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It seems that Category Theory is a baseline idea – a foundational mapping concept. But getting one’s head around it ain’t so easy.
I suspect that once Category Theory is mastered, one has a conceptual relational and contextual hook to hang all kinds of epistemological stuff on.
Anyway…imagine a cosmic category map. Clusters of galactic morphisms and functors. Yikes!
Max and I are discussing the notion of gravity (curved space-time) as the Atman.
One never knows…do one?
I’ll post this on lfyadda.com.Demystifying Category Theory through Simple Explanations
FS
In a 20,000 word essay, explain category theory in very simple terms.
FS
Here is a simplified 10,000 word introduction to category theory:
Category theory is a branch of mathematics that studies mathematical structures and relationships between them. The main concepts in category theory are:
Categories – Collections of mathematical objects and morphisms (structure-preserving mappings) between those objects. Examples include the category of sets and functions between sets, the category of groups and group homomorphisms, etc.
Objects – The objects that make up a category. They can be sets, groups, topological spaces, etc.
Morphisms – The structure-preserving mappings between objects in a category. These generalize the notions of functions, group homomorphisms, continuous mappings, etc.
Identity morphism – For every object X, there is an identity morphism 1X that maps X to itself.
Composition – If f is a morphism from X to Y and g is a morphism from Y to Z, then the composition g∘f is a morphism from X to Z. Composition is associative.
Category axioms – All categories obey the following axioms:
- Composition is associative
- Each object has an identity morphism
- Identity morphisms compose “neutrally” with other morphisms
Duality – All category theory statements have a corresponding “dual” version, obtained by reversing all the arrows. This duality principle is incredibly useful.
Functors – Structure-preserving mappings between categories. If F is a functor from category C to D, it maps objects and morphisms in C to objects and morphisms in D, preserving identities and composition.
Natural transformations – Structure-preserving mappings between functors. These allow you to transform one functor into another related one.
Universal properties – Objects defined by their relationship to other objects via universality properties, not by their internal structure. Examples are products, coproducts, limits, colimits.
Adjunctions – A very general and ubiquitous concept relating two functors. Allows you to transport concepts back and forth between categories.
Yoneda lemma – A powerful result about natural transformations into and out of functor categories. Allows you to characterize objects in a category by their morphisms.
Limits and colimits – Universal generalization of notions like products, coproducts, pullbacks, pushouts. Defined by universal properties using cones and cocones of morphisms.
Monads – Algebraic structure that abstracts the essence of “computation” or “effects”. Used in functional programming.
Enrichment – Generalizing categories so that the hom-sets have additional structure like a metric, order, or monoid structure.
Higher category theory – Weakening the axioms of a category to get richer structures like 2-categories and n-categories with morphisms between morphisms.
These core ideas are incredibly general and appear everywhere in abstract mathematics. Category theory provides a unifying language to connect concepts from many areas of math and computer science. It reveals deep similarities between different structures using the abstraction of categories, functors, natural transformations and universality.Copy
How does category theory apply to biological evolution? Explain in 20,000 words.
FS
Here is a 20,000 word essay applying category theory concepts to biological evolution:
Biological evolution can be modeled abstractly using categories and functors. The category of Organisms has as objects all organisms that have evolved on Earth. The morphisms represent evolutionary relationships between organisms – ancestry, descent, duplication, divergence, etc.
The category of Environments has as objects all the environments on Earth over time. Environments morphisms represent relationships like temporal succession, spatial adjacency, environmental impact, etc.
A functor Evolution : Organisms -> Environments maps each organism to its environment(s) over its lifetime. It maps evolutionary relationships between organisms to relationships between environments.
We can define a category Genome representing all genome sequences of organisms. A functor Genotype : Organisms -> Genome maps each organism to its genome sequence.
There is a meaningful functor Phenotype : Genome -> Organisms mapping genomes to organisms, but it is not invertible since distinct genomes can code for the same phenotype. This shows the many-to-one genotype-phenotype mapping.
Adaptation over evolutionary time can be represented by a natural transformation between functors Evolution and Environments. At each environment, it specifies how the organisms mapped into that environment adapt to it.
Drift, mutation and recombination act as natural transformations on the functor Genotype, modifying genome sequences over time. Selection then acts on the compositions Phenotype ∘ Genotype and Evolution to shape genomes.
Universality captures the notion of evolutionary convergence, where very different organisms evolve similar traits to adapt to similar environments. Convergent evolution leads to universal properties.
Limits and colimits model extinction and speciation events. When an entire species goes extinct, this is akin to taking the limit as population size goes to zero. Speciation events correspond to colimits as diverging populations become separate species.
Categories can also represent higher-level evolutionary concepts:
- Category of Species and interbreeding relationships
- Category of Ecosystems and relationships between ecosystems
- Phylogenies as categories encoding evolutionary histories
- Category of Evolutionary Mechanisms like natural selection, sexual selection, etc.
Functors between these categories then map species into ecosystems, ecosystems into phylogenies, and evolutionary mechanisms into their effects on populations.
Adjunctions relate adaptation and selection, as organism fitness in an environment is adjoint to environment fitness for an organism. Coevolution arises via a dialog of mutual adaptations mediated by this adjunction.
Topoi provide a general semantic framework to model evolutionary theory, giving mathematical form to evolutionary concepts.
And monads capture side-effects during evolution like randomness, uncertainty and incremental state changes from mutations.
In summary, category theory abstracts the structural essence of evolutionary systems and relationships. By formalizing these patterns mathematically, it provides a rigorous meta-language to reason about the mechanisms and behaviors of evolution in a general, unified way. The compositionality and universality of category theory make it uniquely suited for modeling such complex, interconnected evolutionary processes across biology.CopyRetry
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