arXiv subjects

Damjan Pištalo

Publications and source records attributed to Damjan Pištalo.

6 recordsLinked to original sources

arXiv2026

Homotopical algebra of Lie-Rinehart pairs

Dwyer-Kan localization at pairs of quasi-isomorphisms of the category of dg Lie-Rinehart pairs $(A,M)$, where $A$ is a semi-free cdga over a field $k$ of characteristic zero and $M$ a cell complex in $A$-modules, is shown to be equivalent to that of strong homotopy Lie-Rinehart (SH LR) pairs satisfying the same cofibrancy condition. Latter is a category of fibrant objects. We introduce cofibrations of SH LR pairs, construct factorizations, and prove lifting properties. Applying them, we show uniqueness up to homotopy of certain BV-type resolutions. Restricting to dg LR pairs whose underlying cdga is of finite type, and using a different (co)fibrancy condition, we show that the functor $(A,M)\mapsto A$ is a Cartesian fibration with presentable fibers. The two (co)fibrancy conditions yield equivalent $\infty$-categories under Dwyer-Kan localization.

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arXiv2024

Pro-nilpotently extended dgca-s and SH Lie-Rinehart pairs

Category of pro-nilpotently extended differential graded commutative algebras is introduced. Chevalley-Eilenberg construction provides an equivalence between its certain full subcategory and the opposite to the full subcategory of strong homotopy Lie Rinehart pairs with strong homotopy morphisms, consisting of pairs $(A,M)$ where $M$ is flat as a graded $A$-module. It is shown that pairs $(A,M)$, where $A$ is a semi-free dgca and $M$ a cell complex in $\op{Mod}(A)$, form a category of fibrant objects by proving that their Chevalley-Eilenberg complexes form a category of cofibrant objects.

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arXiv2022

The Tor Spectral Sequence and Flat Morphisms in Homotopical $D$-Geometry

Homotopical algebraic $D$-geometry combines aspects of homotopical algebraic geometry of Toen and Vezzosi and $D$-geometry of Beilinson and Drinfeld. It was introduced by the paper's last two authors and di Brino as a suitable framework for a coordinate-free study of the Batalin-Vilkovisky complex and more generally for the study of non-linear partial differential equations and their symmetries. In order to consolidate the foundation of the theory, we have to prove that the standard methods of linear and commutative algebra are available in the context of homotopical algebraic $D$-geometry, and we must show that in this context the étale topology is a kind of homotopical Grothendieck topology and that the notion of smooth morphism is, roughly speaking, local for the étale topology. The first half of this work was done. The remaining part covers the study of étale and flat morphisms in the category of differential graded $D$-algebras and is based on the Tor spectral sequence which connects the graded Tor functors in homology with the homology of the derived tensor product of two differential graded $D$-modules over a differential graded $D$-algebra.

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arXiv2021

Comparison theorems for Kan, faintly universal and strongly universal derived functors

We distinguish between faint, weak, strong and strict localizations of categories at morphism families and show that this framework captures the different types of derived functors that are considered in the literature. More precisely, we show that Kan and faint derived functors coincide when we use the classical Kan homotopy category, and when we use the Quillen homotopy category, Kan and strong derived functors coincide. Our comparison results are based on the fact that the Kan homotopy category is a weak localization and that the Quillen homotopy category is a strict localization.

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arXiv2021

Indeterminacies and models of homotopy limits

In \cite{CompTheo} we studied the indeterminacy of the value of a derived functor at an object using different definitions of a derived functor and different types of fibrant replacement. In the present work we focus on derived or homotopy limits, which of course depend on the model structure of the diagram category under consideration. The latter is not necessarily unique, which is an additional source of indeterminacy. In the case of homotopy pullbacks, we introduce the concept of full homotopy pullback by identifying the homotopy pullbacks associated with three different model structures of the category of cospan diagrams, thus increasing the number of canonical representatives. Finally, we define generalized representatives or models of homotopy limits and full homotopy pullbacks. The concept of model is a unifying approach that includes the homotopy pullback used in \cite{JL} and the homotopy fiber square defined in \cite{Hir} in right proper model categories. Properties of the latter are generalized to models in any model category.

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arXiv2021

A new approach to model categorical homotopy fiber sequences

We propose a simplified definition of Quillen's fibration sequences in a pointed model category that fully captures the theory, although it is completely independent of the concept of action. This advantage arises from the understanding that the homotopy theory of the model category's arrow category contains all homotopical information about its long fibration sequences.

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