V-included categories

V-included categories

[Paul Blain Levy]

Hi,

Let V be a Grothendieck universe.?? A "V-set" is an element of V, and a
"V-class" is a subset of V.

Say that a category C is "V-included" when it has the following two
properties.

(1) ob C is a V-class.

(2) C(x,y) is a V-set for all x,y in ob C.

The advantage of V-inclusion over local V-smallness (i.e. condition (2)
alone) is that V-included categories are W-small for every universe W
greater than V, whereas locally V-small categories are not, in general.

Furthermore, all the standard categories constructed from V are
V-included.?? (Except for the ones that are not even locally V-small,
like the category of V-included categories.)

Is there a standard name for V-inclusion?

Paul





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Re: V-included categories

[Paul Blain Levy]

On 01/01/18 13:10, Paul Blain Levy wrote:
> Hi,
>
> Let V be a Grothendieck universe. A "V-set" is an element of V, and a
> "V-class" is a subset of V.
>
> Say that a category C is "V-included" when it has the following two
> properties.
>
> (1) ob C is a V-class.
>
> (2) C(x,y) is a V-set for all x,y in ob C.
>
> The advantage of V-inclusion over local V-smallness (i.e. condition (2)
> alone) is that V-included categories are W-small for every universe W
> greater than V, whereas locally V-small categories are not, in general.
Another advantage: the category of V-included categories is W-small.

Paul

>
> Furthermore, all the standard categories constructed from V are
> V-included. (Except for the ones that are not even locally V-small,
> like the category of V-included categories.)
>
> Is there a standard name for V-inclusion?
>
> Paul
>
>
>



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Re: V-included categories

[Eduardo Julio Dubuc]

Have you considered the following at the very begining of SGA4:

Remarque 1.1.2. ??? Soit D une cat??gorie poss??dant les propri??t??s suivantes :
(C1) L???ensemble ob(D) est contenu dans l???univers U .
(C2) Pour tout couple (x, y) d???objets de D, l???ensemble HomD(x, y) est un
??l??ment de U .

(Les cat??gories usuelles construites ?? partir d???un univers U poss??dent
ces deux propri??t??s: U -Ens, U -Ab,. . .).

Soit C une cat??gorie appartenant ?? U . Alors la cat??gorie
Fonct(C, D) ne poss??de pas en g??n??ral les propri??t??s (C1) et (C2). Par
exemple la cat??gorie Fonct(C,U-Ens) ne poss??de aucune des propri??t??s
(C1) et (C2).

C???est ce qui justifie la d??finition adopt??e de U-cat??gorie, de
pr??f??rence ?? la notion plus restrictive par les conditions (C1) et (C2)
ci-dessus.

best  e.d.

El 1/1/18 a las 10:10, Paul Blain Levy escribi??:
>
> Hi,
>
> Let V be a Grothendieck universe.?? A "V-set" is an element of V, and a
> "V-class" is a subset of V.
>
> Say that a category C is "V-included" when it has the following two
> properties.
>
> (1) ob C is a V-class.
>
> (2) C(x,y) is a V-set for all x,y in ob C.
>
> The advantage of V-inclusion over local V-smallness (i.e. condition (2)
> alone) is that V-included categories are W-small for every universe W
> greater than V, whereas locally V-small categories are not, in general.
>
> Furthermore, all the standard categories constructed from V are
> V-included.?? (Except for the ones that are not even locally V-small,
> like the category of V-included categories.)
>
> Is there a standard name for V-inclusion?
>
> Paul
>
>
>
>
>
>


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Re: V-included categories

[Paul Blain Levy]

