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WikiProject Mathematics (Rated C-class, High-importance)
WikiProject Mathematics
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 Field: Foundations, logic, and set theory

I was just wondering what the categorification of a topos was, but I don't think I'll find it here though.--SurrealWarrior 00:38, 10 May 2005 (UTC)

I don't think that categorification has ever been rigorously defined. I unfortunately don't have time to speculate on what categorification should mean in this context. - Gauge 22:47, 25 August 2005 (UTC)

A categorified topos would be a 2-category with categories as its objects.--SurrealWarrior 20:37, 22 January 2006 (UTC)

Under Giraud's axioms, I assume "arbitrary" colimits should be taken to mean "small"? I would edit this myself but I don't know for certain.--charleyc 3:54, 2 December 2007 (UTC)

Yes, I think that's what was meant by "arbitrary". Hope it's clearer now. Sam Staton 10:27, 2 December 2007 (UTC)

power-object  \neq exponential[edit]

In the article there is a "wrong link", namely the one titled "power object" which leads to exponentials.

I think the connection comes from the fact that in a topos a power object can be defined as the set of morphisms between an object S and the subobject classifier. I don't know of a definition that doesn't refer to the subobject classifier.--SurrealWarrior 20:44, 22 January 2006 (UTC)
Such a definition does exist. See Categorical Foundations, Cambridge University Press for a discussion of the relationship of power objects and the subobject classifier. Marc Harper 14:52, 11 August 2006 (UTC)
I removed this link, which is misleading and wrong. My reference is Johnstone's Sketches of an Elephant, Section A2.1. Sam Staton 12:48, 5 April 2007 (UTC)

yoga ?[edit]

I realize the sanskrit word 'yoga' comes from a root meaning 'to yoke' -- and I'm certainly open to the possibility that I just haven't encountered this usage, but I find referring to the 'yoking' of topological intuitions with some other mathematical setting a little jarring and overly florid (no offense). Could someone perhaps justify this word choice? Zero sharp

I've done something about this (yoga is jargon of the Grothendieck school). Charles Matthews 11:35, 17 May 2006 (UTC)

Link to intro article[edit]

(Copied from User talk:KSmrq#Topos.)

Carcharoth had suggested to me that Topos was incorrectly tagged with the seeintro template, and after taking a look I agreed with him. In this case there has already been a previous discussion, please do not revert without participating in it. If you take a look at other introduction to articles (like Introduction to special relativity and Introduction to general relativity) you will immediately realise that the background and genesis article does not meet the criteria. It is just what its name claims, an article on background and genesis of topos theory. Please note that the article Introduction to topos does not exist, it is a redirect. In addition, the lead already contains a link to the background and genesis article, we don't need to add another one just before that. I'll appreciate it if this does not turn into another revert war. Loom91 21:06, 18 July 2007 (UTC)

Carcharoth is the editor who had just added the tag only hours before you removed it:
07:43, 2007 July 16 Loom91 (Talk | contribs) (13,559 bytes) (no true introductory article exists)
12:02, 2007 July 15 Carcharoth (Talk | contribs) (13,572 bytes) (add seeintro)
A similar tag was previously added by Filll and then removed by you (with no discussion):
11:07, 2007 January 25 Loom91 (Talk | contribs) (remove unjustified seeintro template)
02:11, 2007 January 14 Filll (Talk | contribs)
Furthermore, it was also Carcharoth who created the redirect:
12:03, 2007 July 15 (hist) (diff) Introduction to topos (←Redirected page to Background and genesis of topos theory)
I see no discussion on the topos talk page (which I watch), nor on your talk page, nor on Carcharoth's talk page about Topos. A broad-ranging search for 'Carcharoth' and 'topos' turned up a brief mention at a completely unrelated FAC discussion, but the statements there do not support your claim. You have acted unilaterally, without explicit endorsement.
I'm sure Carcharoth is quite capable of removing the tag personally. Meanwhile, I'm reverting you a second time, and I suggest you leave it alone. --KSmrqT 04:36, 19 July 2007 (UTC)

Carcharoth did suggest to me that while he had added the tag, he doubted that it was correct and asked me to review. I've asked him to join the discussion. Meanwhile, you have followed your usual practice in not replying to the reasons I gave for removing the tag. Untill and unless you engage in discussion, I'm reverting you. Thank you. Loom91 11:13, 19 July 2007 (UTC)
Loom91 is correct. I failed to tidy up after myself. The redirect should be deleted, or a proper introductory article created. The current compromise hatnote wording of "historical" is fine. Carcharoth 11:29, 19 July 2007 (UTC)
Thanks, Carcharoth. If you're satisfied, I'm satisfied.
The article gets a little attention now and then, but I don't expect to see a "layman's intro" in the near future. Computer science uses topos theory for the semantics of programming languages, and mathematics uses it in a variety of ways. Either way, it tends to be graduate-level material. The naturally-occurring motivations, examples, and definitions already require an experienced reader. At the very least, we depend on a basic knowledge of category theory. --KSmrqT 12:47, 19 July 2007 (UTC)
This is settled then. Loom91 12:49, 19 July 2007 (UTC)
Not necessarily. See if the explanation section I just added helps at all. --Vaughan Pratt 10:04, 27 September 2007 (UTC)

Suggestion about explanation[edit]

Dear Vaughan, I just noticed your recent explanation. I am sure that many will find it helpful, although I am a little concerned that it is quite verbose. I wouldn't like someone to see such a long piece of prose and conclude that toposes are rather complicated.

