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In category theory, a quotient category is a way of constructing a new category from an existing one by identifying certain morphisms or objects according to some equivalence relation. This concept is somewhat analogous to the idea of quotient groups or quotient spaces in algebra and topology, where we partition a set based on an equivalence relation.

In the context of category theory, specifically within the study of abelian categories, the concept of a quotient object is an important one. A quotient object in an abelian category is a way to construct a new object by identifying certain elements or morphisms in a way that reflects the idea of "dividing" the objects by a subobject. ### Definitions 1.

In category theory, refinement generally refers to a process or concept that captures the idea of "smoothing out" or detailing a more general structure to a more precise or specific one. While the term "refinement" might not have a single, universally accepted definition within category theory, it is often used in the context of certain categorical constructs or frameworks.

In category theory, a **section** is a concept that arises in the context of functors, particularly when dealing with object mappings between categories. More formally, a section refers to a right inverse to a morphism. Here’s a more detailed breakdown of what this means: 1. **Categories and Functors**: In category theory, a category consists of objects and morphisms (arrows) between those objects.

A **Segal category** is a concept from higher category theory that serves as a generalization of the notion of a category in the context of higher-dimensional structures. Segal categories are particularly useful in the study of homotopy theory and simplicial sets. They provide a framework for understanding categories where morphisms between objects can themselves have a higher structure.

A **Segal space** is a concept from category theory and higher category theory that generalizes the notion of a space in a way suitable for homotopy theory and higher categorical constructions. It provides a framework for discussing "categories up to homotopy" without relying strictly on the standard notions of topological or simplicial spaces.

A semiautomaton is a concept used primarily in theoretical computer science and automata theory. It refers to a computational model that operates under rules that are less restrictive than those of a full automaton. While traditional automata, such as finite automata, have a complete set of states and transitions, a semiautomaton may not have all transitions defined for each state or may have an incomplete structure.

In category theory, a **sieve** is a concept used in the context of a category, particularly in relation to a given object within that category. It can be thought of as a way to describe certain collections of morphisms (arrows) that reflect a kind of "filtering" process.

Simplicial localization is a concept from algebraic topology and category theory that is concerned with the process of localizing simplicial sets or simplicial categories. The process is usually aimed at constructing a new simplicial set that reflects the homotopical or categorical properties of the original set while allowing one to "invert" certain morphisms or objects. ### Background Concepts 1. **Simplicial Sets:** A simplicial set is a combinatorial structure that encodes topological information.

A **simplicially enriched category** is an extension of the concept of a category that incorporates hom-sets enriched over simplicial sets instead of sets. To unpack this, let's recall a few concepts: 1. **Category**: A category consists of objects and morphisms (arrows) between those objects, satisfying certain composition and identity axioms. 2. **Enrichment**: A category is said to be enriched over a certain structure (like sets, groups, etc.

In category theory, the concept of a **skeleton** is a way to describe a certain kind of subcategory of a given category that retains important structural information while being more "minimal" or "simplified.

In mathematics, a "sketch" typically refers to a rough or informal outline of a mathematical concept, proof, or argument. It helps convey the main ideas without going into exhaustive detail. A sketch might include key steps, important definitions, or significant results, and can serve as a guide for further development into a full, rigorous presentation.

A **spherical category** is a concept that arises in category theory, particularly in the context of higher category theory and homotopy theory. It is generally defined as a type of category that allows for a notion of "spherical" or "n-dimensional" structures, facilitating the study of objects and morphisms in a more flexible way than traditional categories.

Stable model categories are a specific type of model category in which the homotopy theory is enriched with certain duality properties. They arise from the interplay between homotopy theory and stable homotopy theory, and they are particularly useful in contexts like derived categories and the study of spectra. A model category consists of: 1. **Objects**: These can be any kind of mathematical structure (like topological spaces, chain complexes, etc.).

In mathematics, the term "stack" typically refers to a specific kind of mathematical structure used in algebraic geometry and related fields. Stacks are a generalization of schemes that allow for more flexibility, particularly in situations where one needs to control not just global properties but also local symmetries and automorphisms. ### Key Concepts: 1. **Stacks vs.

A subcategory is a specific division or subset within a broader category. It helps to further classify or organize items, concepts, or data that share common characteristics. Subcategories allow for a more detailed and granular classification, making it easier to identify, analyze, or search for specific items within a larger group.

In category theory, a **subterminal object** is a specific type of object that generalizes the notion of a "singleton" in a categorical context. To understand it, let's first define a few key concepts: 1. **Category**: A category consists of objects and morphisms (arrows between objects) that satisfy certain properties (closure under composition, associativity, and identity).

The term "symplectic category" typically refers to a structure in the realm of symplectic geometry and can be related to the study of symplectic manifolds, which are a key concept in both mathematics and theoretical physics, particularly in the context of Hamiltonian mechanics. In the context of category theory, a category may be defined as "symplectic" if its objects and morphisms can be interpreted in terms of symplectic structures.

A T-structure is a concept from the field of category theory, a branch of mathematics that deals with abstract structures and relationships between them. In the context of derived categories, a T-structure provides a way to systematically organize complexes of objects.

Category theory, a branch of mathematics that deals with abstract structures and relationships between them, has a rich history intertwined with various mathematical disciplines. Here’s a timeline highlighting key developments in category theory and its related areas: ### Early Foundations (Pre-20th Century) - **19th Century**: Developments in algebra and topology set the stage for category theory.