article: nJournal-Leinster2011: An Informal Introduction to Topos Theory
article type: expository article
submission type: author submission of arXiv article
submitted: Jan, 2011
refereeing: by expert anonymous referee chosen by the nLab steering committee
status: accepted for publication, 27th June, 2011
referee reports: see below
This is the report of an anonymous expert referee that had been chosen by the nLab steering committee. From April 5, 2011.
The author had reacted to this report with a modeified version by May 2011. The revised article was accepted by the anonymous referee on June 27, 2011.
(The entirety of this report refers to the arXiv submission dated 7 Jan 2011.)
This is an expository article which aims to introduce toposes to readers who are generally (rather) comfortable with categorical methods, but who are not yet acquainted with topos theory. The treatment is light and focuses mostly on motivation. After explaining the definition, the author explains the idea of “topos” in terms of general themes, somewhat along the general philosophic lines of Johnstone’s “Elephant”: a topos can be perceived from one standpoint as a category of generalized sets, from another as a generalized space, and from still another as a type of logical theory.
I believe that publication in the newly formed Journal of the nLab (however it is titled) would be appropriate for this article. Readers who come to the nLab for information are likely to be comfortable with basic category theory, or at least willing to learn. Topos theory is notoriously daunting to beginners; a short article that collects together general themes, as a kind of roadmap, is certainly a welcome addition to the literature.
The present article is engagingly written and I believe succeeds in most of its aims, as indicated already by positive reactions and recommendations it has received on the internet. The article could be published as is, although I would like the author to consider the following points.
As an overall suggestion, please consider linking to nLab or other online articles, by keyword, as the author deems fit. (I am not sure to what degree this has already been done; I included some examples before I petered out. Perhaps the overall suggestion goes without saying for any article submitted to the nLab Journal.)
One general comment is that the article is written with a certain amount of “attitude” (as might be fitting for the nLab and its proclaimed nPOV!?). In one respect this adds to its interest; in another it makes it vulnerable to misunderstanding. I am thinking mostly of the section on connections with set theory, where the author’s declared major theme is that topos theory frees us from “the shackles of ZFC”. This to me sounds like a pretty tendentious way of putting it! (“What shackles?”) I am guessing the author means – although he doesn’t quite say it – that set theory need not be presented by membership-based axioms, and that it is worthwhile to found set theory from the point of view of asking what a category of sets should be like in terms of universal properties. Thus the author argues for ETCS as alternative approach to set theory, independent of and foundationally just as valid as ZFC. I think some readers might wonder, though: if some form of ZFC and some form of ETCS are in some sense equivalent (meaning bi-interpretable and equiconsistent), then why should the general reader pay attention to the abstract nonsense as particularly better? Again, the author doesn’t quite say.
My bigger concern is that this should be the major theme of that section, and “a topos is a generalized category of sets” the minor theme. In my opinion, it should be just the other way around! That is, the real revolution of topos theory as generalized set theory is the huge expansion in semantics, in other words environments in which ordinary set-theoretic arguments (those not involving the axiom of choice or law of the excluded middle) are valid. In my opinion, ETCS is a small chapter of that bigger story, and I wondered why the author wants to place more of the emphasis there. The author is a fine expositor, and I would love it if he could give a light sketch which tells more about the bigger (and frankly, more exciting) story.
The rest of the article is nicely written in my opinion, and I think could basically stand as is – I have only a few minor suggestions to add.
Page 3, line3: “In the category of sets…”. Meaning the category of sets and functions. Some readers may wonder “what do you mean by sets?”, so it might be good to add “however you wish to think of them”, or otherwise acknowledge the fact that yes, there are ways to circumscribe and formalize conceptions of sets, but for now we are proceeding naively.
Please consider adding a link to the word “pullback”, say to the nLab article on pullback.
Page 4: Please consider linking the term “subobject classifier” in definition 1.1 to the nLab article.
Please consider linking lemma 1.2, lemma 1.3, and Fact 1.4 to appropriate nLab articles where readers can look up proofs.
