[Paper Review] Comparative Smootheology

Andrew Stacey|arXiv (Cornell University)|Feb 15, 2008

Homotopy and Cohomology in Algebraic Topology1 references22 citations

TL;DR

This paper compares five foundational definitions of smooth spaces—Chen, Frölicher, Sikorski, Smith, and Souriau—by constructing natural functors between their respective categories. It establishes that Frölicher spaces form a central hub in this network, with all other categories embedding into or relating to them via universal properties, and proves no two categories are equivalent under arbitrary functors, affirming their distinct mathematical identities.

ABSTRACT

We compare various different definitions of "the category of smooth objects". The definitions compared are due to Chen, Frölicher, Sikorski, Smith, and Souriau. The method of comparison is to construct functors between the categories that enable us to see how the categories relate to each other. This produces a diagram of categories with the category of Frölicher spaces sitting at its centre. Our method of study involves finding a general context into which these categories can be placed. This involves considering categories wherein objects are considered in relation to a certain collection of standard test objects. This therefore applies beyond the question of categories of smooth spaces.

Motivation & Objective

• To resolve the proliferation of competing definitions of 'smooth objects' by systematically comparing their categorical structures.
• To determine whether any of the five categories of smooth spaces are equivalent under arbitrary functors, thus assessing their fundamental distinctness.
• To identify natural, non-contrived functors between the categories, particularly those preserving the subcategory of smooth manifolds.
• To develop a general categorical framework for smooth structures based on test objects and forcing conditions, applicable beyond smooth spaces.
• To clarify the role of Frölicher spaces as a central, unifying category in the landscape of generalized smooth spaces.

Proposed method

• The author constructs explicit functors between the categories of smooth spaces defined by Chen, Frölicher, Sikorski, Smith, and Souriau, focusing on those that preserve smooth manifolds.
• A general framework is introduced—'forced virtual T-objects in U'—to unify the definitions, based on an underlying category U, a test category T, and a forcing condition.
• The method uses diagrams of the form X ← T → X' to test smoothness, generalizing the idea of 'testing maps via plots' or 'functionals'.
• Adjunctions and embeddings are used to classify the relationships, identifying reflective or co-reflective subcategory structures.
• The approach is extended to non-set-based theories via the Isbell envelope, generalizing the framework to profunctors and lax factorisations.
• A non-topological reformulation of Sikorski spaces is derived, replacing topological locality with algebraic conditions on functionals.

Experimental results

Research questions

• RQ1 How are the categories of smooth spaces defined by Chen, Frölicher, Sikorski, Smith, and Souriau related through natural functors?
• RQ2 Can any of these categories be embedded into or reflectively embed into others, revealing a hierarchy or central structure?
• RQ3 Are the categories of smooth spaces equivalent under arbitrary functors, or do they represent fundamentally distinct mathematical objects?
• RQ4 What general categorical framework can unify the diverse definitions of smooth structures based on test objects?
• RQ5 Can the concept of smoothness be generalized beyond set-based spaces, particularly to synthetic or non-commutative differential geometry?

Key findings

• Frölicher spaces occupy a central position in the category-theoretic landscape of smooth spaces, with natural functors from all other categories mapping into them.
• The category of Frölicher spaces is both reflective and coreflective in the categories of Chen and Souriau spaces, indicating a strong universal property.
• No two of the five categories (Chen, Frölicher, Sikorski, Smith, Souriau) are equivalent under arbitrary functors, confirming their distinct mathematical identities.
• A general framework for smooth structures is established using test categories and forcing conditions, unifying the five definitions under a common formalism.
• A non-topological version of Sikorski spaces is derived, replacing topological locality with an algebraic condition on functionals and their products.
• The Isbell envelope construction provides a pathway to generalize the framework to non-set-based smooth theories, such as synthetic differential geometry.