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Abstract Algebra & Category Theory

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Abstractions about algebraic and categorical structures, including groups, categories, filtrations and coherence conditions, spanning number-theoretic curiosities like amicable triples, set-theoretic notions like stationary sets, and foundational categories like FinSet.

24 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

  • Accessible category — A category that, for some regular cardinal λ, has λ-filtered colimits and a set of λ-presentable objects from which every object is obtainable as a λ-filtered colimit, making its large object class controllable by bounded presentability data.
  • Additive group — A group whose binary operation is interpreted and written as addition, especially the underlying usually abelian group obtained from a ring, field, vector space, or other multi-operation structure by forgetting its other operations.
  • Alternating group — The subgroup A_n of the symmetric group S_n consisting of all even permutations of n objects, equivalently the kernel of the sign homomorphism, with order n!/2 for n at least two.
  • Amicable triple — Three distinct natural numbers whose aliquot sum for each equals the sum of the other two, equivalently sharing a divisor sum equal to the triple’s total.
  • Automorphic number — A base-b natural number whose numeral reappears as the final k digits of its square, equivalently an idempotent n²≡n modulo b^k for k equal to its digit length.
  • Backward Compatibility — A newer technical version preserves a specified older interface or behavior so a legacy dependent can continue to work within stated limits.
  • Burnside category — An additive category of finite G-sets whose morphisms are group-completed equivalence classes of equivariant spans composed by pullback.
  • CLRg property — A common-limit-in-the-range condition for self-maps f and g: some sequence x_k has both f(x_k) and g(x_k) converge to the same point g(x), placing the shared limit in the range of g without requiring that range to be closed.
  • Coherency (homotopy theory) — In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism".
  • Difference of Two Squares — The algebraic identity a² − b² = (a − b)(a + b), converting a difference of commuting squares into conjugate factors.
  • Direct Product — A product of algebraic structures formed from all factor tuples with coordinatewise operations and structure-preserving projections.
  • Distributive Law Between Monads — A natural transformation λ:TS⇒ST coherent with both monads' units and multiplications, allowing the ordered composite ST to inherit a monad structure.
  • Dixon's identity — A family of binomial and hypergeometric summation identities associated with A. C. Dixon, including a terminating triple-binomial evaluation.
  • Elementary Amenable Group — A group obtainable from finite and abelian groups by isomorphism, subgroups, quotients, extensions, and directed unions.
  • Filtration (algebra) — An ordered family of subobjects of an algebraic structure whose stages are nested with the index order, often with operations respecting degree—for example F_i F_j ⊆ F_{i+j}—so complexity or information accumulates by level.
  • FinSet — The category whose objects are finite sets and whose morphisms are all functions between them, equivalent to the small skeleton FinOrd of finite ordinals and supporting finite products, coproducts, and exponentials.
  • List (computing) — A finite ordered collection whose positions remain distinct even when values repeat, exposed through sequence operations independently of its concrete array or linked representation.
  • Mac Lane's coherence theorem — In a monoidal category, every well-formed diagram composed only of associators and left or right unitors commutes; equivalently, every monoidal category is monoidally equivalent to a strict one.
  • Morley Rank — Morley rank measures recursive infinite definable splitting of a set in a first-order structure.
  • Nilpotent Operator — A linear self-map whose one fixed finite power is exactly the zero operator on its entire space.
  • Stationary set — In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory.
  • Strictification — A coherence-backed replacement of a weak categorical structure by a stricter presentation under a specified structure-preserving equivalence.
  • Supertransitive class — A transitive class A that also contains every subset of each member: whenever x belongs to A, the entire power set P(x) is included in A.
  • Weakly Inaccessible Cardinal — An uncountable regular limit cardinal: a cardinal greater than aleph-zero whose cofinality equals itself and which is not the successor of any smaller cardinal, without requiring the strong-limit property.