[Paper Review] The unreasonable power of the lifting property in elementary mathematics

Gavrilovich Misha|arXiv (Cornell University)|Jul 20, 2017

Homotopy and Cohomology in Algebraic Topology3 citations

TL;DR

This paper demonstrates that the lifting property—a category-theoretic construction—can define fundamental concepts in elementary topology and algebra using only simple morphisms of finite spaces. It shows that compactness, connectedness, separation axioms, and group properties like nilpotency can be captured via iterated orthogonality, revealing a deep, concise combinatorial structure underlying standard mathematical notions.

ABSTRACT

We illustrate the generative power of the lifting property (orthogonality of morphisms in a category) as means of defining natural elementary mathematical concepts by giving a number of examples in various categories, in particular showing that many standard elementary notions of abstract topology can be defined by applying the lifting property to simple morphisms of finite topological spaces. Examples in topology include the notions of: compact, discrete, connected, and totally disconnected spaces, dense image, induced topology, and separation axioms. Examples in algebra include: finite groups being nilpotent, solvable, torsion-free, p-groups, and prime-to-p groups; injective and projective modules; injective, surjective, and split homomorphisms. We include some speculations on the wider significance of this.

Motivation & Objective

• To demonstrate that elementary mathematical concepts in topology and algebra can be defined using the lifting property applied to simple morphisms in finite categories.
• To show that key topological properties—such as compactness, connectedness, and separation axioms—can be expressed as iterated orthogonals of basic morphisms.
• To explore the potential of this formalism as a foundation for a lightweight, formal proof system for elementary topology.
• To investigate whether this approach reveals an underlying combinatorial structure akin to Misha Gromov’s concept of ergostructures.
• To provide a category-theoretic formalism with low Kolmogorov complexity for expressing standard mathematical definitions.

Proposed method

• Apply the Quillen lifting property (orthogonality of morphisms) to generate new classes of morphisms by taking left and right orthogonal closures.
• Use finite topological spaces as a model to represent standard topological properties via morphisms with simple combinatorial structure.
• Express topological notions like compactness as $((\{a\} \to \{a \searrow b\})^{r}_{<5})^{lr}$ and connectedness as $(\{a,b\} \to \{a=b\})^l$.
• Define algebraic properties (e.g., nilpotency, solvability, injectivity) by applying lifting to morphisms in categories of groups and modules.
• Use diagram chasing and orthogonality chains to derive and verify definitions in a purely combinatorial, formal way.
• Leverage known results from general topology (e.g., Urysohn’s lemma, proper maps) to validate the formalism through known characterizations.

Experimental results

Research questions

• RQ1 Can standard topological properties like compactness and connectedness be defined purely through iterated lifting properties applied to simple morphisms?
• RQ2 To what extent can algebraic properties such as nilpotency or injectivity in modules be captured using lifting constructions in finite categories?
• RQ3 Is there a minimal, expressive formalism based on lifting that can reconstruct core notions of elementary topology with low Kolmogorov complexity?
• RQ4 Can the lifting property formalism serve as a foundation for a theorem prover or proof system based on diagram chasing?
• RQ5 Does this approach reveal a combinatorial structure underlying mathematical reasoning, analogous to Gromov’s ergostructures?

Key findings

• The lifting property allows the definition of compactness as $((\{a\} \to \{a \searrow b\})^{r}_{<5})^{lr}$, showing its expression in a formalism of just a few symbols.
• Connectedness is captured by $(\{a,b\} \to \{a=b\})^l$, demonstrating that the property arises from a simple orthogonal construction.
• Dense image is defined as $(\{b\} \to \{a \searrow b\})^l$, showing that the lifting property can formalize basic topological relations.
• Separation axioms such as T1, T2, T3, and T4 are expressible via lifting properties with morphisms between finite topological spaces.
• The formalism captures algebraic properties: e.g., injective modules are defined as $\{0\} \to M$ such that $\{0\} \to M$ is right orthogonal to all surjections.
• The paper provides a combinatorial, low-complexity formalism for topological notions, suggesting that their Kolmogorov complexity is minimal in this framework.