We give an operadic definition of a genuine symmetric monoidal $G$-category, and we prove that its classifying space is a genuine $E_\infty $$G$-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power and Lack to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal $G$-categories to genuine permutative $G$-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When $G$ is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal $G$-categories as input to an equivariant infinite loop space machine that gives genuine $\Omega $-$G$-spectra as output.
This content will become publicly available on June 1, 2025
- Award ID(s):
- 2204297
- PAR ID:
- 10554625
- Publisher / Repository:
- Journal of Topology
- Date Published:
- Journal Name:
- Journal of Topology
- Volume:
- 17
- Issue:
- 2
- ISSN:
- 1753-8416
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
Abstract -
more » « less
In this paper, we present a homotopical framework for studying invertible gapped phases of matter from the point of view of infinite spin lattice systems, using the framework of algebraic quantum mechanics. We define the notion of quantum state types. These are certain lax-monoidal functors from the category of finite-dimensional Hilbert spaces to the category of topological spaces. The universal example takes a finite-dimensional Hilbert space [Formula: see text] to the pure state space of the quasi-local algebra of the quantum spin system with Hilbert space [Formula: see text] at each site of a specified lattice. The lax-monoidal structure encodes the tensor product of states, which corresponds to stacking for quantum systems. We then explain how to formally extract parametrized phases of matter from quantum state types, and how they naturally give rise to [Formula: see text]-spaces for an operad we call the “multiplicative” linear isometry operad. We define the notion of invertible quantum state types and explain how the passage to phases for these is related to group completion. We also explain how invertible quantum state types give rise to loop-spectra. Our motivation is to provide a framework for constructing Kitaev’s loop-spectrum of bosonic invertible gapped phases of matter. Finally, as a first step toward understanding the homotopy types of the loop-spectra associated to invertible quantum state types, we prove that the pure state space of any UHF algebra is simply connected.
-
more » « less
Abstract We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale’s 1967 result: a filtered deformation of a Frobenius algebra over a field is Frobenius as well. Toward the goal, we first construct a monoidal associated graded functor, building on prior works of Ardizzoni and Menini, Galatius et al., and Gwillian and Pavlov. Next, we produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form; this builds on work of Fuchs and Stigner. These two results of independent interest are then used to achieve our goal. As an application of our main result, we show that any exact module category over a symmetric finite tensor category $\mathcal {C}$ is represented by a Frobenius algebra in $\mathcal {C}$. Several directions for further investigation are also proposed.
-
more » « less
We introduce a theory of stratifications of noncommutative stacks (i.e., presentable stable
-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as -monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion. Our main application is to equivariant stable homotopy theory: for any compact Lie group
, we give a symmetric monoidal stratification of genuine -spectra. In the case that is finite, this expresses genuine -spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory.
-
Abstract We describe the universal target of annular Khovanov–Rozansky link homology functors as the homotopy category of a free symmetric monoidal linear category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov–Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types.more » « less
