Closed categories

In category theory, a branch of mathematics, a **closed category** typically refers to a category that has certain characteristics related to products, coproducts, and exponentials. However, the term "closed category" can have different interpretations, so it's important to clarify the context. One common context is in the classification of categories based on the existence of certain limits and colimits. A category \( \mathcal{C} \) is said to be **closed** if it has exponential objects.

An *-autonomous category is a concept from category theory, specifically in the context of categorical logic and type theory. More formally, a category \( \mathcal{C} \) is said to be *-autonomous if it has a structure that allows for a notion of duals and exponential objects that satisfies certain properties.

A **Cartesian closed category** (CCC) is a type of category in the field of category theory, which is a branch of mathematics that studies abstract structures and their relationships. A category is defined by a collection of objects and morphisms (arrows) between these objects, satisfying certain axioms.

In category theory, a **closed category** typically refers to a category that has certain properties analogous to those found in the category of sets with respect to the concept of function spaces.

A **closed monoidal category** is a specific type of category in the field of category theory that combines the notions of a monoidal category and an internal hom-functor. To break it down, let's start with the definitions: 1. **Monoidal category**: A monoidal category \( \mathcal{C} \) consists of: - A category \( \mathcal{C} \).

A **compact closed category** is a concept from category theory, a branch of mathematics that deals with abstract structures and relationships between them. Compact closed categories provide a framework in which one can model concepts from topology, linear logic, and quantum mechanics, among other fields. Here are some key features and definitions related to compact closed categories: 1. **Categories**: A category consists of objects and morphisms (arrows) between those objects, where morphisms must satisfy certain composition and identity properties.