Journal of Homotopy and Related Structures最新文献

We show that the Picard group ({{mathrm{Pic}}}(mathcal {A}_mathbb {C}(1))) of the stable category of modules over (mathbb {C})-motivic (mathcal {A}_mathbb {C}(1)) is isomorphic to (mathbb {Z}^4). By comparison, the Picard group of classical (mathcal {A}(1)) is (mathbb {Z}^2 oplus mathbb {Z}/2). One extra copy of (mathbb {Z}) arises from the motivic bigrading. The joker is a well-known exotic element of order 2 in the Picard group of classical (mathcal {A}(1)). The (mathbb {C})-motivic joker has infinite order.

Homotopical geometry over differential operators is a convenient setting for a coordinate-free investigation of nonlinear partial differential equations modulo symmetries. One of the first issues one meets in the functor of points approach to homotopical ({{{mathcal {D}}}})-geometry, is the question of a model structure on the category ({mathtt{DGAlg({{{mathcal {D}}}})}}) of differential non-negatively graded ({{{mathcal {O}}}})-quasi-coherent sheaves of commutative algebras over the sheaf ({{{mathcal {D}}}}) of differential operators of an appropriate underlying variety ((X,{{{mathcal {O}}}})). We define a cofibrantly generated model structure on ({mathtt{DGAlg({{{mathcal {D}}}})}}) via the definition of its weak equivalences and its fibrations, characterize the class of cofibrations, and build an explicit functorial ‘cofibration–trivial fibration’ factorization. We then use the latter to get a functorial model categorical Koszul–Tate resolution for ({{{mathcal {D}}}})-algebraic ‘on-shell function’ algebras (which contains the classical Koszul–Tate resolution). The paper is also the starting point for a homotopical ({{{mathcal {D}}}})-geometric Batalin–Vilkovisky formalism.

We study the subcategory of topological operads ({mathsf {P}}) such that ({mathsf {P}}(0) = *) (the category of unitary operads in our terminology). We use that this category inherits a model structure, like the category of all operads in topological spaces, and that the embedding functor of this subcategory of unitary operads into the category of all operads admits a left Quillen adjoint. We prove that the derived functor of this left Quillen adjoint functor induces a left inverse of the derived functor of our category embedding at the homotopy category level. We deduce from this result that the derived mapping spaces associated to our model category of unitary operads are homotopy equivalent to the standard derived operad mapping spaces, which we form in the model category of all operads in topological spaces. We prove that analogous statements hold for the subcategory of k-truncated unitary operads within the model category of all k-truncated operads, for any fixed arity bound (kge 1), where a k-truncated operad denotes an operad that is defined up to arity k.

We study the questions of how to recognize when a simplicial set X is of the form (X={text {map}}_{*}({mathbf {Y}},{mathbf {A}})), for a given space ({mathbf {A}}), and how to recover ({mathbf {Y}}) from X, if so. A full answer is provided when ({mathbf {A}}={mathbf {K}}({R},{n})), for (R=mathbb F_{p}) or (mathbb Q), in terms of a mapping algebra structure on X (defined in terms of product-preserving simplicial functors out of a certain simplicially enriched sketch (varvec{Theta })). In addition, when ({mathbf {A}}=Omega ^{infty }{mathcal {A}}) for a suitable connective ring spectrum ({mathcal {A}}), we can recover ({mathbf {Y}}) from ({text {map}}_{*}({mathbf {Y}},{mathbf {A}})), given such a mapping algebra structure. This can be made more explicit when ({mathbf {A}}={mathbf {K}}({R},{n})) for some commutative ring R. Finally, our methods provide a new way of looking at the classical Bousfield–Kan R-completion.

