Hmm, not sure. I’d have to check Dold’s book again. In principle one could say that one has a collection of functions to [0,1] such that at each point only countably many are nonzero, and the sum exists and is 1 at each point.

]]>28, 29: Pareigis uses at some places the notion of adjunction between a covariant and a contravariant functor, calling it “adjoint on the right” in one of the two imaginable cases. For example in his paper with Morris on formal schemes, proposition 1.1 doi pdf. I did not check if he has this notion in his category textbook.

]]>True. But we also have dual adjunctions.

]]>Well, to me the phrase “Galois connection” mainly connotes a *contravariant* adjunction between posets $P, Q$. You *could* just say ’adjunction’, with the normal meaning of covariant functors, but I tend not to like this because of the symmetry-breaking: you have to choose which of $P, Q$ you’re going to attach the $^{op}$ to, and if it’s $P$ say, you have to remember which is the right adjoint: is it in the direction $P^{op} \to Q$ or the direction $Q \to P^{op}$ (or similarly, on which of these posets do we get a *coclosure* operator)?

So for me “Galois connection” says just a tiny bit more than adjunction: it implies a certain array of choices of presentation based on symmetry and convenience. I’m so used to thinking about that that “Galois connection” just gets my neurons firing a certain way and saves me from having to think.

On the other hand, I personally would never use the phrase “monotone Galois connection” – *there* I’d say “adjunction” instead, unhesitatingly. For me, “Galois connection” should imply antitonicity or contravariance of the maps involved.

why multiply terms needlessly for the same concept?

Why indeed? We have perfectly good terms “adjunction” and “equivalence” already, why bother with this “Galois” stuff? (-: It doesn’t really make any sense to name “adjunctions of posets” after Galois just because Galois theory involves a particular adjunction between posets; if we were going to name it after anyone we ought to call it an “Ore connection” if he was the first to study them abstractly. My tongue is in my cheek, of course, but I always *have* wondered. Presumably the term “Galois connection” predates the term “adjunction”, but why continue to use it now?

Mike: okay, interesting. I had *thought* that was *the* standard, yes. And I can say it is *a* standard according to Wikipedia, and I thought this also was the convention used in for example Paul Taylor’s book, although I can’t find my copy at the moment to double-check that.

That being said, the notion of Galois connection seems to be due to Oystein Ore (who spells it ’connexion’), and actually he’s like you: he uses ’connexion’ and ’correspondence’ pretty interchangeably for the adjunction concept in his article. He calls the correspondence or connexion ’perfect’ if it’s an equivalence. Encyclopedia of Mathematics doesn’t even use the term ’connection’; it’s just ’correspondence’.

However, I do like the idea of using ’correspondence’ for the equivalence. First, why multiply terms needlessly for the same concept? (why indeed, Oystein?), and second, the word ’correspondence’ is used in the phrase “bijective correspondence” and so it should be easy to remember – a Galois correspondence would be a bijective correspondence between posets that is contravariant.

]]>I have added to *locally finite cover* statement and proof that *every locally finite refinement induces a locally finite shrinking*.

What’s a “non-point finite partition of unity”? If it is what it sounds like, then how is the sum well defined?

]]>Todd, I meant that I never learned to distinguish between a Galois *connection* meaning an adjunction and a Galois *correspondence* meaning an equivalence. As you can see, in the paper with Moritz we used the two words interchangeably to mean an adjunction. But it could be standard, in some circles at least, and just that neither of us had encountered that standardization before. Did you learn it as “standard”?

I recently put on the arXiv this preprint:

https://arxiv.org/abs/1704.00303

that stemmed from a question I posed on MathOverflow a few months ago (the title is the same, googling gives both the arxiv preprint and the MO-thread), and that received some attention and positive comments (I hope).

We authors are in the phase of polishing some details, and improving the clarity of the discussion. Once this process is finished, I’d like to have it published: what is, in your opinion, a good journal where to send the preprint?

Thanks!

]]>Again the idea of an adjoint 7-tuple appears, here for pointed derivators, in Mike’s

- Generalized stability for abstract homotopy theories, arXiv:1704.08084

and this is infinitely extendable in the stable case with a periodicity 6.