Page 5: Please consider linking the principal terms in definition 1.5 (topos, cartesian closed category, finite limit) to nLab articles.
Page 6: Considering the kind of paper this wants to be, I guess it’s wise not to embark on a discussion on “why these particular axioms” beyond the few brief words given, but I would have been sorely tempted to say something more. Perhaps one could usefully link to Pare’s theorem? Come to think of it, if there is or is to be a good nLab discussion on the topic “why these axioms?”, this would be a good thing to link to.
I certainly agree with the opening paragraph: topos theory has indeed been revolutionary in terms of how we think of “sets”. But the revolution goes way beyond ETCS, and my own taste leads me to wish the author would say just a bit more on topos theory as generalized set theory. I’ll take the liberty to describe one set of ideas (which I’m sure the author knows).
To use Lawvere’s terminology, ETCS is a theory of “constant sets”, which are qualitatively different from the more usual sorts of toposes which can be thought of as universes of “variable sets” (see particularly Lawvere’s Chicago lecture notes, Variable sets etendu and variable structures in topoi ). Typical examples of “variable sets” include various categories of sheaves used throughout topology and in algebraic geometry, where one has sets that continuously vary over a site. But Lawvere argues that even in studies ostensibly about constant sets, the methods typically and fundamentally involve passage through (usually unmentioned) universes of variable sets. He gives two basic examples.
Robinson nonstandard analysis. One starts with some “standard” category of (constant) sets Set, then one passes to a topos of variable or indexed sets like Set/N, and finally one localizes or “freezes the variation”, taking the stalk at an ideal point given by an ultrafilter $F$, to get back down to another category of constant sets. This stalk is usually called an ultrapower of $Set$: one takes the colimit
applying restriction functors to get to smaller and smaller slice toposes $Set/U$ as the $U$ go deeper into the ultrafilter (effectively, over the neighborhood filter of the ideal point). It is not too common to read in the standard texts that the process of taking an ultrapower of an object is the same as taking the stalk at a point of a corresponding object in the topos of sheaves over a Stone-Cech compactification (even if some of the cognoscenti realize it), and thus the role of variable toposes usually passes by hidden from notice, even when it clarifies things.
Cohen’s method of forcing. Here the main character is a poset of forcing conditions, or better yet, the topos of presheaves: sets varying over that poset. The hard bit of this, or of any forcing technique, is to get the forcing conditions right, so that the standard moves of taking double negation sheaves and then localizing at an ultrafilter give a category of constant sets with desired properties. In fact, the usual approach to forcing seems to be saddled with some technicalities (such as starting from a countable transitive model – why?), which can be bypassed through a different approach involving Boolean-valued sets (a certain Boolean topos). Ultimately, the Boolean-valued sets form a sheaf topos, and the most direct expression of the idea of forcing is, arguably, right there in the Cohen topos.
A huge expansion occurs when one takes worlds of variable sets just as seriously as worlds of relatively constant sets represented by ETCS. Another good illustration of this is synthetic differential geometry, as explored by Dubuc, Kock, Lawvere, Moerdijk, Reyes, and others. A primary message is that one can reason set-theoretically in worlds of SDG and obtain notable conceptual simplifications, every bit as much as in say non-standard analysis.
Thus, for the purposes of motivating topos theory, I believe the real emphasis on the connection with set theory ought to be not on the relatively narrow band represented by ETCS, but on the liberating effect of considering toposes as worlds of generalized sets, embracing both constant sets and variable sets. Obviously some of the examples given in section 1 could be put to service here.
Returning to the present paper, ETCS is a categorical theory of “constant sets” par excellence, and the comparison between (bounded) ZC and ETCS is of course a good thing to point out. I personally found some of the meta-discussion, surrounding for example the issue of ‘circularity’, a slight distraction from the proper focus of the article: “why (or whither) topos theory?”. (Evidently the author is setting out to correct a common but wrong belief that ETCS, or category theory generally, is somehow dependent on a prior theory of collections, and therefore on traditional set theory, but I still thought it’s something of a side issue.)