Dear Eduardo,

On 01/01/18 21:14, edubuc@dm.uba.ar wrote:
> Have you considered the following at the very begining of SGA4:
>
> Remarque 1.1.2. ??? Soit D une cat??gorie poss??dant les propri??t??s
> suivantes :
> (C1) L???ensemble ob(D) est contenu dans l???univers U .
> (C2) Pour tout couple (x, y) d???objets de D, l???ensemble HomD(x, y) est
> un ??l??ment de U .
>
> (Les cat??gories usuelles construites ?? partir d???un univers U poss??dent
> ces deux propri??t??s: U -Ens, U -Ab,. . .).
>
> Soit C une cat??gorie appartenant ?? U . Alors la cat??gorie
> Fonct(C, D) ne poss??de pas en g??n??ral les propri??t??s (C1) et (C2). Par
> exemple la cat??gorie Fonct(C,U-Ens) ne poss??de aucune des propri??t??s
> (C1) et (C2).
Thanks for your reply but I am mystified by this statement in SGA4.?? It
appears to me Fonct(C,U-Ens) does satisfy both (C1) and (C2), so I must
be missing something.?? Here is my proof; please would you point out
where I'm going wrong?

Firstly: C is in U, so ob C and every object, every homset and every
morphism of C are in U, by transitivity of U.

For (C1), we must show that any functor F from C to U-Ens is in U.?? Any
such F is an ordered pair (ob F, mor F).??

- ob F is a set of ordered pairs (c,x) where c is a C-object and x is in
U. Such an ordered pair is in U.?? So ob F is a subset of U and its
cardinality is that of ob C so ob F is in U (Proposition 7 in the
Appendix of SGA4, p98).

- mor F is a set of triples (c,d,p) where c and d are C-objects and p is
a map from C(c,d) to Fc->Fd hence a subset of C(c,d) * (Fc -> Fd).?? And
Fc and Fd are in U, so Fc -> Fd is too by the Corollary to Proposition 6
(on p98).?? So C(c,d) * (Fc -> Fd) is in U, so p is in U, so (c,d,p) is
in U.?? So mor F is a subset of U, and its cardinality is that of (ob
C)*(ob C) which is in U.?? By Proposition 7, mor F is in U.

In conclusion F = (ob F, mor F) is in U.

For (C2), let F and G be functors from C to U-Ens.?? The set of natural
transformations F -> G is a subset of Prod_{c in ob C} (Fc --> Gc).?? For
any c in C, we know that Fc and Gc are in U, so Fc -> Gc is in U.?? So by
the Corollary to Proposition 6, Prod_{c in ob C} (Fc -> Gc) is in U, so
the set of natural transformations F -> G is in U.??

Best regards,
Paul




>
> C???est ce qui justifie la d??finition adopt??e de U-cat??gorie, de
> pr??f??rence ?? la notion plus restrictive par les conditions (C1) et
> (C2) ci-dessus.
>
> best?? e.d.
>
> El 1/1/18 a las 10:10, Paul Blain Levy escribi??:
>>
>> Hi,
>>
>> Let V be a Grothendieck universe.?? A "V-set" is an element of V, and a
>> "V-class" is a subset of V.
>>
>> Say that a category C is "V-included" when it has the following two
>> properties.
>>
>> (1) ob C is a V-class.
>>
>> (2) C(x,y) is a V-set for all x,y in ob C.
>>
>> The advantage of V-inclusion over local V-smallness (i.e. condition (2)
>> alone) is that V-included categories are W-small for every universe W
>> greater than V, whereas locally V-small categories are not, in general.
>>
>> Furthermore, all the standard categories constructed from V are
>> V-included.?? (Except for the ones that are not even locally V-small,
>> like the category of V-included categories.)
>>
>> Is there a standard name for V-inclusion?
>>
>> Paul
>>
>>
>>



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Re: V-included categories

[rosicky]

