maths.free › Category Theory › Categories, functors and natural transformations

Categories, functors and natural transformations

Objects and arrows, composition, and maps between categories.

A category is objects plus arrows that compose associatively with identities: sets and functions, groups and homomorphisms, spaces and continuous maps. A functor maps one category to another preserving composition. Picture it: a commutative diagram, a square of arrows where both routes agree. Think it: "naturality" makes precise what it means for a construction to involve no arbitrary choices.

工作范例 · {1,2} union {3}

Set-Theory · Sets

{1,2} union {3}

一步

  1. A = \left\{1, 2\right\},\quad B = \left\{3\right\}

    The two sets.

  2. A \cup B = \left\{1, 2, 3\right\}

    Keep everything in either set.

  3. |A| = 2,\ |B| = 1,\ |A \cup B| = 3

    Cardinalities. Note |A ∪ B| = |A| + |B| − |A ∩ B|.

发送答案
A \cup B = \left\{1, 2, 3\right\}

现在轮到你 选择问题, 输入或绘制您自己的 。 每一步, 图片, 答案隐藏到您询问 。

保持你自己的工作

免费账户在每一课上都添加注解, 记录你完成的, 在一个地方解决的, 以及您可以询问的关于此页的导师。 数学本身向每个人开放, 是否签名 。

签名上签名 登录

此处所用符号

输入完整定义、图片和其中每个字母的含义的符号。

如何: Categories, functors and natural transformations

  1. The two sets.
  2. Keep everything in either set.
  3. Cardinalities. Note |A ∪ B| = |A| + |B| − |A ∩ B|.

人们对人们提出的问题

What is a functor?

A map between categories that sends objects to objects and arrows to arrows while respecting composition. Taking the fundamental group of a space is a functor from spaces to groups.

更多 Category Theory

Universal properties, limits and adjunctions