Finally, as noted above, the reader may wonder why “shackles of ZFC”, when the discussion of the major theme concludes with the essential equivalence between forms of ETCS and forms of ZFC.
Even if the author would prefer to keep the major focus on ETCS, it still might help to expand a bit on what exactly is ‘wrong’ with traditional foundations. I would say, and imagine that the author agrees, that the conception of mathematics as founded on membership and the cumulative hierarchy results in a kind of distortion: mathematicians do typically think of sets not in terms of iterated chains of membership but in terms of “bags of dots”. To put it in a more dignified way, mathematicians are concerned with properties which are isomorphism-invariant, and categorical approaches to foundations achieve this, whereas the study of membership-based set theory per se naturally goes off in a different direction.
I think one could say something like this: traditional set theory as a description of the cumulative hierarchy is more or less a huge study on the theme of well-foundedness, not of enormous interest to anyone but set theorists, whereas the points where set theory has direct impact on core mathematics (e.g., measurable cardinals, Vopenka’s principle) are just those points effectively expressed in structural or isomorphism-invariant terms. Here categorical approaches tend to have something illuminating to say.
Page 7, line 4: again, “according to the working mathematician’s naive conception of sets” or whatever.
Please consider linking “well-pointed” to an appropriate nLab article.
Page 8: Please consider linking “natural numbers object” and “axiom of choice” to nLab articles.
The penultimate paragraph on page 8 deserves more amplification. “There is, however, another type of answer…” (preceded by “at no point… did it feel necessary to call on an axiom system”) – how is the other answer really different? In other words, Lawvere’s idea amounts to another axiom system. In what way might his solution be superior to a membership-based axiom system (one which feels ‘natural’ to many)?
Please consider linking to nLab articles on ETCS or (perhaps better) directly to Lawvere’s reprint at TAC.
Page 10: The paragraph beginning “Temporarily, …”: tendentious? One could equally well say, “if you are more at home with the phrase ‘well-pointed topos with natural numbers and where epis split’ than with ‘model of ZFC’, you might find it less mysterious.” No doubt this will seem sort of apples vs. oranges for many people.
Page 11, line 2: “that can be proved constructively”. I realize this is a kind of shorthand expression, but to remove any impression that this refers to some specific school of constructivism, it might help to link to appropriate articles. Please consider links for references to internal language, Mitchell-Benabou, etc.
Much of this section is attractively written. The elegant ideas of Leinster’s n-Category Café post “Sheaves Do Not Belong to Algebraic Geometry” are apparent here.
On the top of page 16, in addition to the example of the effective topos, there is also the type of topos mentioned above in this report, involving “freezing at an ultrafilter”.
In the section on Locales: one thing I wondered about a little is whether it might be worth pointing out another connection between toposes and locales. A locale is a poset $P$ whose $\mathbf{2}$-enriched Yoneda embedding $P \to \mathbf{2}^{P^{op}}$ has a left exact left adjoint. A Grothendieck topos is (modulo a small lie) a $Set$-enriched category $C$ whose Yoneda embedding $C \to Set^{C^{op}}$ has a left exact left adjoint. (This is due to Street. The small lie involves a moderate size condition.)
If the author admits that the example of fields (as a geometric theory) is “feeble”, then might he consider switching to local rings as a less feeble example? I don’t think that’s much (if at all) more complicated, and in my opinion it’s a pretty important example. (I’m pretty sure that readers who need to know about coends to read the paper will not mind any slight complications this entails.)
I realize that the author wants to keep it short, but a little part of me wanted to hear more about “points” and “models”, with connections back to the localic theme. The general theme harks back to topological interpretations used in model theory, where models (say of a propositional theory as codified by a Lindebaum algebra) correspond to points of a spectrum, and the compactness theorem of propositional or first-order logic has to do with actual topological compactness. “Stone’s representation theorem, blah, blah, blah.”
Ah, so much to say and so little time. I thank the author for a job well done!