In category theory and specifically in topos theory, the notion of inhabited object in a general topos is a generalization of the notion of inhabited set from the archetypal topos Set. It is the semantics of an inhabited type in type theory.
This is equivalent to saying that the unique map is an epimorphism. More generally, we can make the same definition in any regular category, in which case it is equivalent to being a regular epimorphism.
On the other hand, is said to be externally or globally inhabited if there exists a morphism , i.e. a global element. The adjective “externally” is perhaps misleading, since this notion can also be expressed in the internal type theory of the topos (as having an element/term ), though not in its more usual internal higher-order logic. The internal assertion “every internally inhabited object is globally inhabited” is a version of the propositional axiom of choice? (PAC).
Every globally inhabited object is internally inhabited, since every split epimorphism is a regular epimorphism. The converse is true if is projective, as is the case in a well-pointed topos (such as Set). (This may seem to contradict the above, since PAC can fail even in (constructively) well-pointed toposes, but the point is that PAC also says something about families of inhabited objects rather than single “globally defined” objects.)
Some sources use “ is inhabited” in the global sense, while others use the term “inhabited” only internally. Regardless, a pointed object always means one equipped with a global element (which is the same whether interpreted internally or externally), and the terms “global element” and “well-supported” are always unambiguous.
One situation where this plays a role is in the study of certain smooth toposes with objects of invertible infinitesimals . There is an immediate definition of such a topos, the topos called at Models for Smooth Infinitesimal Analysis, for which this object exists, but is not inhabited. Only the weaker internal statement is true.
But for some useful constructions in these toposes, such as for giving an internal definition of distributions as genuine functions (internally), it is desirable to have be inhabited. In the above situation this is achieved by forcing the existence of invertible infinitesimal elements. The result is the refined topos denoted at Models for Smooth Infinitesimal Analysis.
The category Set is a very special sort of topos. A generally useful principle in formulating categorical properties of Set is that “external” and “internal” notions should coincide. A general default understanding across the nLab is that is a well-pointed topos whose terminal object is projective and indecomposable. These latter properties of the terminal object hold for any well-pointed topos if the metalogic is classical; for purposes of constructive mathematics, we usually agree to add them in.
As remarked above, projectivity of easily makes internal and external inhabitedness agree. We also present a constructive/intuitionistic proof of the following result about emptiness.
In a well-pointed topos, an object which is not internally inhabited is empty (is an initial object).
Note that the negation “not” in this statement is an external one, which is what makes it nontrivial. If an object is internally “not inhabited”, then it is empty essentially by definition.
Let be the epi-mono factorization of the unique map . Here is monic but not epic if is not epic, and we use this to prove . In that case, we have a map , whence since in a topos initial objects are strict. (For given , the projection has a section, and it follows that since and retracts of initial objects are initial. This also implies that for any the map is monic: for it is obviously true that for any pair of maps we have that implies , as is initial by strictness and this makes automatic.)
Let be the classifying map of the mono , and let be the classifying map of the mono . There is no map , else would retract it and hence be epic. Hence it is vacuously true that for all , and so by well-pointedness. Hence the subobjects and coincide, forcing . (In other words, the presence of a subobject classifier causes any generator to be a strong generator.)
While writing this page, we had the following discussion about whether or not “ is inhabited” in a topos should be interpreted internally or externally, before deciding that we should mention both.
Mike: I strongly disagree that “inhabited” means “has a global element” in a topos. Intuitionistically, “ is inhabited” means “there exists an ” which when interpreted in the internal logic of a topos means that is well-supported. By contrast, the property of having a global element is not expressible in the internal language at all. “Inhabited” is also universally used in the topos-theoretic literature to mean well-supported.
Toby: Then what is the term for what I have called ‘inhabited’? At least one reference uses the term that way; I see (through Google) that it's used in the Elephant, but it's not in the index and I haven't managed to tell what the definition is. Certainly I'm not in the position to make a good literature search.
Mike: On p618 of the Elephant he uses “inhabited” to mean “there exists an ” in the internal language. What do you think about the change I made above?
Toby: I certainly can't disagree with any of the statements there.