But this is the usual definition of a locally small category in V -
objects form a class (more precisely, are equipotent
to a class) and hom-collections are sets.
Jiri Rosicky
Dne 2018-01-01 14:10, Paul Blain Levy napsal:
> Hi,
>
> Let V be a Grothendieck universe.?? A "V-set" is an element of V, and a
> "V-class" is a subset of V.
>
> Say that a category C is "V-included" when it has the following two
> properties.
>
> (1) ob C is a V-class.
>
> (2) C(x,y) is a V-set for all x,y in ob C.
>
> The advantage of V-inclusion over local V-smallness (i.e. condition (2)
> alone) is that V-included categories are W-small for every universe W
> greater than V, whereas locally V-small categories are not, in general.
>
> Furthermore, all the standard categories constructed from V are
> V-included.?? (Except for the ones that are not even locally V-small,
> like the category of V-included categories.)
>
> Is there a standard name for V-inclusion?
>
> Paul
>



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Re: V-included categories

[Michael Shulman]

I believe that in his paper "Notions of topos" Ross Street used the
name "V-moderate category" for this or a closely related notion.
There the point was that V-moderate categories have another advantage
over locally V-small ones, namely that their objects can (assuming the
axiom of choice) be well-ordered with all initial segments being
V-small.

On Mon, Jan 1, 2018 at 5:10 AM, Paul Blain Levy <P.B.Levy@cs.bham.ac.uk> wrote:
>
> Hi,
>
> Let V be a Grothendieck universe.?? A "V-set" is an element of V, and a
> "V-class" is a subset of V.
>
> Say that a category C is "V-included" when it has the following two
> properties.
>
> (1) ob C is a V-class.
>
> (2) C(x,y) is a V-set for all x,y in ob C.
>
> The advantage of V-inclusion over local V-smallness (i.e. condition (2)
> alone) is that V-included categories are W-small for every universe W
> greater than V, whereas locally V-small categories are not, in general.
>
> Furthermore, all the standard categories constructed from V are
> V-included.?? (Except for the ones that are not even locally V-small,
> like the category of V-included categories.)
>
> Is there a standard name for V-inclusion?
>
> Paul
>
>
>


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Re: V-included categories

[Paul Blain Levy]

On 03/01/18 00:20, edubuc@dm.uba.ar wrote:
> On 1/1/18 19:46, Paul Blain Levy wrote:
>> Dear Eduardo,
>>
>> On 01/01/18 21:14, edubuc@dm.uba.ar wrote:
>>> Have you considered the following at the very begining of SGA4:
>>>
>>> Remarque 1.1.2. ??? Soit D une cat??gorie poss??dant les propri??t??s
>>> suivantes :
>>> (C1) L???ensemble ob(D) est contenu dans l???univers U .
>>> (C2) Pour tout couple (x, y) d???objets de D, l???ensemble HomD(x, y) est
>>> un ??l??ment de U .
>>>
>>> (Les cat??gories usuelles construites ?? partir d???un univers U poss??dent
>>> ces deux propri??t??s: U -Ens, U -Ab,. . .).
>>>
>>> Soit C une cat??gorie appartenant ?? U . Alors la cat??gorie
>>> Fonct(C, D) ne poss??de pas en g??n??ral les propri??t??s (C1) et (C2). Par
>>> exemple la cat??gorie Fonct(C,U-Ens) ne poss??de aucune des propri??t??s
>>> (C1) et (C2).
>> Thanks for your reply but I am mystified by this statement in SGA4.?? It
>> appears to me Fonct(C,U-Ens) does satisfy both (C1) and (C2), so I must
>> be missing something.
>
> It is clear that U-Ens^C satisfy (C1) and (C2) (see the practice of
> category theory by any mathematician).
OK, I mistakenly assumed you were endorsing the statement you quoted.??
Sorry for boring you with this obvious proof.
> Now, it is necessary to see what exactly means "Fonct(C,U-Ens)"
> and/or?? "appartenant ?? U" in SGA4.
:-)???? Alternatively: the authors just made a mistake.?? And evidently,
had they not made this mistake, they would have defined "U-category" by
(C1)--(C2), since they regard these conditions as a priori natural.??
That's good to see.

Paul




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Re: V-included categories

[Steve Vickers]

Dear Thomas,

What I'm curious to understand is how inevitable is a certain practical behaviour of universes. In many situations we know they are foundationally necessary, but we still seek ways to disregard them in everyday mathematics.

