---
date: 2021-10-18 18:54
modification date: Monday 18th October 2021 18:54:00
title: Intro Category Theory Talk
aliases: [Intro Category Theory Talk]
---

Tags: #todo #projects/my-talks  #higher-algebra/category-theory 

# Intro Category Theory Talk

- Definition: **category** (objects, morphisms, composition)
- Big list of examples
	- $\Grp, \mods{R}, \kalg, \Top, \LRS, [\cat C, \cat D]$,
	- The free category on a poset
	- $\Finset_+$, the category [Finset](Unsorted/Finset.md) of finite linearly ordered sets with objects of the form $[n] = \ts{0, \cdots, n}$.
	- $\Delta$ the [simplex category](simplex%20category) : finite *totally ordered* sets.
	- The groupoid associated to a group, general groupoids
	- $\Grpd, \Cat$
	- $\Open(X)$ for $X\in \Top$
	- $\smooth\Mfd$
	- $\Sch$, $\Sch\slice S$
- Definition: **functor**
	- Examples:
		- $\pi_1: \Top \to \Grp$
		- $\pi_*: \Top \to \gr_\ZZ \Grp$
- Definition: **Natural transformations** 
	- Interpretation as morphisms in $[\cat C, \cat D]$.
- Definition: **isomorphism** of objects vs **equivalence** of objects
- Equivalence of categories:
	- Definition [essentially surjective](essentially%20surjective)
	- Definition: [full functor](full%20functor)
	- Definition: [faithful functor](faithful%20functor.md)
	- Definition: [equivalence of categories](equivalence%20of%20categories.md)
	- Philosophy: [isomorphism vs equivalence](isomorphism%20vs%20equivalence)
- Definition: **adjunction**
	- Definition: [unit of an adjunction](unit%20of%20an%20adjunction) and counit
	- Examples:
		- [tensor-hom adjunction](tensor-hom%20adjunction)	
			- [Cartesian closed category](Cartesian%20closed%20category.md)
		- [free-forgetful adjunction](free-forgetful%20adjunction)
		- [Frobenius reciprocity](Frobenius%20reciprocity)
		- [restriction and extension of scalars adjunction](restriction%20and%20extension%20of%20scalars%20adjunction)
		- Definition: [adjoint equivalence](adjoint%20equivalence)
		- [adjoint functor theorem](adjoint%20functor%20theorem.md)
- Useful constructions
	- Initial and terminal objects
	- Universal properties
		- Quotient group or quotient topology
		- Tensor product of modules
	- Product and coproduct
	- Pullback and pushout
	- [slice category](slice%20category) and [under category](under%20category)
		- [cone category](cone%20category.md)
	- [Colimit](Colimit.md) and limit
		- As initial/terminal objects in cone category
		- [Unsorted/RAPC](Unsorted/RAPC.md) and [LAPC](LAPC)
	- [coequalizer](coequalizer.md) and equalizer
		- Example: [sheaf](Unsorted/sheaf.md) and [stacks MOC](Unsorted/stacks%20MOC.md) conditions, [descent](Unsorted/descent.md)
- Definition: [Yoneda embedding](Yoneda%20embedding)
	- Definition: [representable functor](representable%20functor)
	- Philosophy: [functor of points](functor%20of%20points.md)

# Further Topics

- [reflective](reflective) and coreflective subcategories
- [cocontinuous functor](cocontinuous%20functor) and continuity of $\Hom$.
- [cocomplete category](cocomplete%20category)
- [monad](monad.md), [algebra over a monad](algebra%20over%20a%20monad), and [Beck's monadicity theorem](Beck's%20monadicity%20theorem)
- [monoid object](monoid%20object.md)
- [monoidal category](monoidal%20category.md)
- [cofinal functor](cofinal%20functor.md)
- [filtered category](filtered%20category) and [filtered colimits](filtered%20colimits)