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Right, one could regard this is a functor to the 1-category of categories. But if one thinks of as a 2-category, then every sensible map into it out of a category is to be called a pseudofunctor, even if it respects composition on the nose.
(The idea is that up to the respective concept of equivalence, it is not actually possible to say that anything 2-categorical respects composition on the nose, saying so really involves breaking the principle of equivalence a bit and looking at particular representatives. )
I find it inconsistent to have entries called overcategory (one word) and under category (two words). Wouldn’t it be nicer to choose one convention?
I would have changed overcategory to “over category”, but I don’t want to annoy someone.
The naming was even more inconsistent. The entry itself uses ’over category’ and links to over topos and over (∞,1)-category.
I renamed it and fixed up the redirects.
Personally I prefer “slice” and “coslice”, which seem more common among category theorists (or at least the ones I spend time with), but if we’re sticking with “over” then I would prefer “overcategory” or “over-category” to “over category”, since “over” is not an adjective (and likewise for “under”).
I’d be happy with the hyphenated version. (though I also prefer ’slice category’ over all the existing options)
It used to be that a single word such as overcategory was better for being searched but Google has gotten much better at giving the same results for “over category”.
I would actually prefer camel case which is becoming quite common in branding, e.g. “YouTube”, “HarperCollins”, “iPhone” and “FedEx”.
Camel case conveys that the word is a unit and also indicates component boundaries. A word like coopen might be hard to parse while coOpen shows its structure. I wouldn’t be surprised if languages that allow very long compounds, such as German, start switching to camel case.
I also prefer slice and co-slice, but would agree with hyphenation for over- and under-.
I like “over” and “under” because anyone can figure out what those words mean. True, it requires understanding the concept of an object “over” another, like a bundle over a space - but that’s a visually intuitive concept, since objects throw shadows on the ground below. On th other hand, “slice” seems like a randomly chosen word to me. You can learn it, but you have to deliberately decide to learn it.
However, I’m only arguing because I’m stuck in a cafe without enough to do.
Still better than the vague name “Comma Category”.
Actually, “over” and “under” can be confusing. For example, the category of commutative -algebras over a field is an under-category . But in ordinary parlance we speak of an algebra over . So it may not be entirely clear that anyone can figure out what the words are supposed to mean. (The usage is probably biased toward geometric pictures.)
But John raises a good point. It might be good to develop a little story about the choice of word “slice”, to help cement the meaning. I think it can probably be done.
Honestly, I like the term “filter category” instead of slice or overcategory. That is what we are doing, filtering everything to a chosen object, morphism, 2-morphism… n-transfor.
(The usage is probably biased toward geometric pictures.)
Yes, when we speak of a commutative algebra over , we are secretly imagining it as an affine scheme equipped with a map down to the affine scheme corresponding to .
(This is probably not the origin of the term “algebra over ”, but I’ll shamelessly twist history to make the terminology seem logical.)
I really hope that camel-case doesn’t infect mathematics.
I really hope that camel-case doesn’t infect mathematics.
I agree. I remember that Tom Wolfe enjoyed poking fun at the trend in A Man in Full.
Re #15: cf BanAnaMan, though that might be apocryphal. More seriously, don’t a lot of categories go by names that are some version of camel-case?
I guess you’re right David, for example and the like. But aren’t those are by and large abbreviations as opposed to full-blown noun (phrases)?
Right, the “name” of a category in this sense is mathematical notation, like and . Variables in a computer program are also often named with camel-case. That’s totally different in my mind from the English words that we use to describe mathematical concepts, even if the former are sometimes obtained as abbreviations of the latter.
@Mike ah, I see. I also didn’t read the suggestions in #8. I agree that actual names for things should be well thought out. There’s a lot of linguistic constructions that are overused (such as the suffix -oid, in some cases) in naming mathematical things.
I recall that someone (perhaps Frank Adams?) said the ’use of ’over’ was overdone’ or something similar. (This was at the time that Ioan James was talking about ex-homotopies, ex-maps, ex-spaces, and so on.)
