[[Mathematics MOC]]
# Category Theory MOC
Category theory, 
affectionately called "General Abstract Nonsense" by many mathematicians.

[[History of Category Theory]]

## Objects

The central object of category theory is, of course, the [[Category]].
We can reason about objects and morphisms in a category using a [[Commutative diagram]].

### Classification

See [[types of category]].

### Additional structure

- Category + Tensor product = [[Monoidal category]]
- Category + Internal hom = [[Closed category]]
## Internal constructions

- [[Universal construction]]
  - [[Limits and colimits]]
  - [[Initial and terminal objects]]
  - [[Products and coproducts]]

## $n$-morphisms of categories

- [[Functor]], [[Natural transformation]]

## External constructions

- [[Quotient category]]
- [[Free category]]
- [[Product category]]
- [[Subcategory]]
- [[Comma category]]


## Categorification

- [[Categorification]] (Vertical)
- [[Oidification]] (Horizontal)

## Categorical foundations

- [[ETCS]]

### Issues

- [[Russell's paradox for categories]]

## Bibliography

- [[@awodeyCategoryTheory2010]]
- [[@milewskiCategoryTheoryProgrammers2019]]
- [[@maclaneCategoriesWorkingMathematician1978]]

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