Category Theory II 3.1: Examples of Limits and Colimits

[@-ion]

Is a cocone also known as a "ne"?

[@connemignonne]

oh heck that bit at the end with the applications of continuity is the beauty that keeps me interested in mathematics

[@marcusklaas4088]

Definitely. Wonderful stuff.

[@Rozenkrantzz]

Thank you for uploading these so frequently!

[@j.c.m.2181]

Re: pushout examples. If you think of a pullback as a 'specialized' version of a product (i.e. a product, except with an extra condition on top that says the components' morphisms applied to them produce a B), a pushout would be a 'specialized' sum/disjoint union where components are produced by a morphism from B. So, if a pullback is a filter that produces *pairs of outputs* , a pushout is a filter that produces *unions of factorizations* with respect to B. A simple example would be a category of positive integers with addition as morphisms - the pushout produces disjoint unions of 2-step paths to produce a number from any other starting point. For instance, the pushout of 2 from 0 is: "Either" [b=0, f=(+0), a=1, g=(+2)] [b=0, f=(+1), a=1, g=(+1)]. The 'true' pushout colimit in this case is, I think, just adding one factorized as "Either" [(+0) (+1)] [(+1) (+0)] (A & C omitted for brevity). If my understanding and/or that example is correct, I would suspect the pushouts to pop up in pathfinding algorithms somewhere...

[@andreacesaro317]

I think that the constructions of a sphere (as pushout of DD and D*, where S is the circle, D is the Disk, * is the point and S->D is the map from the circle to the border of the disk) are intuitive examples of pushoput. For a programming example I think that the "deadly diamond of death" can be formalized as a pushout.

[@isaacdefrain9595]

Bartosz must've been hungry during this lecture :) Incredible explanations and examples! Thank you.

[@nathanielvirgo]

Why are these the next interesting limits after products and terminal objects? What about the limit of the category 1, or the category with two objects and one morphism between them, or one object and one non-identity arrow? Do those turn out to be trivial in some way? Also, what about the initial object in the category of cones (rather than cocones)? Does that give us anything interesting?

[@zbynekslajchrt8089]

Great lecture(s)! I am just wondering whether as an intuitive example of a pushout could be taken a type Z representing things that are both A and B, while A and B are mutually exclusive types specialized from type C, i.e. the root. Let's says that C is Thing, A could be Animate, B Inanimate and Z something like "Zombie"? In other words, Z is a superposition of A and B.

[@SJohnTrombley]

Are morphisms in the category of sets functions, or binary relations?

[@andreacesaro317]

The category Set is usually defined with ordinary functions as morphisms.

[@ShimshonDI]

I like the examples. So in Set, is a pullback always a subset of a Cartesian product? Also, can you give any examples of coequalizers, or offer intuition about them?

[@lwave9544]

I thought, a Terminal/initial object is unique in a category. But as you said, every singleton set in category set is a Terminal object. So which one is „the“ best to pick as the limit candidate of the „cheeting“ functor from the empty category to set? Are they only unique up to isomorphism?

[@DrBartosz]

AFAIK, all universal constructions are unique up to unique isomorphism

[@eliastandel]

The punchline is mind-blowing

[@ethannguyen2754]

I propose that we start calling cocones “nes”