Category Theory For Beginners: Adjoint Functors

Описание: We start with the homset based definition of an adjunction. We introduce adjoint functors using preordered sets and Galois connections, and hence give examples of adjunctions between monotonic functions. We give an alternative description of adjunctions via universal morphisms, and describe how this works. We describe lots of theory behind adjoint functors, and how they relate to natural transformations like the unit and co-unit. We also give many examples of adjoint functors. In particular we discuss the co-product functor, the diagonal functor, the product functor, the exponential functor, free categories, and adjoint functors related to discrete functors (which relate to connectivity), as well as free/cofree (chaotic) functors for dynamical systems. The later is related to this online question:

https://math.stackexchange.com/questi...

A great guide to the later topics is the second addition of Lawvere's Conceptual Mathematics (although just the first addition seems easily accessible online):

https://img.4plebs.org/boards/tg/imag...

A slight correction for time 2:33:56 is that, the exponential functor is just the right adjoint of the functor which does the product of things by a fixed object, whereas the picture gives the impression that the exponential functor is the right adjoint of the functor that does the product when both input objects are variable.

In the description of my video Category Theory For Beginners: Yoneda Lemma I started to give links to other unlisted youtube videos I made, mostly following my working through Colin Mclarty's Elementary Categories, Elementary Toposes.

I ran out of space in the description, so I put links to the rest of the videos below

Extensions and arrows
   • Extensions and arrows  


Initial objects and negation 1
   • Initial objects and negation 1  


Initial objects and negation 2
   • Initial objects and negation 2  


Initial objects and negation 3
   • Initial objects and negation 3  

Coproducts in Toposes 1
   • Coproducts in Toposes 1  

Coequalizers in Toposes
   • Coequalizers in toposes  

Fuzzy steps towards the fundamental theorem
   • Fuzzy steps towards the fundamental theorem  

First steps in synthetic differential geometry
   • First steps in synthetic differential geom...  

Forgetful and product slice functors
   • Forgetful and product slice functors  

More slice functors and the fundamental theorem
   • More slice functors and the fundamental th...  

More about the fundamental theorem
   • More about the fundamental theorem of topo...  

philosophical implications of topos theory
   • Philosophical implications of topos theory  

complements, decidables, booleans
   • Complements, decidables, boolean toposes, ...  

External semantics early stage
   • External semantics early stage  

Fundamentals of topos logic 1
   • Images and fundamentals of topos logic 1  

Fundamentals of topos logic 2
   • Images and fundamentals of topos logic 2  

Fundamentals of topos logic 3
   • Images and fundamentals of topos logic 3  

Fundamentals of topos logic 4
   • Images and fundamentals of topos logic 4  

Fundamentals of topos logic 5
   • Images and fundamentals of topos logic 5  

Even more external semantics
   • Even more external semantics  

Categories in Toposes 1
   • Categories in toposes 1  

Categories in Toposes 2
   • Categories in a topos 2  

Natural number objects 1
   • Natural number objects 1  

Natural number objects 2
   • Natural number objects 2  

Topos research goals
   • Topos research goals  

Natural number objects 3
   • Natural number objects 3  

Sets and topologies briefly
   • Sets and topologies briefly  

Dynamical systems 1
   • Dynamical system applications 1  

Dynamical systems 2
   • Dynamical systems applications 3  

Dynamical systems 3
   • Dynamical systems applications 3  

Dynamical systems 4
   • Dynamical system applications 4  

Dynamical systems 5
   • Dynamical systems applications 5  

Dynamical systems 6
   • Dynamical system applications 6  

Dynamical systems 7
   • Dynamical system applications 7  

Infinitely rich objects
   • Infinitely rich objects  

Graph neighborhood problem 1
   • Graph neighborhood problem 1  

Graph neighborhood problem 2
   • Graph neighborhood problem 2  

Infinitely rich objects 2
   • Infinitely rich objects 2  

Introducing Heyting algebras
   • Introducing Heyting Algebras  

Topos research goals 2
   • Topos research goals 2  

Introducing Heyting algebras 2
   • Introducing Heyting Algebras 2  

Introducing Heyting algebras 3
   • Introducing Heyting Algebras 3  

Relating the algebras with the fundamental theorem
   • Introducing Heyting Algebras 4