Diagram spaces and symmetric spectra

TL;DR

This paper develops a homotopical framework for structured diagram spaces, relating them to symmetric spectra, and introduces graded units of symmetric ring spectra to analyze periodicity and logarithmic structures in stable homotopy.

Contribution

It provides a unified homotopical analysis of diagram spaces like I-spaces and J-spaces, connecting them to symmetric spectra and introducing graded units for stable homotopy applications.

Findings

I-spaces model homotopy categories with strict E-infinity structures

J-spaces model graded E-infinity spaces and graded units

Graded units detect periodicity in stable homotopy and inform topological logarithmic structures

Abstract

We present a general homotopical analysis of structured diagram spaces and discuss the relation to symmetric spectra. The main motivating examples are the I-spaces, which are diagrams indexed by finite sets and injections, and J-spaces, which are diagrams indexed by the Grayson-Quillen construction on the category of finite sets and bijections. We show that the category of I-spaces provides a convenient model for the homotopy category of spaces in which every E-infinity space can be rectified to a strictly commutative monoid. Similarly, the commutative monoids in the category of J-spaces model graded E-infinity spaces. Using the theory of J-spaces we introduce the graded units of a symmetric ring spectrum. The graded units detect periodicity phenomena in stable homotopy and we show how this can be applied to the theory of topological logarithmic structures.