arXiv subjects

Marion Boucrot

Publications and source records attributed to Marion Boucrot.

3 recordsLinked to original sources

arXiv2024

Homotopy theory of pre-Calabi-Yau morphisms

In this article we study the homotopy theory of pre-Calabi-Yau morphisms, viewing them as Maurer-Cartan elements of an $L_{\infty}$-algebra. We give two different notions of homotopy: a notion of weak homotopy for morphisms between $d$-pre-Calabi-Yau categories whose underlying graded quivers on the domain (resp. codomain) are the same, and a notion of homotopy for morphisms between fixed pre-Calabi-Yau categories $(\mathcal{A},s_{d+1}M_{\mathcal{A}})$ and $(\mathcal{B},s_{d+1}M_{\mathcal{B}})$. Then, we show that the notion of homotopy is stable under composition and that homotopy equivalences are quasi-isomorphisms. Finally, we prove that the functor constructed by the author in a previous article between the category of pre-Calabi-Yau categories and the partial category of $A_{\infty}$-categories of the form $\mathcal{A}\oplus\mathcal{A}^*[d-1]$, for $\mathcal{A}$ a graded quiver, together with hat morphisms sends homotopic $d$-pre-Calabi-Yau morphisms to weak homotopic $A_{\infty}$-morphisms.

math.KT↗

arXiv2024

Morphisms of pre-Calabi-Yau categories and morphisms of cyclic $A_{\infty}$-categories

In this article we prove that there exists a relation between $d$-pre-Calabi-Yau morphisms introduced by M. Kontsevich, A. Takeda and Y. Vlassopoulos and cyclic $A_{\infty}$-morphisms, extending a result proved by D. Fernández and E. Herscovich. This leads to a functor between the category of $d$-pre-Calabi-Yau structures and the partial category of $A_{\infty}$-categories of the form $\mathcal{A}\oplus\mathcal{A}^*[d-1]$ with $\mathcal{A}$ a graded quiver and whose morphisms are the data of an $A_{\infty}$-structure on $\mathcal{A}\oplus\mathcal{B}^*[d-1]$ together with $A_{\infty}$-morphisms $\mathcal{A}[1]\oplus\mathcal{B}^*[d]\rightarrow \mathcal{A}[1]\oplus\mathcal{A}^*[d]$ and $\mathcal{A}[1]\oplus\mathcal{B}^*[d]\rightarrow \mathcal{B}[1]\oplus\mathcal{B}^*[d]$.

math.KT↗

arXiv2023

Homotopy Transfer Theorem and minimal models for pre-Calabi-Yau categories

In this article, we extend results of J. Leray and B. Vallette on homotopical properties of pre-Calabi-Yau algebras to the case of pre-Calabi-Yau categories. We give direct proofs of the results adapting techniques used by D. Petersen for the case of coalgebras, instead of using properadic calculus. More precisely, we prove that given quasi-isomorphic dg quivers and a pre-Calabi-Yau structure on one of them there exists a pre-Calabi-Yau structure on the other as well as a pre-Calabi-Yau morphism between the two. Moreover, we show that a pre-Calabi-Yau morphism whose first component is an isomorphism of dg quivers is an isomorphism in the category of pre-Calabi-Yau categories. We incidentally show that any pre-Calabi-Yau category has a minimal model. Finally, we prove that any quasi-isomorphism of pre-Calabi-Yau categories admits a quasi-inverse.

math.AT↗