---
title: Functors
description: Functors are structure-preserving mappings between different categories. Learners will understand how to translate objects and morphisms from one mathematical context to another, enabling the transfer of theorems and proofs across systems.
category: mathematics
subcategory: category-theory
difficulty: beginner, intermediate, advanced
url: /subject/functors
---

# Functors

Functors are structure-preserving mappings between different categories. Learners will understand how to translate objects and morphisms from one mathematical context to another, enabling the transfer of theorems and proofs across systems.

## Available Resources

1 Courses • 4 Websites

## Websites

### 1. Category Theory for Programmers

Free online book by Bartosz Milewski introducing category theory through Haskell and C++ examples, starting from types and functions and building to functors, natural transformations, adjunctions and monads. Readers learn to recognize categorical structures in everyday functional programming code.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/

**Tags:** category-theory, functors, functional-programming, haskell, monads

### 2. nLab: Functor

Reference entry from the nLab wiki defining functors between categories, with variants such as contravariant, full, faithful and essentially surjective functors, plus examples and properties. Useful for checking precise definitions and seeing how functors connect to wider higher-category theory.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/functor

**Tags:** category-theory, functors, natural-transformations, higher-category-theory

### 3. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

### 4. nLab

Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://ncatlab.org

**Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki

## Courses

### 1. Category Theory for Programmers

Learn functors and other category theory concepts applied to programming! This course by Bartosz Milewski makes abstract math practical.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PLVFrD1dmDdvcjCQDPhExqP56jqxp0Ssn_

**Tags:** category-theory, functors, functional-programming, haskell, natural-transformations

---

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