I don't think you need to go as far as the category of all categories to find an example of a large category. Most math students and even most working mathematicians may have never met that example, and will not find it particularly motivating. Why not consider the collection of all groups, or the collection of all vector spaces. It is too big to be a set, for the same reason as Russell's paradox. But it is a good setting to do a big chunk of mainstream mathematics, and especially to talk about invariants in those "collections" or categories. It saves a lot of work of get results at the level of abstract vector spaces, with linear transformations as morphisms, rather than get bogged down in the overwhelming detail of matrix algebra. Looking at math from the outside (I work on computational linguistics, which gets me interested in category theory without having much background in analysis), it seems that a lot of interesting modern work that pushes the boundaries is done at the more general level of categories and their morphisms and functors, rather than fixing a particular set down to elements and extensional equality. If math can discover nuggets of insight from sturctures "up to isomrophism," this has a good chance of being useful across a range of applications. Working at that level of generality is not something mathematicians, or scientists, will not want to give up for the sake of staying grounded in a particular set (or hierarchical universe of sets, such as ZFC). Grounding can come later, when you have some interesting insights and you are wondering to what extent it is applicable in a certain domain. More work, beyond category-theoretical results, will be needed in a particular domain (e.g. to construct an isomorphism that actually works in your problem domain), but the top-down approach is interesting and useful, even if it doesn't address every aspect (or even most aspects) of math and applications. An interesting video, please come back to these issues in future posts.
In Algebra by Hungerford, he also claims that this paradox can be omitted by identifying a proper class as a collection of objects that is not a set. So then we can say that a collection of all sets is a proper class.
A slightly more categorical implementation of this approach is to use Grothendieck universes, a bit like adding a few large cardinal axioms. There is a somewhat advanced category theory video by Mike Shulman that introduces Grothendieck universes in its early part ua-cam.com/video/9MLQDOpy190/v-deo.html. Or you can do what the Stacks projects does, and just assume that a few specific large categories exist, if that is all you need in the math you are developing. No need to solve the problem in general, it might not be the right place and time to address such general issues.
No. I think there's a substantive point to be made here regarding (for lack of a better word) containerization in mathematics. Namely, you start with sets but if you want a mathematical object that includes "all sets", you need a new mathematical construct or else you run into a contradiction. Similarly with small/large categories (i.e. you can create a category that includes all small categories, but it itself is no longer a small category). I admit perhaps we're sweeping the difficulties under the rug, because what kind of mathematical object can include all large categories?
@@infinitedimensions9436 Yeah I was wondering the exact same thing. We're playing a semantic trick IMO in the sense that we're just moving up the problem one level.
@@alganpokemon905 Not really. A set and a category would have two different natures. Something cannot contain itself, thus a "set of all sets", which would be a set, meaning that it contains itself, cannot exist.
Excellent. Please make more videos on category theory.
What software are you drawing on?
I'm using Microsoft whiteboard with a Wacom tablet.
I don't think you need to go as far as the category of all categories to find an example of a large category. Most math students and even most working mathematicians may have never met that example, and will not find it particularly motivating. Why not consider the collection of all groups, or the collection of all vector spaces. It is too big to be a set, for the same reason as Russell's paradox. But it is a good setting to do a big chunk of mainstream mathematics, and especially to talk about invariants in those "collections" or categories. It saves a lot of work of get results at the level of abstract vector spaces, with linear transformations as morphisms, rather than get bogged down in the overwhelming detail of matrix algebra.
Looking at math from the outside (I work on computational linguistics, which gets me interested in category theory without having much background in analysis), it seems that a lot of interesting modern work that pushes the boundaries is done at the more general level of categories and their morphisms and functors, rather than fixing a particular set down to elements and extensional equality. If math can discover nuggets of insight from sturctures "up to isomrophism," this has a good chance of being useful across a range of applications. Working at that level of generality is not something mathematicians, or scientists, will not want to give up for the sake of staying grounded in a particular set (or hierarchical universe of sets, such as ZFC). Grounding can come later, when you have some interesting insights and you are wondering to what extent it is applicable in a certain domain. More work, beyond category-theoretical results, will be needed in a particular domain (e.g. to construct an isomorphism that actually works in your problem domain), but the top-down approach is interesting and useful, even if it doesn't address every aspect (or even most aspects) of math and applications.
An interesting video, please come back to these issues in future posts.
In Algebra by Hungerford, he also claims that this paradox can be omitted by identifying a proper class as a collection of objects that is not a set. So then we can say that a collection of all sets is a proper class.
A slightly more categorical implementation of this approach is to use Grothendieck universes, a bit like adding a few large cardinal axioms. There is a somewhat advanced category theory video by Mike Shulman that introduces Grothendieck universes in its early part ua-cam.com/video/9MLQDOpy190/v-deo.html. Or you can do what the Stacks projects does, and just assume that a few specific large categories exist, if that is all you need in the math you are developing. No need to solve the problem in general, it might not be the right place and time to address such general issues.
@@fbkintanar Thanks!
Is this not just a semantic trick?
No. I think there's a substantive point to be made here regarding (for lack of a better word) containerization in mathematics. Namely, you start with sets but if you want a mathematical object that includes "all sets", you need a new mathematical construct or else you run into a contradiction. Similarly with small/large categories (i.e. you can create a category that includes all small categories, but it itself is no longer a small category).
I admit perhaps we're sweeping the difficulties under the rug, because what kind of mathematical object can include all large categories?
@@infinitedimensions9436 Yeah I was wondering the exact same thing. We're playing a semantic trick IMO in the sense that we're just moving up the problem one level.
@@alganpokemon905 Not really. A set and a category would have two different natures. Something cannot contain itself, thus a "set of all sets", which would be a set, meaning that it contains itself, cannot exist.
Yes I agree category theory does resolve the paradox. And this is codified explicitly in Lawvere's fixed point theorem.