]]>Thanks. I have added a line relating to Eilenberg-MacLane. Noticed that we didn’t even have that reference listed at *category*, so I put in there (Eilenberg-MacLane 45). One day maybe somebody finds the energy to bring our basic category-theoretic entries into better shape…

Re #21, the paper there

[GS17a] Moritz Groth and Michael Shulman. Abstract stabilization: the universal absolute.

sounds like you’re picking up on the Hegelian vibe.

]]>There’s something for now. Hopefully more later. The idea of an exhaustive list of categories seems lost to us (even Martin-Löf). But it’s interesting to reflect on whether there are basic types. One of the ways, I guess, is to think about joining implicit inclusions (join of maps). If you can’t quite see something, you say can say “it’s either a rabbit or a rock”, presumably as they’re both included in some basic kind of discrete, middle-sized entity. But we don’t go across the categories and say something is “cold, being beaten or an elephant”.

]]>how do we square the idea of “types are concepts” with “propositions as types”?

That’s immdiate. Proposition “X is blue”, Concept of blueness.

]]>how do we square the idea of “types are concepts” with “propositions as types”? I take it we don’t want to lump all 3 together, and even following propositions as some types, propositions would appear as a species of concept, which seems wrong…

]]>As I wrote before by email: I think it would be a good idea. If it is heavy on the notation, then best to split it off as a separate entry, for instance, *separation axioms in terms of lifting properties*, and then link to that from the main entry.

I have now merged the first example from Todd’s #13 into the entry, here.

(So I copied it over, then added numbered environmens, adopted the notation a little to fit with the rest of the entry, added cross-pointers to previous definitions/propositions that the example refers to or which it re-proves.)

Didn’t do it for the second example in #13 yet.

]]>Would it be a good idea to add lifting property reformulations of the separation axioms to the nlab page? In fact, the appendix of the notetex sort of does that: it is mostly the text of the wikipedia/nlab entry mixed with the lifting property reformulations.

However, I’m not sure how to go about it, as these reformulations require introducing somewhat cumbersome notation for maps of finite topological spaces. The easiest would be to add the appendix as a separate section; but that would lead to repeatition of text so is probably a bad idea….

]]>Our entry *category (philosophy)*, being the stub that it is, lacks any mentioning of Aristotle. I don’t feel qualified. Could you maybe add a few lines to provide a minimum of substance for the entry? And then we should add that quote by Martin-Löf which you found!

concept logichas been highlighted long before…He would probably say ’Yes, by Aristotle’,

Good, that’s just what is argued for in the *Science of Logic* article here: the sentences in Aristotle’s logic are typing judgements. And that’s the jumping-off point for Hegel’s “concept logic”.

Then he laments Mac Lane taking the word ’category’ for another purpose.

Luckily we now know that it is in fact the same purpose, just generalized to allow for “directedness”.

There *is* some net progress even in philosophy.

Mike: shouldn’t I say “We say” instead? (Yes, I believe “Galois connection for an adjunction is standard. Isn’t it?)

]]>Another example of a Galois connection on the arxiv today: abstract stability.

Todd #17: You say “I say”. Is that terminology standard?

]]>

concept logichas been highlighted long before…

He would probably say ’Yes, by Aristotle’, but he’d prefer to retain the word ’category’:

Given how basic the very idea of type is, it is of course unthinkable that there shouldn’t be a word already in the tradition for what we here call types. You will all know that the traditional word for what I here call type is category. It was introduced by Aristotle and heavily used by Kant. I will show that the traditional use of the word coincides with the way I am using it here. What I call here the doctrine of types, the idea that an object is always an object of a certain type, really goes back to Aristotle.

Then he laments Mac Lane taking the word ’category’ for another purpose. (This is all in that Boldini article cited above. It’s a shame we can’t have access to notes from the lectures he’s given over the years.)

If you were hoping for mention of Hegel, I think this is unlikely. He strikes me as someone who follows other lines out of Kant. And yet, how close is the relationship between a type theorist’s types and Kant’s twelve categories?

]]>