Core Concepts

The category of topological spaces with open maps does not possess binary products, meaning a universal product space cannot always be constructed within this category.

Abstract

**Bibliographic Information:**Bezhanishvili, G., & Kornell, A. (2024). The category of topological spaces and open maps does not have products.*arXiv preprint arXiv:2407.13951v3*.**Research Objective:**To determine whether the category of topological spaces with open maps has binary products.**Methodology:**The authors employ a proof by contradiction. They assume the existence of a product in this category, particularly for the Sierpiński space with itself. By constructing embeddings of arbitrarily large posets into this hypothetical product, they arrive at a contradiction with set-theoretic principles.**Key Findings:**The category of topological spaces with open maps does not have binary products. This result extends to various subcategories, including T0-spaces, sober spaces, preordered sets, posets, and well-founded posets. The authors also demonstrate that the category of complete Heyting algebras lacks binary coproducts, and the category of Kripke frames lacks binary products.**Main Conclusions:**The non-existence of binary products in the category of topological spaces with open maps resolves the long-standing Esakia problem. This finding has implications for related areas such as pointfree topology and modal logic.**Significance:**This paper makes a significant contribution to category theory and topology by resolving a fundamental question about the structure of the category of topological spaces with open maps.**Limitations and Future Research:**The paper focuses on binary products. Exploring the existence of products for larger cardinalities within this category could be an area for future research. Additionally, investigating alternative categorical constructions or weaker notions of products in this context might yield interesting insights.

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by Guram Bezhan... at **arxiv.org** 10-07-2024

Deeper Inquiries

The result that the category TopOpen lacks binary products has interesting implications for other areas of mathematics where open maps are significant, particularly when categorical constructions are involved:
Algebraic Geometry: In algebraic geometry, the Zariski topology, which is crucial for defining important concepts like irreducible varieties, uses open sets extensively. While the category of algebraic varieties with regular maps is not directly comparable to TopOpen, the absence of products in TopOpen might suggest exploring alternative categorical frameworks for studying open maps between algebraic varieties. This could lead to new insights into the structure of algebraic varieties and their morphisms.
Functional Analysis: Open mapping theorems are fundamental in functional analysis, connecting the topological and algebraic structures of vector spaces. The lack of products in TopOpen raises questions about the categorical behavior of open mappings between topological vector spaces. It might be worthwhile to investigate whether this absence has any consequences for the construction of products or coproducts in categories of topological vector spaces or if it necessitates considering different categorical settings.
Overall, this result encourages mathematicians to carefully consider the categorical properties of open maps in their respective fields. It highlights that the existence of categorical products, often taken for granted in other contexts, cannot be assumed when working with open maps. This prompts the search for alternative categorical frameworks or a deeper understanding of the limitations imposed by the absence of products.

While the standard categorical definition of a product fails in TopOpen, exploring alternative notions of "product" that capture some desired properties is possible:
Weak Products: One could consider weaker notions of products, such as weak products, where the universal property is relaxed. Instead of requiring a unique morphism into the product, we might allow for a family of morphisms satisfying certain coherence conditions. However, finding a suitable weak product notion that aligns with the intuition of "combining" topological spaces under open maps remains a challenge.
Different Categories: Another approach is to embed TopOpen into a larger category where products exist and then study the properties of these products when restricted to TopOpen. For instance, as mentioned in the paper, the category of topological spaces with local homeomorphisms as morphisms admits binary products. Investigating the relationship between these products and the objects in TopOpen might offer valuable insights.
Internal Products: Category theory offers the concept of internal products, which generalize the usual notion of products to categories equipped with additional structure. It might be fruitful to explore whether TopOpen, potentially with some additional structure, admits internal products and how these relate to the intuitive understanding of combining spaces under open maps.
The search for a different "product" definition for TopOpen is an open question. It requires carefully balancing the desired properties of a product with the constraints imposed by open maps. This investigation could lead to a deeper understanding of the category TopOpen and potentially reveal new categorical constructions relevant to other areas of mathematics.

The lack of binary products in TopOpen has significant implications for understanding the interaction and composition of continuous processes modeled by open maps:
Limitations on Combining Processes: The absence of products suggests that there might not always be a natural way to combine two continuous processes represented by open maps into a single process that projects onto the original ones. This limitation highlights that the intuitive idea of running two processes "in parallel" might not always be well-defined in this categorical framework.
Restrictions on Universality: The universal property of products guarantees that any other object admitting morphisms from the factors uniquely factors through the product. The absence of products in TopOpen implies that we lose this universality when working with open maps. This restricts our ability to reason about and construct new continuous processes from existing ones using categorical tools.
Alternative Composition Mechanisms: The lack of products prompts the search for alternative mechanisms for composing continuous processes in TopOpen. This could involve exploring different categorical constructions, such as coproducts, pushouts, or more general limits and colimits, to capture different ways of combining and relating continuous processes.
Focus on Specific Subcategories: The paper mentions that products exist in certain subcategories of TopOpen, such as the category of hyperstonean spaces with open maps. This suggests that focusing on specific subcategories with additional properties might provide a more suitable framework for studying the interaction and composition of certain types of continuous processes.
In conclusion, the absence of binary products in TopOpen presents a challenge for studying the interaction and composition of continuous processes modeled by open maps. It encourages mathematicians to explore alternative categorical frameworks, focus on specific subcategories, or develop new tools and techniques to overcome the limitations imposed by this absence. This exploration could lead to a deeper understanding of the nature of open maps and their role in representing continuous processes.

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