Journal of Homotopy and Related Structures最新文献

Motivated by the work of Gerstenhaber-Voronov and that of Malvenuto-Reuternauer, we define on pointed multiplicative operads in the category of vector spaces over an arbitrary ground field (mathbb {K}), a cosimplicial vector space structure. This permits us to construct on such operads some algebraic structures such as the homotopy G-algebra and the bicomplex algebra structures. Moreover we illustrate our constructions through some examples and explain or extend some well-known results.

Cazanave proved that the set of naive (mathbb {A}^1)-homotopy classes of endomorphisms of the projective line admits a monoid structure whose group completion is genuine (mathbb {A}^1)-homotopy classes of endomorphisms of the projective line. In this very short note we show that, over a field which is not quadratically closed, such a statement is never true for punctured affine space (mathbb {A}^nhspace{-0.1em}smallsetminus {0}) for (nge 2).

We investigate the real cycle class map for singular varieties. We introduce an analog of Borel–Moore homology for algebraic varieties over the real numbers, which is defined via the hypercohomology of the Gersten–Witt complex associated with schemes possessing a dualizing complex. We show that the hypercohomology of this complex is isomorphic to the classical Borel–Moore homology for quasi-projective varieties over the real numbers.

We develop the theory of nilpotent G-spaces and their localisations, for G a compact Lie group, via reduction to the non-equivariant case using Bousfield localisation. One point of interest in the equivariant setting is that we can choose to localise or complete at different sets of primes at different fixed point spaces—and the theory works out just as well provided that you invert more primes at (K le G) than at (H le G), whenever K is subconjugate to H in G. We also develop the theory in an unbased context, allowing us to extend the theory to G-spaces which are not G-connected.

We study the composition of Bousfield localizations on a tensor triangulated category stratified via the Balmer-Favi support and with noetherian Balmer spectrum. Our aim is to provide reductions via purely axiomatic arguments, allowing us general applications to concrete categories examined in mathematical practice. We propose a conjecture which states that the behaviour of the composition of the localizations depends on the chains of inclusions of the Balmer primes indexing said localizations. We prove this conjecture in the case of finite or low dimensional Balmer spectra.

Answering a question of Randal-Williams, we show the natural maps from split Steinberg modules of a Dedekind domain to the associated Steinberg modules are surjective.

For a finite cyclic group (C_n), we identify Greenlees’ equivariant connective K-theory (kU_{C_n}) as an (RO(C_n))-graded localization of the actual connective cover of (KU_{C_n}).

We introduce categorical models of (N_infty ) spaces, which we call normed symmetric monoidal categories (NSMCs). These are ordinary symmetric monoidal categories equipped with compatible families of norm maps, and when specialized to a particular class of examples, they reveal a connection between the equivariant symmetric monoidal categories of Guillou–May–Merling–Osorno and those of Hill–Hopkins. We also give an operadic interpretation of the Mac Lane coherence theorem and generalize it to include NSMCs. Among other things, this theorem ensures that the classifying space of an NSMC is an (N_infty ) space. We conclude by extending our coherence theorem to include NSMCs with strict relations.

For a positive integer k, the k-cut complex of a graph G is the simplicial complex whose facets are the ((|V(G)|-k))-subsets (sigma ) of the vertex set V(G) of G such that the induced subgraph of G on (V(G) setminus sigma ) is disconnected. These complexes first appeared in the master thesis of Denker and were further studied by Bayer et al. (SIAM J Discrete Math 38(2):1630–1675, 2024). In the same article, Bayer et al. conjectured that for (k ge 3), the k-cut complexes of squared cycle graphs are shellable. Moreover, they also conjectured about the Betti numbers of these complexes when (k=3). In this article, we prove these conjectures for (k=3).

A duoidal category is a category equipped with two monoidal structures in which one is (op)lax monoidal with respect to the other. In this paper we introduce duoidal (infty )-categories which are counterparts of duoidal categories in the setting of (infty )-categories. There are three kinds of functors between duoidal (infty )-categories, which are called bilax, double lax, and double oplax monoidal functors. We make three formulations of (infty )-categories of duoidal (infty )-categories according to which functors we take. Furthermore, corresponding to the three kinds of functors, we define bimonoids, double monoids, and double comonoids in duoidal (infty )-categories.