Categories, Sheaves & Homotopy

← Back to Domain-Specific Families

Abstractions that use category theory to organize algebraic, geometric and homotopical structure — higher and monoidal constructions (2-categories, Day convolution, categorical trace), toposes and sheaf-theoretic geometry (ringed spaces, closed immersions, Gabriel–Rosenberg reconstruction), homological functors (delta, effaceable functors), and model-category formalisms (Joyal model structure, Quillen adjunction).

21 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.

• 2-Category — A strict higher category with objects, arrows between objects, and arrows between parallel arrows, composed by compatible vertical and horizontal laws.
• Age (Model Theory) — The isomorphism-closed class of finitely generated structures that embed into a fixed model, recording exactly its finite or finitely generated local patterns.
• Auslander–Reiten theory — A representation-theoretic framework organizing indecomposable modules and morphisms through almost-split sequences and Auslander–Reiten quivers.
• Categorical set theory — Set-theoretic foundations formulated or analyzed through category-theoretic objects, morphisms, and axioms such as ETCS.
• Categorical Trace — Close an endomorphism of a dualizable object in a symmetric monoidal category into an endomorphism of its unit.
• Closed Immersion — A scheme morphism that identifies its source with a closed subscheme through locally surjective maps from ambient functions.
• Day Convolution — Lift an indexing category's tensor to a product of functors by left Kan extending their value-wise tensor along it.
• Delta Functor — A graded family of additive functors whose natural connecting maps turn short exact sequences into long exact sequences.
• Effaceable Functor — An additive functor for which every object embeds by a map whose induced functor morphism is zero.
• Exceptional Inverse Image Functor — Exceptional Inverse Image Functor is a recurring identity in formal models and representations, mathematics, logic, and statistics defined by this frozen evidence: In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves.
• Functor — A structure-preserving map between categories that carries objects and morphisms while respecting identities and composition — the two axioms that make it a licence to transport theorems from one category to another along its rails.
• Gabriel–Rosenberg Reconstruction Theorem — A categorical reconstruction theorem recovering a suitably separated scheme, including its topology and structure sheaf, from the abelian category of its quasi-coherent sheaves.
• Homotopy Category — A category that retains objects while replacing maps by homotopy classes or, more generally, formally inverting a designated class of weak equivalences.
• Initial and terminal objects — Initial and terminal objects are category-theoretic universal endpoints: an initial object has exactly one morphism to every object, while a terminal object has exactly one morphism from every object.
• Joyal Model Structure — The model structure on simplicial sets whose cofibrations are monomorphisms, weak equivalences are categorical equivalences, and fibrant objects are quasi-categories.
• Normal Morphism — A normal monomorphism is a subobject map that actually arises as a kernel; dually, a conormal epimorphism arises as a cokernel.
• Quillen Adjunction — An adjoint pair between model categories whose left member preserves cofibrations and trivial cofibrations, equivalently whose right member preserves fibrations and trivial fibrations.
• Ringed Space — A topological space equipped with a sheaf of rings assigning locally compatible ring-valued algebraic data to every open set and restriction maps to inclusions.
• Simplicial Presheaf — A simplicial presheaf is a contravariant functor from a category to simplicial sets, equivalently a simplicial object in set-valued presheaves, combining sectionwise homotopy data with functorial restriction.
• Synthetic differential geometry — A topos-theoretic formalization of differential geometry that encodes smooth infinitesimal behavior synthetically rather than through classical limit analysis.
• Topos — A category with finite limits, exponentials and a subobject classifier, providing a categorical universe in which objects, maps and internal predicates can be studied.