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Stream: community: general

Topic: for beginners

John Baez (May 20 2021 at 18:41):

For any beginners lurking around here: I wrote two series of tweets, one explaining some basic intuitions behind colimits and the other explaining some basic intuitions behind limits.

To understand category theory one thing you need is an intuitive grasp of "colimits". So: A colimit of a bunch of objects is a way of "sticking them together" to get a new object. We often use colimits to assemble interesting objects from basic pieces. (1/n)

- John Carlos Baez (@johncarlosbaez)

To understand category theory you need to understand "limits". It's easiest to get a feeling for these by looking at limits in the category of sets. For example, the graph of a function {(x,y): x ∈ X, y ∈ Y, y = f(x)} is a limit! (1/n)

- John Carlos Baez (@johncarlosbaez)

John Baez (May 20 2021 at 18:43):

Both these series focuses on categories that resemble the category of sets: limits in a category are colimits in its opposite, so they're really not two different concepts, but in categories like Set they behave very differently.

Tom Hirschowitz (May 21 2021 at 08:43):

(I thought you had given up on twitter???)

Henry Story (May 21 2021 at 08:55):

In the second tweet on colimits you use the example of a graph as the colimit of the walking vertext and walking edge. I wonder how that compares to graphs as functors from the small category $s, o: A \to N$ to Set?

Henry Story (May 21 2021 at 09:11):

Let me check if I have the idea for the third tweet on colimits right.
Wikipedia entry on colimits has the well known definition of colimits as a diagram. I think that maps to your image of the green and blue circle as follows: that Z is the blue&green intersection region, X and Y are the blue and green circles, and P is the union of both circles. So if one takes a $z: Z$ then $i_1(f(z)) = i_2(g(z))$.
If instead of Z being the intersection, we took say $\empty$ the empty set for Z, then we would have the usual Either structure in programming (disjoint union) so we would have a different P where the intersection region was duplicated somehow.... Would we need the circles to be on two planes?

A more interesting kind of colimit is a "pushout". Here you stick together two things with a specified "overlap". For example, here we are taking the pushout of a green and a blue disk, where we specify how the blue-green region is included in each one. (4/n) https://twitter.com/johncarlosbaez/status/1393699213729406979/photo/1

- John Carlos Baez (@johncarlosbaez)

Matteo Capucci (he/him) (May 21 2021 at 09:49):

One thing I was obsessed with when I was learning this stuff is that you can build every limit/colimit if you have just terminal objects and pullbacks/initial objects and pushouts, or alternatively if you have products and equalizers/coproducts and coequalizers

Matteo Capucci (he/him) (May 21 2021 at 09:49):

I think this helps intuition a lot, because you can develop intuitions for just two cases and they scale pretty well

John Baez (May 21 2021 at 14:05):

Tom Hirschowitz said:

(I thought you had given up on twitter???)

I've relapsed.

John Baez (May 21 2021 at 14:09):

Henry Story said:

In the second tweet on colimits you use the example of a graph as the colimit of the walking vertex and walking edge. I wonder how that compares to graphs as functors from the small category $s, o: A \to N$ to Set?

They're closely connected. The category of graphs is a "presheaf category" - the category of functors from (the opposite of) some small category X to Set. Every object of X gives an object of the presheaf category called a "representable" - in this case they're the walking vertex and the walking edge. And every object in a presheaf category is a colimit of representables.

This is an important fact, and the category of graphs shows that this fact is all about building up structures in a tinker-toy like fashion from sticking basic pieces together.

John Baez (May 21 2021 at 14:13):

Henry Story said:

Let me check if I have the idea for the third tweet on colimits right.
Wikipedia entry on colimits has the well known definition of colimits as a diagram. I think that maps to your image of the green and blue circle as follows: that Z is the blue&green intersection region, X and Y are the blue and green circles, and P is the union of both circles. So if one takes a $z: Z$ then $i_1(f(z)) = i_2(g(z))$.

Right, exactly - that's the idea I was trying to convey in a very nontechnical way.

If instead of Z being the intersection, we took say $\empty$ the empty set for Z, then we would have the usual Either structure in programming (disjoint union) so we would have a different P where the intersection region was duplicated somehow.... Would we need the circles to be on two planes?

Or just sitting next to each other not overlapping, or something. Instead of thinking of the circles as subsets of some pre-existing space, you should really think of them more abstractly as "free-floating" entities, namely topological spaces. Then doing a pushout lets you stick them together with any desired amount of overlap.

Doing a pushout where $P$ is the empty set lets you say there's no overlap, and then your pushout is a coproduct. And remember, the empty set is the initial object in the category of topological spaces. There's a little theorem saying that if you do a pushout where P is the initial object you just get a coproduct.

John Baez (May 21 2021 at 14:34):

Matteo Capucci (he/him) said:

One thing I was obsessed with when I was learning this stuff is that you can build every limit/colimit if you have just terminal objects and pullbacks/initial objects and pushouts, or alternatively if you have products and equalizers/coproducts and coequalizers

Yes, I like these things. To be pedantic:

You can build every finite colimit from pushouts and an initial object, or from finite coproducts and coequalizers.

We can build arbitrary (small) colimits from wide pushouts and an initial object, or from coproducts and wide coequalizers.

John Baez (May 21 2021 at 14:36):

I think this helps intuition a lot, because you can develop intuitions for just two cases and they scale pretty well.

Yeah. There's this stuff about the infinite case one has to ponder, but it's just a generalization of the finite case.

Martti Karvonen (May 21 2021 at 14:58):

If you have arbitrary small coproducts, you don't need wide coequalizers - ordinary coequalizers are sufficient to get all colimits

John Baez (May 21 2021 at 20:39):

Oh, great! I keep wondering about that.

John Baez (May 21 2021 at 20:40):

Like, for one minute every year.