Category Theory For Beginners: Adjoint Functors

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:

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):

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

Initial objects and negation 1

Initial objects and negation 2

Initial objects and negation 3

Coproducts in Toposes 1

Coequalizers in Toposes

Fuzzy steps towards the fundamental theorem

First steps in synthetic differential geometry

Forgetful and product slice functors

More slice functors and the fundamental theorem

More about the fundamental theorem

philosophical implications of topos theory

complements, decidables, booleans

External semantics early stage

Fundamentals of topos logic 1

Fundamentals of topos logic 2

Fundamentals of topos logic 3

Fundamentals of topos logic 4

Fundamentals of topos logic 5

Even more external semantics

Categories in Toposes 1

Categories in Toposes 2

Natural number objects 1

Natural number objects 2

Topos research goals

Natural number objects 3

Sets and topologies briefly

Dynamical systems 1

Dynamical systems 2

Dynamical systems 3

Dynamical systems 4

Dynamical systems 5

Dynamical systems 6

Dynamical systems 7

Infinitely rich objects

Graph neighborhood problem 1

Graph neighborhood problem 2

Infinitely rich objects 2

Introducing Heyting algebras

Topos research goals 2

Introducing Heyting algebras 2

Introducing Heyting algebras 3

Relating the algebras with the fundamental theorem

[simka]

Today I wished someone would explain adjoin functors to me like I am 5 years old (or like I am an undergrad) . Thank you for that, I liked that style of the lecture :)

[connorfrankston]

Thank you, Richard, you’re a patient and insightful man, and your work is much appreciated.

[unsightedmath]

adjoint explanation is start from 11:50, thanks for such work sir. :)

[Fluffyunny]

Thank you so much for all of your work. This is extremely helpful.

[dmitrynik]

Thanks a lot! Great explanation. Appreciate your work in making this and other videos on your channel.

[ryanjbuchanan]

I have searched everywhere for an explanation... ripped my hair out. Then I watch this video. Thank you Sir Southwell.

[him]

Thanks so much for the huge list of examples!

[TwoDogSay]

Consider life as an animating functor on matter and death as a functor working in reverse;
The notions of adjunct functors and terminal morphism could be interesting to define by asking the question:
Is life a Left or Right adjunct functor to death? - where Life is a functor animating a set of inert and random molecules into some order that dissipates back into another set of inert molecules, disorderly and random, under the functor Death? Clearly, they cannot be isomorphisms when they operate not simply on different sets but also on different things. So, they are arrows between different categories, where in one the organisms behave purposefully and seek order in change to ensure survival, and in the other the molecules act randomly in the direction of more disorder. If Life defines some kind of terminal morphism from the perspective of the living toward inert matter (in the category, say, particle motion), then I would be inclined to define life as a Right adjunct functor to death.

[msk.nfazal]

i am so much thankful to you for this video sir. may God bless you.

[opendiagrams]

Thank you so much for this series of videos and your book, they helped me a lot !
At 3:24:27 you describe how the coproduct is the left adjoint to the diagonal which is the left adjoint to product which in turn is the left adjoint to the exponential.
Is there something like a co-exponential ? (which might be a left adjoint to the coproduct, to complete the symmetry of the diagram ?)
Thanks

[adityaprasad]

Hmm at 19:00, L(q) <= L(p) but q <= p is false. I think you want a <=b => L(a) <= L(b). (That is, one-directional implication.)

Thanks for your excellent videos! They've been really helpful. Especially the one on Yoneda!

[ianc]

At 1:22 (for example), where you have (pi1, pi2) o delta(h) = (f o g) o (pi1bar, pi2bar), is that (f o g) in fact meant to be (f, g) ?
(Later edit: Oh, ok, I see at 1:31 you fixed that - thanks).

[jeffreyhowarth]

after watching these videos, Mathproofsable's videos on topos theory look less intimidating.

[newlandsvalley]

At 1:16:10, I don't understand how (𝝅1, 𝝅2) acting on (a*b, a*b) can produce (a, b), Isn't this begging the question? Or are these just arbitrary labels at the moment?

[him]

Could you please leave a comment to describe what you mean by a dynamical system, and how the morphisms are defined? It’s clear you don’t mean a topological or measure-preserving system such as we might consider in ergodic theory so what else?

[hyperduality]

Symmetry is dual to anti-symmetry, duality is being conserved.
Duality = self intersection = Klein bottle ( The left handed mobius loop is dual to the right handed mobius loop).
Energy is duality, duality is energy.
The future is dual to the past, time asymmetry or time duality.