It’s far from clear to me that someone hearing the word “overcategory” without having previously encountered the notion would be able to figure out what it means. And if they have encountered the notion, then they probably encountered it with a name, and calling it by the name they learned will make it unnecessary for them to do any figuring at all. Moreover, when talking about a particular slice category, one usually says “the slice category of over ”, so the word “over” is there anyway.
The fact that David and Todd also prefer “slice” somewhat supports my impression that that name is commoner among modern category theorists. CWM says “the category of objects over ”, but Sheaves in geometry and logic says “slice category”, as do the Elephant and Riehl and Spivak and Leinster. One might make an argument that “overcategory” would have been a better name, but I don’t think “slice category” is a bad name (in particular, once you learn it, it’s totally unambiguous), so I think it’s hard to justiify going against the trend.
Just as a reminder: the name of an entry on the Lab is somewhat secondary, it mainly serves to produce a URL. For human users what matters more is that 1) all imaginable synonyms redirect to the entry and 2) the Idea section explains the status of the terminology convention in the literature, so that the reader knows what’s going on.
Therefore I suggest that instead of looking for a winner among the arguments on terminology exchanged above, these arguments should be added to the Idea-section of the entry, to the extent that they have not already.
I think the name of the entry does matter. It suggests to newcomers what they should call a concept, or at least what the “nLab opinion” about its name is.
I like the over/under terminology, but I never (at least in writing) use just bare “overcategory” but rather “category of objects over ”. I don’t really have an opinion on what makes the best entry name, but I want to point out that there is a number of related notions that could use a consistent naming pattern.
Here is an example of what I mean. Given a category and a -shaped diagram in some other category, I like to call its extensions to “cones under ” and extensions to “cones over ”. This is a direct generalization of objects over/under when . This seems non-standard, but I find it concise, unambiguous and fairly self-explanatory (at least to people who are already used to objects over/under). I haven’t had a need to go beyond that, but there is a number of cone-like and slice-like notions out there that I think may benefit from this naming convention.
I think “cone under” and “cone over” are pretty standard, actually, although the former are sometimes called “cocones under” instead.
Are they really standard? I thought that “cones” and “cocones” are standard, but it seems that I always see people write “cones over” as well as “cocones over”.
I suppose some people might say “cocones over”, but I’m sure I’ve seen “cocones under” too.
6 How about many compounds like overflow, underground, including verbs underwrite, underestimate, overestimate, overheat…? It seems to me that the rules for their meaning in compounds is standard and different than function in the sentence. I personally also prefer slice to a bit strange name overcategory (it actually confused me when I first few times saw it written in Lab/Forum community as I never heard it before I entered this circle of activities; by that time I had quite some reading in category theory, stack/sheaf theory and Grothendieck school of algebraic geometry). It was abolsutely not clear what it meant from the word, before I figured it out from the texts in Lab (what I did not do in the first encounter as I thought it was about some strange construction not of my concern).
As for ’why’ the word ’slice’: it might be something so simple as “slice” suggesting pictorially an inverse image. For example, for functions between sets, think of as slicing a set into individual slices (making the slice a category of “-parametrized slicings”, interestingly). The slice itself is the inverse image of over , so in other words a slice of according to the inverse image interpretation.
Well, this is one idea that occurred to me; maybe others have different suggestions.
It would be interesting to know who first used the word “slice”. Your description of the domain as sliced up into the fibers is one that also occurred to me.
Isn’t it also the pun on the slash notation ?
It never occurred to me that it might be, but I suppose it might. Maybe the categories mailing list would know who invented the term?
The notation might be from Grothendieck, I have a vague feeling that the pure category theorists in old times preferred the comma notation (which is more general).
I was just editing a manuscript and the ’over’ v. ’slice’ terminological point was relevant so let me say that I have always found that ’slice’ was not clear as it gave no intuition as whetehr it was ’a/A’ or ’A/a’. Oh I know it is properly defined in the usual sources, but the image of ‘slicing’ was vague. I think the use of the comma category notation is less ambiguous, whilst still talking about objects over a or under a.
