Category Theory & Homotopical Algebra

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Abstractions about categorical structure and its homotopical refinements, spanning core categorical notions (functors, sections, adequate subcategories, isomorphism theorems), monoidal and operadic structures (Cartesian monoidal category, A∞-operad, strong monad), and homotopy-theoretic constructions (homotopy fiber, simplicial space, directed algebraic topology).

18 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.

• Adequate subcategory — In category theory, a branch of mathematics, an adequate subcategory of a category X is an analog of a dense subspace in topology for presheaves: namely, a subcategory i : A \hookrightarrow X such that the restriction of the Yoneda embedding X \hookrightarrow \mathbf{P}(X) along i is still fully faithful.
• A∞-operad — In mathematics, an A ∞ -operad is a type of operad used in algebraic topology and homotopy theory to describe algebraic structures where the property of associativity is loosened.
• Cartesian monoidal category — In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category.
• Compactly supported homology — In mathematics, a homology theory in algebraic topology is compactly supported if, in every degree n, the relative homology group H n (X, A) of every pair of spaces.
• Convergence Group — A group action on a compact space whose induced action on distinct triples is properly discontinuous, equivalently exhibiting subsequential collapse away from a repelling point in the metrizable case.
• Directed algebraic topology — In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction.
• Homotopy associative algebra — There is a notion of algebras, called A_\infty -algebras, which still have a property on the multiplication which still acts like the first relation, meaning associativity holds, but only holds up to a homotopy, which is a way to say after an operation "compressing" the information in the algebra, the multiplication is associative.
• Homotopy fiber — In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f:A \to B .
• Homotopy group with coefficients — In topology, a branch of mathematics, for i \ge 2 , the i-th homotopy group with coefficients in an abelian group G of a based space X is the pointed set of homotopy classes of based maps from the Moore space of type (G, i) to X, and is denoted by \pi_i(X; G) .
• Irreducible Component — Identify a maximal topological piece that cannot be expressed as the union of two proper closed pieces.
• Isomorphism theorem — In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients, homomorphisms, and subobjects.
• Lightface Pointclass — In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to be an element of some perfect Polish space.
• Partially Ordered Space — A partially ordered space is a topological space whose partial-order relation is closed in the product topology.
• Quillen's Theorem A — The two theorems play central roles in Quillen's Q-construction in algebraic K-theory and are named after Daniel Quillen.
• Scattered order — In mathematical order theory, a scattered order is a linear order that contains no densely ordered subset with more than one element.
• Section (category theory) — In category theory, a branch of mathematics, a section is a right inverse of some morphism.
• Simplicial space — In mathematics, a simplicial space is a simplicial object in the category of topological spaces.
• Strong monad — In category theory, a strong monad is a monad on a monoidal category with an additional natural transformation, called the strength, which governs how the monad interacts with the monoidal product.