I would like us to be a bit bolder with the terminology if it's safe and useful (neither of which condition has been established, of course). The Elephant has its share of terminological changes (like ‘cartesian category’ and ‘cartesian morphism’, which I remember got a lot of complaints on the categories mailing list), so I'd want to check its references (which I can do later).
The wiki saved a previous version of your comments; since you changed it, I won't hold you to it. But having read it does inspire me to say that the terminology that comes naturally to me is indeed ‘inhabited’ for having a global element and ‘internally inhabited’ for being well supported; the latter seems at least as well justified as ‘internal axiom of choice’ (as used in, say, Mac Lane \& Moerdijk). That's just me, of course.
Mike: Yeah, I wasn’t sure if it would (there seems to be a certain timeout within which multiple edits by the same person overwrite each other?). I also wasn’t sure if it was kosher to remove/change my comment; perhaps I shouldn’t have. The reason I changed it was that “inhabited” and “internally inhabited” did start to make a little sense. My current feeling is that “inhabited” is a set-theoretic term, and as such should only be used in set-theoretic-like situations. This includes (1) constructive set theory, (2) IHOL and hence the internal language of a topos, and (3) a well-pointed topos. If we are talking about an arbitrary topos, and it is not clear that our statements are to be interpreted in the internal logic, I would rather use “has a global element” and “is well-supported” since they are both unambiguous.
You are certainly right about the Elephant and terminology. “Cartesian category,” “prone morphism,” and “effective regular category” are the ones that come to mind that seem to have been rejected by much of the categorical community.
Mike: Another thought: one could argue that just as a “ring” in a topos means a model of the theory of rings, and likewise for many other concepts, so should an “inhabited object” mean a model of the theory of inhabited objects, which is the same as a well-supported object. Of course this fails for “projective object,” but I don’t think there is a “theory of projective objects,” at least not one interpetable in the usual internal logic of a topos. And I suppose maybe it fails for “choice object” too, although that probably depends on whether you formulate the theory of choice objects to be equipped with a choice function or merely assert that one exists. Perhaps the literature is not very consistent in its use of “internally” or lack thereof.
Toby: I see ‘inhabited’ as just a convenient abbreviation of ‘that has an element’ (convenient enough that ‘is inhabited’ is still nicer than ‘has an element’). So an inhabited object should naturally be an object with an element (a global element, that is; every object has a generalised element, and we must at least reproduce the situation in ). And after all, ‘inhabited’ is hardly more of a set-theoretic term than ‘axiom of choice’! (^_^)
Maybe life would be simpler if we always internalised using the internal language, but there's a lot of precedent that we don't, probably because it's a lot easier not to. And anyway, that's what the word ‘internalised’ is for!
As for the timeout, I think that it's an hour, although I haven't timed it carefully. (It definitely exists.) I think that it's fine to remove or change old comments, certainly if they haven't been replied to, but one should be aware that they can still be read. (And if you're not sure if they can still be read, try ‘See changes’ or ‘Back in time’ below.)
Mike: Well, I think that I will continue using “has a global element” myself for clarity, but you’ve convinced me that it’s not entirely unreasonable to use “is inhabited” to mean the same thing. Though I reserve the right to reopen the discussion if we discover precendent to the contrary. (-:
A different question, as you mentioned earlier, is whether it is useful. How often do we want to talk about objects that have a global element? We may frequently care about pointed objects, which are equipped with a global element, but there isn’t any dispute about what to call those.
Toby: You've got a good point there. Probably ‘pointed object’ and ‘well-supported object’ are the only really useful notions. Actually, I think that a lot of constructivists (the ones that are really think that mathematics should talk about constructions, like Bishop and Coquand) would say that an inhabited set and a pointed set are really the same thing. We can distinguish them, of course, by their morphisms (or even isomorphisms), but that doesn't mean that we need two words (just as we don't use two different words for metric spaces with, say, continuous maps between them and uniformly continuous maps bewteen them). So as you move towards my position, I move towards yours ….
Mike: Does that mean you might be satisfied with the way it’s written now? (I added a note about pointed objects.)
Toby: Yes, I'm happy now for now.