You think is less ambigous than ? Or are you talking about ? I find it perfectly clear in that is “over” , just like in the 1 is “over” the 2.
The only notation that I find really execrable (apart from that no one seems to use any more, thankfully, but has left its trace in the name “comma category”) is , in which it looks like is still “over” but what is meant is the co-slice under .
’slice’ was not clear as it gave no intuition as whetehr it was ’a/A’ or ’A/a’.
More confusingly is that one of those things is a coSlice.
under category rather disparagingly says
Given a category and an object , the under category (also called coslice category) (also written and sometimes, confusingly, ) is the category whose …
I think it is confusing to use the same operator symbol for two different operations where you have to disambiguate by from the type of the arguments.
I’ve been looking at the category of bi-pointed edge labeled quivers on a fixed set of labels which involve slice and coslice in its definition.
( is the one object quiver with edges, is the free rooted action of on a set - the infinite edge labeled tree)
At least with different symbols you know what things are supposed to be categories and what are objects.
[ people like me with camelCased last names are more sympathetic to camelCase ]
and are not two different operations; they are both special cases of the same operation , usually called a comma category.
I was meaning the notation or . They are cumbersome but unambiguous. I like A/a and probably a/A is not too bad, but the / and \ based notations mix poorly with other contexts sometimes e.g. in groups. My problem is with ’slice’ and ’coslice’ as they do not conjure up an image (unlike ‘over’ and under’, as you say.) I am not suggesting any change but those names are not that clear to me.
and are not two different operations; they are both special cases of the same operation , usually called a comma category.
Yes, formally, Often informally one speaks of being an object in and doesn’t explicitly state that the two symbols in really stand for an inclusion of into and the identity on . It is this rotational convention and elision where the two distinct operations exist.
I’ve never ever been confused by this – it’s always obvious to me from context what is a category and what is an object, especially because people tend to use separate typefaces for categories and their objects, such as or . Whereas I have frequently been confused by .
My problem is with ’slice’ and ’coslice’ as they do not conjure up an image
I was addressing precisely that in #30, and offering a possible image to keep in mind.
Todd: That does help.
it’s always obvious to me from context what is a category and what is an object
is not machine parsible into something that adds inclusions and identities without knowing the types of its arguments which makes the parsing much more complicated. For parsing the common abbreviated notation one really needs 3 operators - slice, coslice, and real comma category where the latter has arguments that are explicitly functors and doesn’t imply additional structure.
Fortunately, most mathematicians are not machines.
Proposition 3.4 in the article overcategory suddenly starts talking about limits in undercategories. Yet there is a separate article undercategory. Should this proposition be moved there?
Yeah, I guess so, although I guess the author meant to dualize and then forgot?
Yes I think this was laziness on my part. The bit on (co)limits was written for undercategories, but the entry for overcategories was the more comprehensive one which should contain such information.
We should (copy it over to “undercategory”) and edit it to apply to overcategories.
I have copied over the statement and proof to undercategory, then dualized the statement here and added pointer to there.
(Being interrupted now, hope all is consistent.)
Added in examples
The fundamental theorem of topos theory states that the slice category for any object in a topos is itself a topos.
For a monoidal category the slice category over any monoid object is monoidal.
Is there more to be said about the latter case?
Anonymous
A special case of the second example is the construction of monoidal toposes as slice toposes over monoid objects. I have added (here) pointer to that example.
It would be nice to have a reference for where slice categories were introduced. Does anyone have an idea where this might be?
Re #53: do you mean a reference predating Lawvere’s introduction of comma categories in his thesis?
Re #53: do you mean a reference predating Lawvere’s introduction of comma categories in his thesis?
Yes. I notice that in Marta Bunge’s thesis, she attributes the concept both to Grothendieck and Beck, but does not give an explicit reference.
Added a cross-reference to slice of presheaves is presheaves on slice. At the moment, this content is duplicated here.
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