arXiv subjects

Miguel Barata

Publications and source records attributed to Miguel Barata.

4 recordsLinked to original sources

arXiv2024

A dendroidal approach to operadic right modules and manifold calculus

In this work we study the homotopy theory of the category $\mathsf{RMod}_{\mathcal{P}}$ of right modules over a simplicial operad $\mathcal{P}$ via the formalism of forest spaces $\mathsf{fSpaces}$, as introduced by Heuts, Hinich and Moerdijk. In particular, we show that, for $\mathcal{P}$ a simplicial closed $\Sigma$-free operad, there exists a Quillen equivalence between the projective model structure on $\mathsf{RMod}_{\mathcal{P}}$, and the contravariant model structure on the slice category $\mathsf{fSpaces}_{/N\mathcal{P}}$ over the dendroidal nerve of $\mathcal{P}$. As an application, we comment on how this result can be used to simplify the computation of derived mapping spaces between operadic right modules, and use this formalism to analyse the components and layers of the Goodwillie--Weiss tower coming from embedding calculus.

math.AT↗

arXiv2024

The right cancellation property for certain classes of dendroidal anodynes

We generalize a previous result of Stevenson to the category of dendroidal sets, yielding the right cancellation property of dendroidal inner anodynes within the class of normal monomorphisms. As an application of this property, we show how to construct a symmetric monoidal $\infty$-category $\mathsf{Env}(X)^\otimes$ from a dendroidal $\infty$-operad $X$, in a way that generalizes the symmetric monoidal envelope of a coloured operad.

math.CT↗

arXiv2022

Loops in the fundamental group of $\mathrm{Symp} (\mathbb C\mathbb P^2\#\,5\overline{ \mathbb C\mathbb P}\,\!^2)$ which are not represented by circle actions

We study generators of the fundamental group of the group of symplectomorphisms $\mathrm{Symp}({\mathbb C\mathbb P}^2\#\,5\overline{\mathbb C\mathbb P}\,\!^2, ω)$ for some particular symplectic forms. It was observed by J. Kȩdra that there are many symplectic 4-manifolds $(M, ω)$, where $M$ is neither rational nor ruled, that admit no circle action and $π_1 (\mathrm{Ham} (M,ω))$ is nontrivial. On the other hand, it follows from previous results that the fundamental group of the group $\mathrm{Symp}_h({\mathbb C\mathbb P}^2\#\,k\,\overline{\mathbb C\mathbb P}\,\!^2, ω)$, of symplectomorphisms that act trivially on homology, with $k \leq 4$, is generated by circle actions on the manifold. We show that, for some particular symplectic forms $ω$, the set of all Hamiltonian circle actions generates a proper subgroup in $π_1(\mathrm{Symp}_h({\mathbb C\mathbb P}^2\#\,5\overline{\mathbb C\mathbb P}\,\!^2, ω)).$ Our work depends on Delzant classification of toric symplectic manifolds, Karshon's classification of Hamiltonian $S^1$-spaces and the computation of Seidel elements of some circle actions.

math.SG↗

arXiv2022

On the additivity of the little cubes operads

We give a new proof of Dunn's additivity for the little $n$-cubes operads $C_n$, which has the advantage of being considerably shorter than the ones in the literature. At the end we remark on how our proof can be adjusted to work for the tensor product of a finite number of factors.

math.AT↗