Perhaps this is only my impression, and I leave you to decide whether it is is worth anything. But to illustrate my point:

  • The terminology "functional action" is introduced but never really used. The same goes, to a lesser extent, for the terminology "concrete".
  • The discussion about why equalizers suffice is beside the point. It wouldn't hurt just to say "a topos ... having all limits". Also, I assume you explictly include 1 in the components of a topos to avoid chosing it later. Some would say that is a bit pernickety. In any case, a definition of topos already appears a few lines above; shouldn't it only be defined once on the page?
  • The business about "chairs" and "tables" is a bit poetic for me (!).
  • The problems about equivalence classes of monos are something that some topos theory teachers would sweep under the carpet for the newcomer.

Anyway, I leave it to you to decide what, if anything, can be done for my complaints. Perhaps paragraph headings would make it more approachable. But I admit that you have a considerable amount more experience than me, both in teaching and research. With best regards, Sam Staton 17:46, 28 September 2007 (UTC)

Thanks Sam, excellent feedback, which helped me focus on what changes were needed. Hopefully it addresses all your concerns. Regarding equivalence classes of monos, I reversed the order of the first and second order classifications -- I agree the latter can be deferred but I don't see how a teacher can get away with diverting attention away from the implicit classification when it's staring the students in the face. If they can't see that the f's classify the monics then they're missing the whole point. --Vaughan Pratt 02:44, 2 October 2007 (UTC)

Vaughan, I like what you've done there. Thank you! And I now agree with you on the point of classification. Sam Staton 09:59, 2 October 2007 (UTC)

Thanks, Sam. In retrospect I think my ideal organization of the elementary topos material would be to spin off a separate article from the Grothendieck toposes (as per the proposal in the section below) and merge the subobject classifier article with the elementary topos material of this article. Alternatively, have three separate articles for cartesian categories, cartesian closed categories, and subobject classifiers, and then a much shorter article on toposes that defines the notion with reference to the CCC and SOC articles, lists some examples (including FinSet) and nonexamples (including FinSet^J for a suitable infinite J), and mentions some properties. --Vaughan Pratt 19:48, 3 October 2007 (UTC)

The second direction might be more sensible, since there are some (though perhaps not many) interesting categories with subobject classifiers that are not toposes -- for example, Boolean extensive categories. On the other hand, this subtlety might be something that can be explained on the topos theory page, to which "subobject classifier" would redirect. I think I am in favour of keeping a separate CCC page; I think many people study CCCs without thinking of toposes.

There is also the important point of "power objects", which are not explained (or defined) here, and really ought to be explained somewhere. I think a topos theory article would be an appropriate place for this. Sam Staton 16:31, 8 October 2007 (UTC)

Meanwhile I changed my mind on the order and moved the more general second-order definition of "subobject" back up to the beginning. This has the benefit of allowing the reader to follow the explanation with at least some goal in mind, even if it's not everyone's preferred motivation for toposes (is the characteristic feature of an elephant its trunk or its size?). --Vaughan Pratt (talk) 08:40, 14 March 2008 (UTC)

Separating Grothendieck and Elementary Toposes?[edit]

What do people think of spinning off the elementary topos section as an article in its own right? The developments of the two kinds of topos are qualitatively different, and the connection while certainly present is not elementary, making it inappropriate to maintain the appearance of an elementary connection by treating them under the one heading. The Boolean algebra (structure) and Boolean algebra (logic) articles have a more immediately obvious connection but even those have been split into separate articles. --Vaughan Pratt 20:48, 27 September 2007 (UTC)

This once was such a stubby article we were happy to see any substance added. I get the impression that writing on the logic aspect entices you. If a major expansion is brewing, I would support a split. We'd need to be careful, maybe converting Topos into a disambig page, so we don't cause "wrong link" problems. --KSmrqT 10:37, 28 September 2007 (UTC)
This makes sense to me. Some books, like that of Mac Lane and Moerdijk, do a good job of unifying the two accounts, but perhaps it is too ambitious to try to do this in an article. Sam Staton 17:46, 28 September 2007 (UTC)

Finite limits in elementary topos[edit]

