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Stream: learning: reading & references

Topic: Riehl, Category Theory in Context, Ch. 3

Julius Hamilton (Jun 08 2024 at 21:39):

I will try to read all or part of this chapter and would like to discuss the material here.

D215B0E4-B5C9-48FA-AD63-5F206ABB0DC6.png

Julius Hamilton (Jun 08 2024 at 22:29):

From a very simple topological space, the real line $\mathbb{R}$ with its usual metric, one can build a wide variety of new topological spaces.

Just reminding myself that a topology is a collection of subsets which are closed under finite union and infinite intersection. Still haven’t figured out why the union must be finite and the intersection infinite.

I will guess that the “usual metric” is the Euclidean norm: $d(x, y) = |x - y| = |y - x|$.

I had the feeling that topology allows us to express the idea of continuity, without having to use metrics. But in this case, I guess the Euclidean norm induces topological structure.

Taking products of $\mathbb{R}$ with itself, one defines the Euclidean spaces $\mathbb{R}^n$ in finite or infinite dimensions.

The space $\mathbb{R}^n$ has interesting subspaces, defined to be the solutions to certain continuous polynomial functions $\mathbb{R}^n \to \mathbb{R}$ including the $(n − 1)$-sphere $S^{n−1}$, which bounds the closed $n$-disk $D^n$.

If the sphere bounds the disk, shouldn’t it have one dimension higher, just as the 3-sphere bounds the 2-disk?

From the sphere $S^n$, one can define real projective space $\mathbb{R}P^n$ as a quotient by identifying each pair of antipodal points.

A quotient is generally a map from elements to sets of elements defined by an equivalence relation (equivalence classes). I haven’t yet thought about what quotients “look like” when we look at them as morphisms in categories. I know that projective space has a point at positive and negative infinity, but I don’t see how one gains that property by taking the quotient of a sphere.

From spheres and disks one can build tori, the Möbius band, the Klein bottle, and indeed any cell complex through a sequence of gluing constructions, in which disks are attached to an existing space along their boundary spheres.

I don’t know what a cell complex is or how attaching disks to a boundary sphere could create a Mobius strip.

In each case, the newly constructed object is a particular set equipped with a specific topology. In this chapter, we will explain how all of these topologies can be defined in a uniform way, via a universal property that characterizes the newly constructed space either as a limit or a colimit of a particular diagram in the category of topological spaces.

Then I can read on and not get bogged down in details.

John Baez (Jun 08 2024 at 22:45):

Julius Hamilton said:

Just reminding myself that a topology is a collection of subsets which are closed under finite union and infinite intersection.

It's the other way around, unless you're taking an unorthodox approach and defining a topology in terms of closed sets instead of the usual open sets.

Kevin Carlson (Jun 08 2024 at 23:22):

Noticing all those unfamiliar things in the introduction to this chapter is a hint that the intended audience of this book is really somebody who’s had at least a couple of graduate-level courses in pure math, particularly in algebra and topology (which I think I have the impression you haven’t, sorry if I’m mistaken.) You may of course still find the book useful but I don’t think it’s the one I’d recommend to you, in case you want other suggestions!

Eric M Downes (Jun 09 2024 at 10:43):

IMO a good sequence of undergraduate classes, and a willingness to ask questions and go slowly, would even be sufficient to approach Emily Reihl's book!

Since many folks may be unaware, JHU has online versions of their core math classes! Here I'm thinking specifically of 110.301, 401, 412, 413, with analysis added depending on one's taste and blood pressure. All told I think that'd net someone already holding an undergraduate degree ~$6-7k. No category theory yet, sadly, but I love that we live in an age where you can get decent quality math instruction online!

Julius Hamilton (Jun 09 2024 at 12:12):

I want to read this chapter and make explicit the questions I have as a learning exercise.

Limits and colimits can be defined in any category.

Special cases include constructions
of the infimum and supremum

I think of these concepts in relation to orders and/or to lattices. I believe the supremum of a set $S$ to be an element $s$ - which can be in $S$ or not - for which, for all $x \in S$, $x \leq s$, where $\leq$ is any order relation. There are different binary relations we associate with the concept of ‘order’: preorder, partial order, total order. IIRC, some of the most relevant properties regarding an order include:

• reflexivity: $\forall x, x \leq x$
• anti-symmetry: $x \leq y \rightarrow \neg (y \leq x)$
• transitivity: $x \leq y \wedge y \leq z \rightarrow x \leq z$
• and connectedness: $\forall x, y, x \leq y \vee y \leq x$

The other part of being a supremum is that it is a least upper bound. I wonder if it therefore can be expressed elegantly in terms of two applications of set-comprehension: first, you designate the set $S_{\geq}$ of all elements which are greater-than-or-equal-to all elements in a set of interest, $S$. Then, within that set, you select the element $y$ which is less-than-or-equal-to them all.