Our objective is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of a simplicial set embedded in ({mathbb {R}}^{n}) induces a multiplicative filtration on the dg algebra of simplicial cochains, we use a result by Kadeishvili to get a unique (A_{infty })-algebra structure on the complete persistent cohomology of the filtered simplicial set. We then construct of a (pseudo)metric on the set of all barcodes of all cohomological degrees enriched with the (A_{infty })-algebra structure stated before, refining the usual bottleneck metric, and which is also independent of the particular (A_{infty })-algebra structure chosen. We also compute this distance for some basic examples. As an aside, we give a simple proof of a result relating the barcode structure between persistent homology and cohomology, that was observed by de Silva, Morozov, and Vejdemo-Johansson under some restricted assumptions which we do not suppose.

This paper is about a correspondence between monoidal structures in categories and n-fold loop spaces. We developed a new syntactical technique whose role is to substitute the coherence results, which were the main ingredients in the proof that the Segal–Thomason bar construction provides an appropriate simplicial space. The results we present here enable more common categories to enter this delooping machine. For example, such as the category of finite sets with two monoidal structures brought by the disjoint union and Cartesian product.

A finite ringed space is a ringed space whose underlying topological space is finite. The category of finite ringed spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed space, endowed with a finite open covering, produces a finite ringed space. We study the homotopy of finite ringed spaces, extending Stong’s homotopy classification of finite topological spaces to finite ringed spaces. We also prove that the category of quasi-coherent modules on a finite ringed space is a homotopy invariant.

Let ({mathcal {A}}) be an abelian category. In this paper we study monoform objects and atoms introduced by Kanda. We classify full subcategories of ({mathcal {A}}) by means of subclasses of ({mathrm{ASpec}}{mathcal {A}}), the atom spectrum of ({mathcal {A}}). We also study the atomical decomposition and localization theory in terms of atoms. As some applications of our results, we study the category Mod-A where A is a fully right bounded noetherian ring.

Using a suitable notion of normal Galois extension of commutative rings, we develop the relative theory of the generalized Teichmüller cocycle map. We interpret the theory in terms of the Deuring embedding problem, construct an eight term exact sequence involving the relative Teichmüller cocycle map and suitable relative versions of generalized Brauer groups and compare the theory with the group cohomology eight term exact sequence involving crossed pairs. We also develop somewhat more sophisticated versions of the ordinary, equivariant and crossed relative Brauer groups and show that the resulting exact sequences behave better with regard to comparison of the theory with group cohomology than do the naive notions of the generalized relative Brauer groups.

Let S be a commutative ring and Q a group that acts on S by ring automorphisms. Given an S-algebra endowed with an outer action of Q, we study the associated Teichmüller class in the appropriate third group cohomology group. We extend the classical results to this general setting. Somewhat more specifically, let R denote the subring of S that is fixed under Q. A Q-normalS-algebra consists of a central S-algebra A and a homomorphism (sigma :Qrightarrow mathop {mathrm{Out}}nolimits (A)) into the group (mathop {mathrm{Out}}nolimits (A)) of outer automorphisms of A that lifts the action of Q on S. With respect to the abelian group (mathrm {U}(S)) of invertible elements of S, endowed with the Q-module structure coming from the Q-action on S, we associate to a Q-normal S-algebra ((A, sigma )) a crossed?2-fold extension (mathrm {e}_{(A, sigma )}) starting at (mathrm {U}(S)) and ending at Q, the Teichmüller complex of ((A, sigma )), and this complex, in turn, represents a class, the Teichmüller class of ((A, sigma )), in the third group cohomology group (mathrm {H}^3(Q,mathrm {U}(S))) of Q with coefficients in (mathrm {U}(S)). We extend some of the classical results to this general setting. Among others, we relate the Teichmüller cocycle map with the generalized Deuring embedding problem and develop a seven term exact sequence involving suitable generalized Brauer groups and the generalized Teichmüller cocycle map. We also relate the generalized Teichmüller cocycle map with a suitably defined abelian group of classes of representations of Q in the Q-graded Brauer category of S with respect to the given action of Q on S.