Hi! I read in John Bell's "Development of categorical logic", in a note at page 4th that: "Lawvere and Tierney’s definition of elementary topos originally required the presence of finite limits and colimits—these were later shown to be eliminable." Anyone knows where it was demonstrated? Should the article be changed then? Thanks. Cyb3r (talk) 19:40, 14 December 2007 (UTC)


  • In the article it says: "The main utility of this notion is in the abundance of situations in mathematics where topological intuition is very effective but an honest topological space is lacking". I couldn't find examples of those situations in the article or hints to them. Can someone provide them?  franklin  20:12, 10 January 2010 (UTC)
    The étale-topos is the main motivating example: you want some sort of "topology" on a variety that encodes the étale structure, in order to define étale-cohomology, but such a topology simply doesn't exist (the Zariski-topology is not useful). So you define the étale-topos of the variety instead, and the cohomology machinery of toposes works just like the cohomology machinery of topological spaces. So you can use topological intuition even though there's no topology there. AxelBoldt (talk) 19:38, 17 December 2012 (UTC)
  • Also, in a different place it says: "Topos theory is, in some sense, a generalization of classical point-set topology. One should therefore expect to see old and new instances of pathological behavior. For instance, there is an example due to Pierre Deligne of a nontrivial topos that has no points (see below)." Where is that example? A "see below" is very dangerous in Wikipedia since is the destination is removed or moved it can be hard to notice that the see below is pointing no-where. If the example is actually there, can someone link it in a better way? If it is not there, can someone include or at least give a reference?  franklin  20:48, 10 January 2010 (UTC)
    The example is indeed not in the article; the "see below" was probably intended to point the reader to the definition of a point of a topos. I'll clarify it. AxelBoldt (talk) 19:38, 17 December 2012 (UTC)

The general topic 'TOPOS', needs a disambiguation page as it is used in several other disciplines aside from mathmatics.Kaeote (talk) 10:48, 29 October 2010 (UTC)

Elementary vs. Grothendieck toposes[edit]

It would be nice if the article could explain the relation between elementary and Grothendieck toposes. Is every Grothendieck topos an elementary one, or vice versa? If the two concepts are distinct, it might be better to have two articles, to allow for more precise linking from other articles. AxelBoldt (talk) 19:24, 17 December 2012 (UTC)

The former. Toposes categorify Boolean algebras in the sense that the proportion topos:Grothendieck topos:presheaf topos does for categories what the proportion Boolean algebra:field of sets:power set does for sets. In both cases the three are listed in order of decreasing generality (strictly so except for Boolean algebra:field of sets).
Just as a power set has the form 2X for a set X, making it a Boolean algebra, so does a presheaf category have the form SetC for a small category C, making it a presheaf topos. And just as a field of sets is any subalgebra of a power set (as a Boolean algebra), so is a topos of sheaves or Grothendieck topos any subtopos of a presheaf topos (Giraud's theorem). Lastly, just as a Boolean algebra is an axiomatically defined abstraction of a field of sets, so is a topos an axiomatically defined abstraction of a Grothendieck topos.
If people think it would be helpful and there are no objections, I may put this relationship in the lead. --Vaughan Pratt (talk) 21:33, 15 September 2013 (UTC)
Clarifying this relation in the lead would be useful, thanks. --Mark viking (talk) 00:43, 16 September 2013 (UTC)

What is "Presh"? I see here that you write "presheaf", is that what you mean? Please make it more clear. Thanks! Cammy169 (talk) 17:25, 24 April 2014 (UTC)

Why doesn't the next sentence, "Here Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a presheaf", explain this? Vaughan Pratt (talk) 07:37, 2 May 2014 (UTC)

Separate the math part and the logic part into two articles[edit]

I want to make it clear that I don't have much background in this area, but doesn't it make more sense to separate this article into two articles? The one is about the topos in mathematics and the other in logic. By the way, I personally want to see more discussion on infinity topos and Grothendieck's homotopy hypothesis. -- Taku (talk) 05:06, 28 November 2014 (UTC)

I support the split. The article is currently written as two separate articles, so the should be separated.Brirush (talk) 00:33, 30 November 2014 (UTC)

"Grothendieck logic"?[edit]

This page refers to "Grothendieck logic". Is this related to this topic? Can anyone go into more detail on the subject? -- Impsswoon (talk) 14:31, 21 March 2015 (UTC)

It's not a term I've heard before, and it would be historically inaccurate; but you might be right, and this page might be about the intended meaning. A good book on this subject is Mac Lane and Moerdijk, Sheaves in Geometry and Logic. Ozob (talk) 16:12, 21 March 2015 (UTC)
In the context you pointed out, I think it is more of an analogous construction than an actual topos. It is a construction over multiple logic systems in the form of Grothendeick institutions, for which we have an Institution (computer science) article. The construction is talked about in Foundations of Software Science and Computation Structures: 5th International Conference, FOSSACS 2002., p.334-335. --Mark viking (talk) 18:48, 21 March 2015 (UTC)