My favorite quote in this one (8:35): "So this is what picking means... our hunter-gather ancestors didn't realize that they were dealing with functors."
Thank you so much for putting these online - been watching them as a complement to your book and I love the joy and passion that you give to the subject - makes it so much easier to understand and intuit.
I love how excited you get towards the end of this video! I was feeling the same way, and I'm going to hurry along to the next lecture and have my mind blown some more! Thank you so much for these videos!
38:07 you're saying the morphisms in the category of cones are only those morphisms between the apices that make those triangles commute... but do they not already commute just due to the composition law in the category? Is that what you meant by the remark afterward, or is there some more specific condition that needs to be met?
Thanks for these great lectures! I feel like a "mind astronaut" being shot through higher and higher levels of abstraction :-). This is huge fun! One thing in this lecture still puzzles me: In the universal construction of the product, you emphasized that it is important that morphism m is unique. However, in universal construction of the limit, it seems the uniqueness constraint has vanished somehow... there even could be multiple ms (and cones) for each apex c, right? Which part did I get wrong?
Thank you for making this comprehensible. One note. In your earlier lecture on products, you used 'c' along with 'p' and 'q' as the "perfect" product, with c', p', and q' as the less optimal product and projections. In this lecture you used 'axb', 'fst' and 'snd' as the perfect product, and 'c', 'p', and 'q' as the less optimal. While it really shouldn't matter, switching notation can be confusing for those of us struggling to grasp the concepts :) Can't wait for the next one.
At 26:07, why is it necessary to explicitly state those triangles in the cone commute? Since the constant functor delta C collapses all structure of the source category into C I would assume naturality would always hold. Is there an example where it doesn't?
There's no reason why naturality would automatically hold. What happens is that the constant functor collapses one side of the naturality square to a single object and the identity morphism; but the result is a triangle, which still has to commute.
Does the "best" apex of a cone correspond to an arbitrary record type (or N-tuple) in Haskell? Where all the morphisms from the index category is just functions, and the identity morphism of the apex are lifted functions that operate instead on the record type? Edit: and conversely the cocone would be an arbitrary sum type Edit2: so for example if we have only Int and String in the index category (which would be (Int, String) as delta c) and the single function show :: Int -> String The identity morphism of delta c in C would contain the function showDC :: (Int, String) -> (Int, String) which would show the int and "store" it in the second part of the tuple. I.e. showDC :: (Int, String) -> (Int, String) showDC = \(i, s) -> (i, show i) And of course show will also be mapped to showDCD :: (Int, String) -> String and showD :: Int -> String as well
I am trying to understand this in the context of en.wikipedia.org/wiki/Universal_property#Limits_and_colimits If I think of Delta_() as a diagonal functor from C to [I,C] (so that Delta_a is the constant functor (which sends all objects to a) that results from applying Delta_() to object a in C), then am I correct in thinking that an arrow g in C(a,b) gets lifted by Delta_() to become Delta_g, which is a natural transformation which has each component equal to g ?
I've enjoyed all the lectures so far, but there was something particularly special about this one. I think it was your demeanor. Very enthusiastic and inspiring. I dread the moment that I finish this series and there'll be none more to see..
I just had a question about the base of the cones: since we are trying to make the commuting condition work, all the cones that we are talking about should have a common base right? I mean since we use the functor D to define the base, it would have to be that way ... right? I mean what would it look like to have a mapping between bases? Perhaps that's another chapter... An observation I had about the notation "Lim D": D is a Functor Lim D is an object in the target category C that is the apex of the limiting cone. I seem to associate capital letters with functors and categories, lower case letters with objects and morphisms, and Greek letters with natural transformation. The notation "Lim D" seems to grind some gears in that notation-to-concept association.
I still didn't understand limits after watching this video but I do think I understand now after reading some other resources and *really *studying the commutative diagram. Here's the idea that caused it to click for me: if a category is a C++ namespace of functions that derives a bunch of information from some initial value, the "limit" is the algebraic product of everything that can be derived from that initial value. It is literally the limit of everything you could infer from that value by using the category. So in this example, the limit could be represented as a data structure that accepts the initial value in its constructor and stores in its attributes all the values that were derived from knowing that initial value. It gives you all the information that you could get from the namespace without requiring you to work with the namespace directly, so it is effectively an abstraction layer that wraps the namespace.
