Strictly Commutative Complex Orientations of $Tmf_1(N)$

Motivic stable homotopy theory is strictly commutative at the characteristic

We show that mapping spaces in the p-local motivic stable category over an Fp-scheme are strictly commutative monoids (whence HZ-modules) in a canonical way.

Strictly commutative models for E-infinity quasi-categories

In this short note we show that E-infinity quasi-categories can be replaced by strictly commutative objects in the larger category of diagrams of simplicial sets indexed by finite sets and injections. This complements earlier work on diagram spaces by Christian Schlichtkrull and the second author.

A strictly commutative model for the cochain algebra of a space

The commutative differential graded algebra $A_{\mathrm{PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\mathcal{I}}(X)$ of $A_{\mathrm{PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\mathcal{I}$ to model $E_{\infty}$ differential graded algebras by strictly commutative objects, called commutative $\mathcal{I}$-dgas. We define a functor $A^{\mathcal{I}}$ from simplicial sets to commutative $\mathcal{I}$-dgas and show that it is a…

Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2

Previous work constructed a generalized truncated Brown-Peterson spectrum of chromatic height 2 at the prime 2 as an E_infinity-ring spectrum, based on the study of elliptic curves with level-3 structure. We show that the natural map forgetting this level structure induces an E_infinity-ring map from the spectrum of topological modular forms to this truncated Brown-Peterson spectrum, and that this orientation fits into a diagram of E_infinity-ring spectra lifting a classical diagram of modules over the mod-2 Steenrod algebra. In an appendix we document how to organize Morava's forms of K-theor…

严格交换代数在正特征下的障碍理论

An obstruction theory for strictly commutative algebras in positive characteristic

本文研究了严格交换代数在正特征下的障碍理论，定义了余三元积并探讨了其与E_∞-代数的关系，最终得出E_∞-代数可修正当且仅当更高阶Steenrod操作相干地消失。

Strictly commutative complex orientation theory

For a multiplicative cohomology theory E, complex orientations are in bijective correspondence with multiplicative natural transformations to E from complex bordism cohomology MU. If E is represented by a spectrum with a highly structured multiplication, we give an iterative process for lifting an orientation MU -> E to a map respecting this extra structure, based on work of Arone-Lesh. The space of strictly commutative orientations is the limit of an inverse tower of spaces parametrizing partial lifts; stage 1 corresponds to ordinary complex orientations, and lifting from stage (m-1) to stage…