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Stream: learning: questions

Topic: Distributive laws in Cartesian closed categories

Nick Smith (Jun 11 2021 at 08:27):

nLab says that in a locally Cartesian closed category "pullbacks preserve all colimits". Does this mean $X \times_{P} (Y +_{Q} Z) \cong (X \times_{P} Y) +_{Q} (X \times_{P} Z)$ , for all pullbacks and pushouts? There's some detail missing about the morphisms here, but it seems like the correspondence between morphisms might be an "obvious" one that can be explained by a few commutative diagrams.

Or am I misunderstanding :innocent:?

Jens Hemelaer (Jun 11 2021 at 09:00):

Yes, but you have to apply the functor $X \times_P -$ to $Q$ as well, so you get:
$X \times_P (Y +_Q Z) \cong (X \times_P Y) +_{X \times_P Q} (X \times_P Z)$ .

In the notation of the nLab article, $f$ would be the map $f : X \to P$ , and then $f^*$ is the functor $f^* : \mathcal{C}/P \to \mathcal{C}/X$ defined as $f^*(A) = X \times_P A$ .

Does this help?

Nick Smith (Jun 11 2021 at 10:35):

Oh wow! Yes that helps. That equivalence looks so fun :stuck_out_tongue:, I wonder if there’s an intuitive explanation of what it “means” in term of sets and functions in Set...

Fawzi Hreiki (Jun 11 2021 at 11:23):

Well it just means that inverse images preserve sums and quotients.

Fawzi Hreiki (Jun 11 2021 at 11:26):

If you interpret pullback as base change (i.e. shifting from one over category regarded as a relative universe of discourse to another over category) then the stability of a property under pullback tells you something very strong about it - namely that it's 'absolute' or 'universal' (i.e. it doesn't depend on the base space over which you're working).

Nick Smith (Jun 11 2021 at 11:32):

I meant something intuitive like how David Spivak explains pullbacks and pushouts in terms of his “ologs”. An explanation in terms of everyday concepts, rather than more math :smile:.

The “base change” perspective sounds interesting though. I need to read more into it. Is this discussed in simple terms in any books?

Fawzi Hreiki (Jun 11 2021 at 11:35):

Hmm, I'm not sure about any simple books. I think the first step is convincing yourself that for a set $X$ , you have an equivalence of categories between the functor category $\text{Set}^X$ and the slice category $\text{Set}/X$ . This idea that you can parametrise in two different ways (forwards via indexing and backwards via preimages) is a pretty important one.

Fawzi Hreiki (Jun 11 2021 at 11:37):

Its a pretty inocuous idea at first sight but leads to all kinds of interesting mathematics (like the Grothendieck construction on presheaves, fibred categories, stacks, etc...)

Fawzi Hreiki (Jun 11 2021 at 11:41):

I suppose a good book for this kind of thing is 'Generic Figures and Their Glueings' which deals with simple examples of presheaf categories.

Nick Smith (Jun 11 2021 at 11:47):

Ooh I haven’t heard of that book. I’ll check it out!

Nick Smith (Jun 11 2021 at 23:30):

Interestingly I can’t find a single review or discussion of “Generic Figures and their Glueings” online. Seems to be open access though. Has anyone else read it? Is it a well-written book? 😇

Ralph Sarkis (Jun 12 2021 at 00:09):

I just finished a reading group on this book. We got to chapter 11 before deciding to stop because of summer and the following review (most opinions were shared):

The content of this book is nice to see as a first intro to topos theory because it has lots of examples and pictures that can convey intuition about presheaf toposes. However, it is presented in a very different way from the rest of the literature and no effort is made to make connections between their way and the standard way. While their way was relatively helpful for understanding the presheaf examples, I found it lost its edge when we arrived at more abstract stuff (adjoints and internal logic). Here are other issues that make the presentation so poor that I would not recommend reading it.

There are several times where they refer to concepts defined later in the book with no warning.
There are several symbols they use but do not define.
There isn't enough explanations about how the theory is developed.
I found the proofs way drier than I am used to in category theory (that may be true about topos theory in general for me, idk).
The typography is very bad.

All in all, if you want to get comfortable working with presheaf categories, I could suggest to read the first 6 chapters and do the exercises and complete the proofs to do actual computations. If you are more interested in toposes in general, do not bother.

Nick Smith (Jun 12 2021 at 00:34):

Oh wow! Thank you for that detailed review! I’ll treat the book with caution then. I’ll probably explore the first few chapters as you suggest.

John Baez (Jun 12 2021 at 01:33):

Nick Smith said:

nLab says that in a locally Cartesian closed category "pullbacks preserve all colimits". Does this mean $X \times_{P} (Y +_{Q} Z) \cong (X \times_{P} Y) +_{Q} (X \times_{P} Z)$ , for all pullbacks and pushouts?

That's not what it means, but it does imply some sort of isomorphism like that. (Jens fixed it.)

If you're trying to check this in the category of sets, it's easier to check that pullbacks preserve coproducts and coequalizers.... since you can build pushouts from coproducts and coequalizers.

Mike Shulman (Jun 12 2021 at 14:31):

I think a good way to understand this is via the equivalence $\mathrm{Set}/P \simeq \mathrm{Set}^P$ : a set over $P$ is equivalent to a $P$ -indexed family of sets. Under this equivalence, the pullback functor $f^*$ is just reindexing, replacing a $P$ -indexed family of sets $(A_p)_{p\in P}$ by an $X$ -indexed one $(A_{f(x)})_{x\in X}$ . Now the fact that it preserves colimits follows from the fact that colimits in $\mathrm{Set}^P$ are pointwise.