PSV),enl diastolic velocity (EDV),resistance index (RI).To analyze the differences among the blood flow parameters were significant,and analyze the correlation between the blood flow parameters and the clinical characteristics.Results (1)The PSV and EDV in normal control group were higher than in TAO group,and the RI in normal group was lower than in TAO group,P ＜0.05; The constituent ratio of the eyes whose CRA spectrum was interrupted was higher in TAO group than in normal control group,P ＜0.05.(2)The CRA PSV in spectrum interruption group was lower than in spectrum without interruption group,P ＜0.05; The EDV of OA and CRA in spectrum interruption group was lower than in spectrum without interruption group,P ＜0.05; The RI of OA and CRA in spectrum interruption group was higher than in spectrum without interruption group,P ＜0.05.There was statistical significance for the differences of orbital pressure and exophthalmos between the spectrum inte,rruption group and spectrum without interruption group; (3)Either the PSV or the EDV of OA and CRA in normal control were higher than in spectrum interruption group,P＜0.05; The RI of OA and CRA in normal control were lower than in spectrum interruption group,P＜0.05.(4)The EDV of OA in normal control group was higher than in spectrum without interruption group,P ＜0.05; The RI of OA in normal control group was lower than in spectrum without interruption group,P ＜0.05; The PSV and EDV of CRA in normal control group were higher than in spectrum without interruption group,P ＜0.05.Conclusions In the TAO patients,the increase of the orbital pressure influences the hemodynamic of OA and CRA and leads to the decrease the blood flow velocity of eye vessels,so that the Color Doppler Ultrasound can not detect the blood flow,and the blood supply of the eye will be influenced. Key words: Thyroid-Associated Ophthalmopathy; Color Doppler blood flow imaging; Hemodynamics; Blood flow parameter; Spectrum