I am afraid what abstraction levels to expect at the end of the third course :D Is getting quite crazy, but very interesting, with your lecturing approach it's like watching some movie with plot twists
At first I had trouble seeing how there could be different cones between the same ∆c and D. These cones would look the same on a diagram, unlike how ∆c and D themselves look different on a diagram (even though they're functors between the same pair of categories). But remember that different polymorphic functions between the same pair of data types are an example of different natural transformations between the same pair of functors. Ok I feel a bit silly now for not forcing myself to think of some common examples of different cones between the same ∆c and D. Here's part of Alon Amit's answer on Quora: "In the category of Sets, for example, D could simply be a diagram with two objects and no morphisms, so it’s just two sets X and Y, and A could be some other arbitrary set, and a cone from A to D is then just two set-functions A→X and A→Y."
Are you worried about sizes? As long as the indexing category is small and the target category is locally small, the set of natural transformations exists. For an example where it doesn't work, google "Freyd's adjoint functor theorem".
I love how much you enjoy this. Thank you for sharing your knowledge in such a joyful way :)
My favorite quote in this one (8:35): "So this is what picking means... our hunter-gather ancestors didn't realize that they were dealing with functors."
Means picking an object is equivalent to shooting an arrow to this object.
Mic drop moment
There is an isomorphism between this video and the movie Inception such that dreams are mapped to levels of abstraction.
Thank you so much for putting these online - been watching them as a complement to your book and I love the joy and passion that you give to the subject - makes it so much easier to understand and intuit.
I love how excited you get towards the end of this video! I was feeling the same way, and I'm going to hurry along to the next lecture and have my mind blown some more! Thank you so much for these videos!
Really thank you for these great lectures. I'm so eager to see next lecture.
38:07 you're saying the morphisms in the category of cones are only those morphisms between the apices that make those triangles commute... but do they not already commute just due to the composition law in the category? Is that what you meant by the remark afterward, or is there some more specific condition that needs to be met?
Thanks a lot for this lecture Bartosz. It is extremely transparent. I would love to have you as a professor.
Thanks for these great lectures! I feel like a "mind astronaut" being shot through higher and higher levels of abstraction :-). This is huge fun!
One thing in this lecture still puzzles me: In the universal construction of the product, you emphasized that it is important that morphism m is unique. However, in universal construction of the limit, it seems the uniqueness constraint has vanished somehow... there even could be multiple ms (and cones) for each apex c, right? Which part did I get wrong?
The key is multiple ms _and_ cones. But still only one m per cone.
Thank you for making this comprehensible.
One note. In your earlier lecture on products, you used 'c' along with 'p' and 'q' as the "perfect" product, with c', p', and q' as the less optimal product and projections. In this lecture you used 'axb', 'fst' and 'snd' as the perfect product, and 'c', 'p', and 'q' as the less optimal. While it really shouldn't matter, switching notation can be confusing for those of us struggling to grasp the concepts :)
Can't wait for the next one.
Thank you. I feel like I finally have an intuitive understanding of limits
I've been grappling with this idea in Awodey's textbook, and seeing it play out on the board helped *so* much. Thank you for this!
At 26:07, why is it necessary to explicitly state those triangles in the cone commute? Since the constant functor delta C collapses all structure of the source category into C I would assume naturality would always hold. Is there an example where it doesn't?
There's no reason why naturality would automatically hold. What happens is that the constant functor collapses one side of the naturality square to a single object and the identity morphism; but the result is a triangle, which still has to commute.
Got it now. I misread the naturality condition.
Thanks for these great lectures. My most successful attempt at CT so far.
Does the "best" apex of a cone correspond to an arbitrary record type (or N-tuple) in Haskell? Where all the morphisms from the index category is just functions, and the identity morphism of the apex are lifted functions that operate instead on the record type?
