---
date: 2022-01-15 21:49
modification date: Wednesday 16th February 2022 18:44:15
title: modern category theory
aliases: [category theory, category, categories]

---


---

- Tags 
	- #cats #higher-algebra #MOC 
- Refs: #resources 
	- <https://arxiv.org/pdf/1912.10642.pdf> #resources/papers 
	- <http://web.stanford.edu/~truax/notes/Category_Theory.pdf> #resources/notes 
	- From categories to homotopy: <https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf> #resources/books 
	- Chapter 1 <https://mandal.ku.edu/M996QuillenKTheory/0TheKBooK.pdf> #resources/books 
	- [Morgan Opie's wonderful notes](https://www.math.ucla.edu/~mopie/Cats1.pdf) #resources/notes 
		- Local copy: [attachments/Cats1.pdf](attachments/Cats1.pdf) #resources/notes 
	- [attachments/cattheory_partiii_notes.pdf](attachments/cattheory_partiii_notes.pdf) #resources/notes 
	- [attachments/Category Theory Course.pdf](attachments/Category%20Theory%20Course.pdf) #resources/notes 
	- <https://hal.archives-ouvertes.fr/hal-02184208/document> #resources/notes 
	- <https://www.asc.ohio-state.edu/green.2116/homotopyTheoryReadingCourse2020.pdf> #resources/notes 
- Links:
	- [classical category theory](Unsorted/classical%20category%20theory.md)
	- [Unsorted/internal category](Unsorted/internal%20category.md)

---

# modern category theory


For DAG:

- [compact](Unsorted/compact%20object%20of%20a%20category.md)
- [reflexive pair](Unsorted/reflexive%20pair.md)
- [filtered colimits](filtered%20colimits)
- [conservative functor](conservative%20functor)

# Topics

- Constructions of new categories:
	- [slice category](slice category)
	- [comma category](comma category)
- Structure on categories:
	- [triangulated categories](triangulated%20categories.md)
	- [Tensored Category](Tensored Category.md)
	- [DG category](DG%20category.md)
	- [Unsorted/enriched category](Unsorted/enriched%20category.md)
	- [pointed category](pointed category)
- Extending functors:
	- [Kan extension](Kan%20extension.md)
	- [coend](coend.md)
	- [Day convolution](Day%20convolution.md)
- Structure on functors
	- [continuous functor](continuous functor)
	- [continuous functor](continuous functor)
- [span](span)
- Object function of an equivalence of categories?

## Internal Objects

- [monoid object](monoid%20object.md)
- [Unsorted/group object](Unsorted/group%20object.md)
- [ring object](ring object)
- [module object](module object)
- [vector space object](vector space object)
- [algebra object](algebra object)
- [simplicial object](simplicial%20object)


## Monoidal
- [dualizable object of a category](dualizable object of a category.md)
- [invertible object of a category](invertible object of a category.md)
- [center in a category](center in a category)
- [compact object of a category](compact%20object%20of%20a%20category.md)
	- [rigid (objects in a category)](Unsorted/rigid%20(objects%20in%20a%20category).md)
- [Generators of a category](Generators%20of%20a%20category.md)
- [linear category](linear%20category.md)
- [fiber functor](fiber%20functor.md)
- [tannakian](tannakian)
- [filtered objects of a category](filtered objects of a category)
- [graded objects of a category](graded objects of a category)
- [Calabi-Yau object of a category](Calabi-Yau object of a category.md)

## Homotopy 

^72ca2f
- [homotopy colimit](homotopy colimit)
- [homotopy quotient](homotopy quotient)
- [homotopy fixed points](homotopy fixed points)

## Higher Cats
- [higher category](higher%20category.md)
- [infinity categories](infinity%20categories.md)
- [stable infinity category](stable infinity category)
- [presentable category](presentable category)
- [DG category](DG%20category.md)
- [graded category](graded category)
- [whiskering](whiskering.md)

## Representation Theory

- [blocks of a category](blocks of a category)
- [semiorthogonal decomposition](semiorthogonal decomposition)
- [quivers](quivers)
- [semisimple category](semisimple%20category.md)