Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0806.3983

Localization of algebras over coloured operads

Carles Casacuberta·Javier J. Gutierrez·Ieke Moerdijk·Rainer M. Vogt
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and provides new results of the same kind. For instance, under suitable assumptions, homotopical localizations preserve ring spectra (in the strict sense, not only up to homotopy), modules over ring spectra, and algebras over commutative ring spectra, as well as ring maps, module maps, and algebra maps. It is principally the treatment of module spectra and their maps that led us to the use of coloured operads (also called enriched multicategories) in this context.

Explore related subjects

math.ATmath.CT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Carles Casacuberta, Javier J. Gutierrez, Ieke Moerdijk, Rainer M. Vogt. 2008-06-24. Localization of algebras over coloured operads. https://doi.org/10.1112/plms%2Fpdp034

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Chromatic higher semiadditivity via redshift

We give a new proof of the $\infty$-semiadditivity of $K(n)$-local spectra. The proof proceeds by induction on the height via algebraic K-theory, utilizing recent advances in the redshift conjecture, instead of using the Ravenel-Wilson computation of the Morava K-theory of Eilenberg-MacLane spaces. Along the way, we prove the higher semiadditivity of $T(n)$-local modules over the $T(n)$-localized K-theory of $K(n-1)$-local ring spectra, which suffices for the applications of higher semiadditivity to the telescope conjecture.

math.AT↗
arXiv2026

Geometric realization via unoriented bordism and a counterexample to Singer's conjecture for the sixth algebraic transfer

Let $\mathscr{A}$ be the mod-2 Steenrod algebra acting in the usual way on $P_q = \mathbb{F}_2[x_1, \ldots, x_q]$, and let $QP_q = \mathbb{F}_2 \otimes_{\mathscr{A}} P_q$. Singer's algebraic transfer $Tr_q$ sends the dual of $[(QP_q)_n]^{GL(q, \mathbb{F}_2)}$ to $\operatorname{Ext}_{\mathscr{A}}^{q,q+n}(\mathbb{F}_2,\mathbb{F}_2)$; Singer conjectured that $Tr_q$ is always injective. We disprove this nearly forty-year-old conjecture at rank $q=6$, degree $n=36$. Verifying this requires computing $[(QP_6)_{36}]^{GL(6, \mathbb{F}_2)}$ exactly; to handle the resulting combinatorial complexity, we build a new Julia package \texttt{AlgebraicTransfer.jl}, coupling modular invariant theory with bit-level linear algebra over $\mathbb{F}_2$ via Steenrod-hit reductions and Kameko homomorphisms. We prove this source space is two-dimensional, strictly exceeding the known one-dimensional target $\operatorname{Ext}_{\mathscr{A}}^{6,42}(\mathbb{F}_2,\mathbb{F}_2)$, so $Tr_6$ is not injective. We also interpret the transfer kernel geometrically via unoriented bordism: $Tr_q$ factors through bordism classes over $B(\mathbb{Z}/2)^q$ whose Thom images are primitive, characterized by the vanishing of all mixed Wu numbers. Thom's representability theorem guarantees closed $36$-manifolds realizing the homological duals of the source generators, yet we show that standard models (such as the indecomposable Milnor hypersurface $H_{4,33}$, projective products, and Dold manifolds) cannot represent them. We further interpret the inverse Kameko map via Thom spaces of universal real line bundles. Validated by recovering classical Dickson invariant dimensions, this work delivers both a counterexample to Singer's conjecture and a scalable methodology for the Peterson hit problem.

math.AT↗
arXiv2026

Grothendieck-Teichmüller Symmetries of Cyclic Operads and Tangles

We identify the profinite Grothendieck-Teichmüller group $\widehat{\mathsf{GT}}$ with the group of homotopy automorphisms of the profinite completion of the cyclic operad of parenthesised ribbon braids. We transport this action to a profinite cyclic $\infty$-operad of framed configuration spaces. Using the metric prop associated to the cyclic ribbon-braid operad, we also obtain a $\widehat{\mathsf{GT}}$-action on a category of framed unoriented profinite tangles, and compare its prounipotent analogue with the action of Kassel and Turaev. Finally, the rational cyclic Grothendieck-Teichmüller action gives an alternative proof of the rational formality of the cyclic framed little-disks operad.

math.AT↗