When we visually diagram a partial order, it looks like a tree. My intuition says that we can only have a supremum/infimum for a total order, or a strict total order.

Based on what I know about the relationship of orders to categories, why would the supremum/infimum be a limit? I will answer this question later.

Julius Hamilton (Jun 09 2024 at 12:19):

free products, cartesian products, direct sums, kernels, cokernels, fiber products, amalgamated free products, inverse limits of sequences, unions,
and even the category of elements, among many others.

I would say all of these constructions are unfamiliar to me in a categorical setting except Cartesian products.

We then turn our focus to “computational” results, which describe how more complicated limits and colimits can be built out of simpler ones.

To ground intuition for limits and colimits, these abstract constructions are first introduced in §3.2 in the special case of limits of diagrams valued in the category of sets.

Indeed, as a consequence of the Yoneda lemma, the set-theoretical constructions of limits suffice to prove general formulae for limits and colimits in any category.

I think this is because what we tend to call “category theory” is also “set-enriched category theory”, which is why sets play “a special role”, as one person once mentioned.

To understand this, we consider a variety of possible interactions between functors and limits in §3.3, presenting functors that preserve, reflect, or create limits and colimits.

Such functors can be used to recognize cases in which limits or colimits in one category can be constructed from limits or colimits in another.

This vocabulary is used in §3.4 to describe the representable nature of limits and colimits and extend the constructions of general limits from simple limits from Set to any category.

David Egolf (Jun 09 2024 at 15:57):

I like Riehl's book! I've spent a decent chunk of time with it, and have been able to learn some things from it. However, it's still a bit on the advanced side for me. Here are some alternative books which I've found faster to learn from, and which you might potentially also find useful:

• Notes on Category Theory by Perrone
• Basic Category Theory by Leinster
• Seven Sketches in Compositionality by Spivak
• My First Category Theory Textbook by Sarkis

David Egolf (Jun 09 2024 at 16:00):

Julius Hamilton said:

The other part of being a supremum is that it is a least upper bound.... When we visually diagram a partial order, it looks like a tree. My intuition says that we can only have a supremum/infimum for a total order, or a strict total order.

In my opinion, thinking about limits or colimits in a partial order is a really good place to start.

David Egolf (Jun 09 2024 at 16:05):

Here is a partial order, viewed as a category, that you may find helpful to think a bit about:
partial order

I've put a partial order on elements $a,b,c$ with $a \leq c$ and $b \leq c$ (and of course $a \leq a$, $b \leq b$ and $c \leq c$). Then, to make the above picture, I've formed a category where:

• the objects are the elements of the partial order (so, we have three objects $a,b,c$)
• we put a morphism from some element $x$ to some element $y$ exactly if $x \leq y$ [so there is at most one morphism from an element $x$ to an element $y$]

David Egolf (Jun 09 2024 at 16:08):

Here are some questions you may find helpful to think over, for purpose of gaining intuition for colimits:

• Is this partial order a total order?
• Is there a smallest element that is at least as large as $a$ and $b$? In other words, does the set $\{a,b\}$ have a supremum?
• In the category pictured above, does the diagram with just the objects $a$ and $b$ have a colimit? In other words, do $a$ and $b$ have a coproduct in this category?

John Baez (Jun 09 2024 at 17:08):

Nice! At no extra cost we have two more puzzles:

• Is there a largest element that is at least as small as $a$ and $b$? In other words, does the set $\{a,b\}$ have an infimum?
• In the category pictured above, does the diagram with just the objects $a$ and $b$ have a limit? In other words, do $a$ and $b$ have a product in this category?

Jacob Zelko (Jun 09 2024 at 21:58):

I had never heard of @Ralph Sarkis 's book before! Just wanted to drop a quick comment to say that the cover of your text is absolutely delightful Ralph! Thanks for sharing @David Egolf ! I'll also strongly second Leinster's textbook and also suggest Category Theory by Awodey being nice to read too. Additionally, Paolo Aluffi's categorical treatment of abstract algebra is quite enjoyable and once you have had exposure to the basics of Category Theory like you have @Julius Hamilton , you might like Aluffi's Algebra: Chapter Zero quite a bunch.

Ralph Sarkis (Jun 10 2024 at 05:22):

I rarely advertise it because it is not finished (as suggested in the file name). The only thing better than the other books is the fact that I use the knowledge package to have links for most of the terminology and symbols defined.