Edit: and conversely the cocone would be an arbitrary sum type
Edit2: so for example if we have only Int and String in the index category (which would be (Int, String) as delta c) and the single function show :: Int -> String
The identity morphism of delta c in C would contain the function showDC :: (Int, String) -> (Int, String) which would show the int and "store" it in the second part of the tuple. I.e.
showDC :: (Int, String) -> (Int, String)
showDC = \(i, s) -> (i, show i)
And of course show will also be mapped to showDCD :: (Int, String) -> String and showD :: Int -> String as well
The limit of the diagram Int--show->String is isomorphic to Int. Notice that the walls of the cone must commute. Also, identity morphism is \x->x.
I am trying to understand this in the context of en.wikipedia.org/wiki/Universal_property#Limits_and_colimits
If I think of Delta_() as a diagonal functor from C to [I,C] (so that Delta_a is the constant functor (which sends all objects to a)
that results from applying Delta_() to object a in C), then am I correct in thinking that an arrow g in C(a,b) gets lifted by Delta_()
to become Delta_g, which is a natural transformation which has each component equal to g ?
A week is too long to wait for each video. These are great, thank you for putting these up on TH-cam.
lucky that I can watch them all at once :)
Thank you !! excellent presentation and great energy!!!
I don't even know how I got here, but I just watched the whole video. It was amazing! Thanks for sharing.
I've enjoyed all the lectures so far, but there was something particularly special about this one. I think it was your demeanor. Very enthusiastic and inspiring. I dread the moment that I finish this series and there'll be none more to see..
I felt that too, this must be the first lecture where we're _really_ doing category theory.
I just had a question about the base of the cones: since we are trying to make the commuting condition work, all the cones that we are talking about should have a common base right? I mean since we use the functor D to define the base, it would have to be that way ... right?
I mean what would it look like to have a mapping between bases? Perhaps that's another chapter...
An observation I had about the notation "Lim D":
D is a Functor
Lim D is an object in the target category C that is the apex of the limiting cone.
I seem to associate capital letters with functors and categories, lower case letters with objects and morphisms, and Greek letters with natural transformation. The notation "Lim D" seems to grind some gears in that notation-to-concept association.
Right!
My favorite explanation is that limits are basically relational joins generalized to any mathematical objects.
I still didn't understand limits after watching this video but I do think I understand now after reading some other resources and *really *studying the commutative diagram. Here's the idea that caused it to click for me: if a category is a C++ namespace of functions that derives a bunch of information from some initial value, the "limit" is the algebraic product of everything that can be derived from that initial value. It is literally the limit of everything you could infer from that value by using the category. So in this example, the limit could be represented as a data structure that accepts the initial value in its constructor and stores in its attributes all the values that were derived from knowing that initial value. It gives you all the information that you could get from the namespace without requiring you to work with the namespace directly, so it is effectively an abstraction layer that wraps the namespace.
I am afraid what abstraction levels to expect at the end of the third course :D Is getting quite crazy, but very interesting, with your lecturing approach it's like watching some movie with plot twists
At first I had trouble seeing how there could be different cones between the same ∆c and D. These cones would look the same on a diagram, unlike how ∆c and D themselves look different on a diagram (even though they're functors between the same pair of categories). But remember that different polymorphic functions between the same pair of data types are an example of different natural transformations between the same pair of functors.
Ok I feel a bit silly now for not forcing myself to think of some common examples of different cones between the same ∆c and D. Here's part of Alon Amit's answer on Quora: "In the category of Sets, for example, D could simply be a diagram with two objects and no morphisms, so it’s just two sets X and Y, and A could be some other arbitrary set, and a cone from A to D is then just two set-functions A→X and A→Y."
Really wonderful! Can't wait for more :D
Isn't m a natural transformation from Δc →Δd?
Dear Bartosz, thank you for the great lecture! However, I am stgruggling to prove that Nat(delta_c, D) form a set.
Are you worried about sizes? As long as the indexing category is small and the target category is locally small, the set of natural transformations exists. For an example where it doesn't work, google "Freyd's adjoint functor theorem".
For every category you could think of, there exists a terminal object. That’s a fun game to play
My brain just reached a limit and melted. I guess hence the name :D