A formal characterization of discrete condensed objects (2410.17847v3)

Abstract: Condensed mathematics, developed by Clausen and Scholze over the last few years, proposes a generalization of topology with better categorical properties. It replaces the concept of a topological space by that of a condensed set, which can be defined as a sheaf on a certain site of compact Hausdorff spaces. Since condensed sets are supposed to be a generalization of topological spaces, one would like to be able to study the notion of discreteness. There are various ways to define what it means for a condensed set to be discrete. In this paper we describe them, and prove that they are equivalent. The results have been fully formalized in the Lean proof assistant.

• The paper demonstrates that a condensed set is discrete when it lies in the essential image of the left adjoint functor mapping sets to locally constant sheaves.
• It proves that discrete objects exhibit an isomorphism between colimits over profinite sets and their evaluations, deepening our understanding of their structure.
• The study extends these equivalences to condensed modules by leveraging rigorous Lean formalization to validate categorical proofs.

A Formal Characterization of Discrete Condensed Objects

This paper presents a comprehensive investigation into the nature of discrete condensed objects within Clausen and Scholze’s framework of condensed mathematics. By substituting the notion of a topological space with a condensed set, this theory achieves refined categorical properties. A condensed set, specifically, is a sheaf on a site of compact Hausdorff spaces, which forms a bridge between algebraic techniques and topological constructs. A primary inquiry explored is the definition and characterization of discrete condensed sets, alongside proving the equivalency of these definitions. The paper’s results are proven formally using the \Lean proof assistant and contribute to the mathematical library \mathlib.

Main Contributions

The paper introduces the concept of a discrete condensed set through various conditions that are proven to be equivalent. Noteworthy results include:

• Characterization through Left Adjointness: The paper defines a condensed set as discrete if it is in the essential image of a functor $L : \Set \to \CondSet$, which is left adjoint to the underlying set functor $U : \CondSet \to \Set$. This functor $L$ maps a set $X$ to the sheaf of locally constant maps $\LocConst(-, X)$. The paper constructs a natural isomorphism between $L$ and the constant sheaf functor $\underline{(-)}$, thereby establishing a fundamental link between these notions.
• Colimit Characterization: Another insightful result shows that a condensed set is discrete if and only if, for every profinite set $S = \varprojlim_i S_i$, the canonical map $\varinjlim_i X(S_i) \to X(S)$ is an isomorphism. This colimit characterization provides a deeper understanding of how such objects interact within the framework.
• Extension to Condensed Modules: The paper extends the characterization by proving that a condensed module over a ring $R$ is discrete if and only if its underlying condensed set is discrete. This is achieved by showing the full faithfulness of the constant sheaf functor for modules and leveraging general results about sheafification.

Methodology and Formalization

The methodology relies heavily on categorical techniques and the formalized approach via \Lean. The authors construct adjunctions and establish equivalences through careful checking of unit and counit natural transformations. The \Lean formalization ensures rigor and correctness, providing machine-verifiable proofs of the established theorems. The use of techniques like Kan extensions and explicit construction of initial and final functors adds further theoretical depth.

Implications and Future Directions

These results significantly enhance our comprehension of discrete sets within the field of condensed mathematics. The robust equivalence between different characterizations fosters an adaptable framework that can be further explored in areas such as solid abelian groups or when developing new algebraic approaches leveraging the condensed mathematics framework.

The formalization aspect indicates a pathway toward formalizing more complex mathematical theories using proof assistants, potentially transforming how future mathematical work is conducted. By integrating these results into \mathlib, the authors pave the way for subsequent formal research and applications in algebraic topology, homological algebra, and related fields.

Future directions might explore the extension of these results to other types of sheaf categories, investigation into applications in computational homotopy theory, or further development of the \Lean library to handle even more intricate categorical constructs.

Through a detailed formalization and categorical exploration, this paper contributes essential insights into the structure of condensed objects, advancing both theoretical mathematics and the computational methods underpinning it.