What an amazing video! I'm studying category theory for the first time and it is so hard to find intuitive notions behind the definitions but this video made limits more understandable and less scary.
I had the same experience when I was studying CT, and maybe most people have. Abstract = easy 😁 but its so difficult to learn since our brains are wired differently 🧐 I am glad that the video was helpful. Gook luck and a lot of fun on your CT journey!
I know you brushed over this but I just have a question. When you talk about diagrams, (I --> C under F), what are the conditions/restrictions on I? Does this diagram just say that C is a template that you want to represent and that F is some kinda choice from I that picks out the specific shape (no the maths term) that C represents? For example say I take C to be a triangle then does this just say that F will choose a particular triangle from I and map it into C? Sorry for being quite unclear. If this is the case then I guess for restrictions, the indexing category must at least contain the shape C (as if it were like a subgraph, i.e. if I could look like a fully connected graph of four vertices, then it contains 4 triangles, four shapes like C) so that the arrows and objects are able to be mapped functorially into C from it? As in the example I gave, would I then be able to contain many shapes like C in it?
Sorry I realised I was being silly, we don't need the triangles to be a shape In I right just that the arrows and compositions are preserved? Sorry I forgot about functoriality.
@@harrisonbennett7122 Right, but also, hmm, its sounds like you have the association backwards: I is the diagram (e.g. a triangle) and C is the “honest” category (e.g. vector spaces). In the triangle example, the functor F just picks a specific triangle in C. Does that make more sense and answers the remainder of your questions? If not, then let me know: I am happy to help 😄
@@VisualMath Ahh I see but is there any restriction at all on I? Like is it discrete etc? I'm guessing it can be empty since we would probably then get a terminal object or something like this (analogous to an empty family for sets)?
@@harrisonbennett7122 The most exciting I are finite categories (like the triangle), but for the general definition you need no restrictions. Note however that for difficult I it is also difficult to find a functor.
@VisualMath sorry last question. You say that the functor picks out the the shape in C. I guess in general that's the aim right? We want to pick out the shape. Like construct a template of a triangle to choose that in C, but I can't seem to find this being described in the formal definitions, it just a functor, this seems to kind of eliminate the purpose of having a template in I, like I can just always use the constant functor? I.e send the triangle I to one object, then I don't really use the triangular property of I.
What an amazing video! I'm studying category theory for the first time and it is so hard to find intuitive notions behind the definitions but this video made limits more understandable and less scary.
I had the same experience when I was studying CT, and maybe most people have. Abstract = easy 😁 but its so difficult to learn since our brains are wired differently 🧐
I am glad that the video was helpful. Gook luck and a lot of fun on your CT journey!
it's been 2 years but thank you for making this, was really helpful
I blink twice and two years are gone...😰
Anyway, thanks for the comment, you are very welcome. I hope you enjoy categories!
I know you brushed over this but I just have a question. When you talk about diagrams, (I --> C under F), what are the conditions/restrictions on I? Does this diagram just say that C is a template that you want to represent and that F is some kinda choice from I that picks out the specific shape (no the maths term) that C represents? For example say I take C to be a triangle then does this just say that F will choose a particular triangle from I and map it into C? Sorry for being quite unclear. If this is the case then I guess for restrictions, the indexing category must at least contain the shape C (as if it were like a subgraph, i.e. if I could look like a fully connected graph of four vertices, then it contains 4 triangles, four shapes like C) so that the arrows and objects are able to be mapped functorially into C from it? As in the example I gave, would I then be able to contain many shapes like C in it?
Sorry I realised I was being silly, we don't need the triangles to be a shape In I right just that the arrows and compositions are preserved? Sorry I forgot about functoriality.
@@harrisonbennett7122 Right, but also, hmm, its sounds like you have the association backwards: I is the diagram (e.g. a triangle) and C is the “honest” category (e.g. vector spaces). In the triangle example, the functor F just picks a specific triangle in C. Does that make more sense and answers the remainder of your questions? If not, then let me know: I am happy to help 😄
@@VisualMath Ahh I see but is there any restriction at all on I? Like is it discrete etc? I'm guessing it can be empty since we would probably then get a terminal object or something like this (analogous to an empty family for sets)?
@@harrisonbennett7122 The most exciting I are finite categories (like the triangle), but for the general definition you need no restrictions. Note however that for difficult I it is also difficult to find a functor.
@VisualMath sorry last question. You say that the functor picks out the the shape in C. I guess in general that's the aim right? We want to pick out the shape. Like construct a template of a triangle to choose that in C, but I can't seem to find this being described in the formal definitions, it just a functor, this seems to kind of eliminate the purpose of having a template in I, like I can just always use the constant functor? I.e send the triangle I to one object, then I don't really use the triangular property of I.