Me two lectures ago: In what twisted world is THIS "natural"?! Me after this one: Bloody hell. The categorical helicopter has just ascended into the stratosphere. Thank you, this is incredibly entertaining and, surprisingly, useful for loads of topics, homology for instance is way more understandable now. Also, "cocone" is my new favourite word :-)
When I understood (from the end of this video) that a limit is a terminal object in the comma category (Δ_↓D), I was overwhelmed! All the notions from your previous lectures (natural transformations, universality, etc.) joined together in limits. This construction is so powerful and versatile. Thank you for the impressive work on explaining these complex categorical concepts! I would not understand them only from books (I mean without your videos).
this is so much fun to watch! For the initial object exercise (as a limit to the Identity functor) Don't know if I did it right, but ended up with every other candidate being isomorphic to the initial object itself. Thanks again for those great videos.
When we take initial objects as limit of diagram when index category is C, do we assume that C is locally small as was required for Index category to be ?
Wow, these categorical constructions are mind-blown! I really like how they become more and more general. Are there any "limits" of this generalisation? :-) Thank you for sharing your knowledge with us!
Yes they really are amazing. In a way universal morphisms could be considered to be generalizations of limits. I am sure there are plenty of other mind blowing ways to generalize these ideas further. Maybe one could get nearly self referential by considering things like limits of the diagonal functor used in the demonstration that a limit is a universal morphism (just a random idea off the top of my head). There are also higher dimensional analogs of categories (which have objects (0-cells), arrows (1-cells), arrows between arrows (2-cells) [like natural transformations in Cat], etc. The ideas also appear there in a generalized setting. There is so much going on here already, so I've done my next video `Category Theory For Beginners: Everyday Language', which describes how limits appear in a variety of everyday life situations. Adjunctions are another high level concept involving a very pleasing interplay of the notions of limits, universal morphisms and functors,.
So natural transformations are transformations between parallel functors. I wonder now if there are transformations between anti parallel functors or what would be a not natural transformation. This was also the first part of the series where I really had the feeling that I start to understand what is going on.
Here are three. (1) Show that the equalizer and pullback (as I have defined them as limits), act like I claim they do, in the category set. (2) Create your own ontology log using ideas from `Category Theory For Beginners: Everyday Language', and identify some limits within. (3) Show that an initial object is a limit of an identity functor.
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Me two lectures ago: In what twisted world is THIS "natural"?!
Me after this one: Bloody hell. The categorical helicopter has just ascended into the stratosphere.
Thank you, this is incredibly entertaining and, surprisingly, useful for loads of topics, homology for instance is way more understandable now.
Also, "cocone" is my new favourite word :-)
I swear I had like four 'aha!' moments just in the first 15 minutes! Beautiful!
When I understood (from the end of this video) that a limit is a terminal object in the comma category (Δ_↓D), I was overwhelmed! All the notions from your previous lectures (natural transformations, universality, etc.) joined together in limits. This construction is so powerful and versatile. Thank you for the impressive work on explaining these complex categorical concepts! I would not understand them only from books (I mean without your videos).
this is so much fun to watch! For the initial object exercise (as a limit to the Identity functor) Don't know if I did it right, but ended up with every other candidate being isomorphic to the initial object itself. Thanks again for those great videos.
This section and the section on natural transformations are reminiscent to me of Grassmann’s description of “extensions.” Very cool.
When we take initial objects as limit of diagram when index category is C, do we assume that C is locally small as was required for Index category to be ?
Thank you so much for your video series! You are a great teacher!
I believe I found a typo around 39:00: Shouldn't it read βᵢ = πᵢ º h?
Wow, these categorical constructions are mind-blown! I really like how they become more and more general. Are there any "limits" of this generalisation? :-) Thank you for sharing your knowledge with us!
Yes they really are amazing. In a way universal morphisms could be considered to be generalizations of limits. I am sure there are plenty of other mind blowing ways to generalize these ideas further. Maybe one could get nearly self referential by considering things like limits of the diagonal functor used in the demonstration that a limit is a universal morphism (just a random idea off the top of my head). There are also higher dimensional analogs of categories (which have objects (0-cells), arrows (1-cells), arrows between arrows (2-cells) [like natural transformations in Cat], etc. The ideas also appear there in a generalized setting. There is so much going on here already, so I've done my next video `Category Theory For Beginners: Everyday Language', which describes how limits appear in a variety of everyday life situations. Adjunctions are another high level concept involving a very pleasing interplay of the notions of limits, universal morphisms and functors,.
rhank you so much for all the work you put into this. I'm really grateful
So natural transformations are transformations between parallel functors. I wonder now if there are transformations between anti parallel functors or what would be a not natural transformation.
This was also the first part of the series where I really had the feeling that I start to understand what is going on.
Are there any exercises i could complete to help me internalize/remember your lessons better?
Here are three. (1) Show that the equalizer and pullback (as I have defined them as limits), act like I claim they do, in the category set. (2) Create your own ontology log using ideas from `Category Theory For Beginners: Everyday Language', and identify some limits within. (3) Show that an initial object is a limit of an identity functor.