Functors

In programming, particularly in functional programming and type theory, a **functor** is a type that implements a mapping between categories. In simpler terms, it can be understood as a type that can be transformed or mapped over. ### Key Aspects of Functors 1. **Mapping**: Functors allow you to apply a function to values wrapped in a context (like lists, option types, etc.).

In category theory, a **representable functor** is a functor that is naturally isomorphic to the Hom functor between two categories. To understand this concept more fully, let's first break down some key elements. ### Basic Concepts 1. **Categories**: In category theory, a category consists of objects and morphisms (arrows) between those objects, satisfying certain properties.

In category theory, a **2-functor** is a generalization of a functor that operates between 2-categories. To understand what a 2-functor is, we need to break down some concepts. ### Categories A **category** consists of: - Objects - Morphisms (or arrows) between these objects that satisfy certain composition and identity properties. ### Functors A **functor** is a map between two categories that preserves the structure of those categories.

In category theory, an **amnestic functor** is a type of functor that exhibits a specific relationship with respect to the preservation of certain structures. The concept may not be as widely recognized as other notions in category theory, and it's important to clarify that terms might differ slightly based on the context in which they are used.

In mathematics, particularly in the field of category theory and homological algebra, derived functors are a way of extending the notion of a functor by capturing information about how it fails to be exact. ### Background In general, a functor is a map between categories that preserves the structure of those categories. An exact functor is one that preserves exact sequences, which are sequences of objects and morphisms that exhibit a certain algebraic structure, particularly in the context of abelian categories.

In category theory, a **diagram** is a mathematical structure that consists of a collection of objects and morphisms (arrows) between these objects that are organized in a specific way according to a directed graph. Diagrams capture relationships between objects in a category and can represent various mathematical concepts. ### Key Components of Diagrams: 1. **Objects**: In category theory, these are the entities or points that the diagram is composed of.

A dinatural transformation is a concept in category theory, specifically in the context of functors and natural transformations. It generalizes the notion of a natural transformation to situations involving two different functors that are indexed by a third category. In more detail, consider two categories \( \mathcal{C} \) and \( \mathcal{D} \), along with a third category \( \mathcal{E} \).

In category theory, a **dominant functor** is a specific type of functor that reflects a certain degree of "size" or "intensity" of structure between categories.

An **effaceable functor** is a concept from category theory, specifically within the context of derived categories and triangulated categories. Although the term may not be widely known, it generally relates to functors that, under certain conditions, can be "ignored" or "factored out" in some sense without losing too much structure or information.

In category theory, the concept of an **end** is a particular construction that arises when dealing with functors from one category to another. Specifically, an end is a way to "sum up" or "integrate" the values of a functor over a category, similar to how an integral works in calculus but in a categorical context.

In category theory, an **essentially surjective functor** is a specific type of functor that relates to the structure of the categories involved. Let \( F: \mathcal{C} \to \mathcal{D} \) be a functor between two categories \( \mathcal{C} \) and \( \mathcal{D} \).

In category theory, a **final functor** is a specific type of functor that relates to the concept of final objects in a category. In more basic terms, a functor is a mapping between categories that preserves the structure of the categories.

In category theory, a **forgetful functor** is a type of functor that "forgets" some structure of the objects it maps from one category to another. More specifically, it typically maps objects from a more structured category (e.g., a category with additional algebraic or topological structure) to a less structured category (like the category of sets). ### Examples 1.

In category theory, the concepts of full and faithful functors relate to the ways in which a functor preserves certain structures between categories.

In the context of computer science, a **functor** is a design pattern that originates from category theory in mathematics. It is a type that can be mapped over, which means it implements a mapping function that applies a function to each element within its context. ### In Programming Languages 1.

In category theory, the Hom functor is a fundamental concept used to describe morphisms (arrows) between objects in a category. Specifically, given a category \(\mathcal{C}\), the Hom functor allows us to examine the set of morphisms between two object types. ### Definition 1.

Ind-completion is a concept from the field of category theory, specifically related to the completion of a category with respect to a certain type of structure or property. In mathematical contexts, "ind-completion" often refers to a way of completing a category by formally adding certain limits or colimits.

In category theory, a natural transformation is a concept that describes a way of transforming one functor into another while preserving the structure of the categories involved.

A **polynomial functor** is a concept from category theory, particularly in the field of algebraic structures in categories. It provides a structured way to describe functors that have a form similar to polynomial expressions. ### Definition In simple terms, a polynomial functor can be viewed as a functor that combines different types of "operations" such as sums and products, much like a polynomial combines variables with coefficients using addition and multiplication.

In category theory, a presheaf is a structure that assigns data to the open sets of a topological space (or more generally, to objects in a category) in a way that respects the relationships between these sets (or objects). More formally, a presheaf can be defined as follows: ### Definition: Let \( C \) be a category and \( X \) a topological space (or a more abstract site).

A **profunctor** is a concept that arises in category theory, which is a branch of mathematics. It is a generalization of a functor. Specifically, a profunctor can be understood as a type of structure that relates two categories. You can think of a profunctor as a functor that is "indexed" by two categories.

A **pseudo-functor** is a generalization of the concept of a functor in category theory, designed to handle situations where some structure is retained but strictness is relaxed. In formal category theory, functors map objects and morphisms from one category to another while preserving the categorical structure (identity morphisms and composition of morphisms). Pseudo-functors, however, allow for certain flexibility in this structure.

In mathematics, particularly in the field of representation theory and algebra, a **Schur functor** is an important concept that arises in the context of polynomial functors. Schur functors are used to construct representations of symmetric groups and to study tensors, modules, and various other algebraic structures.

In category theory, a **smooth functor** often refers to a functor that preserves certain structures in a way analogous to smooth maps between manifolds, though the term can vary based on context. In the context of differential geometry, a smooth functor is typically one that operates between categories of smooth manifolds and smooth maps. A functor between two categories of smooth manifolds is called smooth if it preserves the smooth structure of the manifolds and the smoothness of the maps.

In category theory, the term "span" refers to a particular type of diagram involving two morphisms that "span" a common object. More formally, a span consists of two objects \( A \) and \( B \) and a third object \( C \) along with two morphisms \( f: A \to C \) and \( g: B \to C \).

In category theory, a **subfunctor** is a concept that extends the idea of a subobject to the context of functors. While subobjects represent "parts" of objects in a category, subfunctors represent "parts" of functors in a more structured manner. ### Definition Let \( F: \mathcal{C} \to \mathcal{D} \) be a functor.

In the context of category theory, a translation functor is not a standard term, and its meaning might depend on the specific field of mathematics involved. However, we can interpret it in a few related contexts: 1. **Translation in Topology or Algebra**: In a topological or algebraic setting, one might consider a functor that shifts or translates structures from one category to another.

The Zuckerman functor, often denoted as \( Z \), is a construction in the realm of representation theory, particularly in the context of Lie algebras and their representations. It is named after the mathematician Greg Zuckerman, who introduced it in relation to the study of representations of semisimple Lie algebras. The Zuckerman functor is a method for producing certain types of representations from a given representation of a Lie algebra.