If we try to use generalized spaces as our prime notion of collection, instead of sets, it begins to look different. We still have universes of a kind. For example the object classifier is a universe whose elements are sets (discrete spaces). But one would not make the mistake of thinking it is a set itself, as it has a non-discrete topology - it even has non-identity specialization morphisms in functions between sets. There are various other universes  for various other kinds of spaces, such as the Boolean algebra classifier for Stone spaces. The different universes represent qualitative distinctions,  not just one of "size".

As you know, I'm trying to do something along those lines but based on arithmetic universes instead of Grothendieck toposes. 

All the best,

Steve.

> On 4 Jan 2018, at 11:00, streicher@mathematik.tu-darmstadt.de wrote:
> 
> Dear Steve at al.
> 
> I think in one or the other form universes are inevitable be they type
> or set theoretical. Of course, you can do category over fairly general
> bases like finite limit categories (as Benabou developed some time ago).
> But then you are bound to work externally since one can speak about a
> fibration in the internal language of its base in an only very
> restricted way.
> 
> Thomas



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Re: V-included categories

[Thomas Streicher]

Dear Eduardo,

thanks for your attempts to clarify the situation!

> Aparently the universes are not closed in the sense that:
>
> 1) X ~ Y, Y belongs U  ===> X belongs U
>
> which is currently accepted (as we see in Thomas posting above) in the naive
> practice of category theory with universes.

I certainly did not incline this (despite my partial involvement into
HoTT :-))
I rather tacitly assumed that a locally U-small means that all homsets
are elements of U. This I find natural though I read that Grothendieck
and Verdier formulated it much more liberally.
However, using AC every locally U-small category is isomorphic to some
category where all hom-sets are elements of U. Moreover, every category
equivalent to such a category is locally U-small in the liberal sense
of Grothendieck and Verdier.

Now, for a category C which is locally U-small in the restricted sense
[C^op,U] is locally U-small in the restricted sense as shown by Paul's
argument.
But if C' is locally U-small in the liberal sense then [C'^op,U] \cong
[C^op,U] and the latter is locally U-small in the restricted sense and
thus [C'^op,U] is locally U-small in the liberal sense.

Thus, I think in SGA4 they made a mistake. No question that both guys
were great mathematicians but that doesn't prevent them from making
little mistakes in logic.

Thomas


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Re: V-included categories

[Thomas Streicher]

Dear Steve,

I do understand your intentions. But I think one cannot mess with
foundations and in particular not with Russell's insight that there is
not a set of all sets. Most mathematicians either don't care and
consider these kind of things as a nuisance.

In type theory one definitely doesn't try to work with a type of
all types. Universes have to be taken seriously but one want's to keep
them in the background (i.e. avoid "universe cycles" as was done in LEGO
as implemented by Randy Pollack).

I personally think that all mathematics needs a foundation. ZFC set theory
with the axiom that all sets are elements of some Grothendieck universe
is a formidable foundation of mathematics.
Set theory allows one to formulate irrelevant (i.e. not iso-invariant)
questions one may want to stick to some type theory where they can't
even be formulated. But even there cannot exist a top most (all
inclusive) universe.

Happy New Year,
Thomas


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Re: V-included categories

[Richard Williamson]

Hello,

> The authors are Verdier and Grothendieck, I doubt they made
> mistakes, and specially in the very basic definitions of the
> whole theory.

This made me smile, but I agree!

> There is something odd here but I am not inclined to take time to
> clarify it, I will sit and wait to see if some in the list come out with
> an explanation.

I think the explanation is rather simple. The first point is that the
foundations here are Bourbaki set theory: everything is a set,
including a category.  When they say that C belongs to U, they
literally mean that C, when rigorously formalised as a set, is an
element of U. The same goes for a functor.

The second point is that U-smallness of a set X is defined to be a set
which is isomorphic to an element of U, not necessarily an actual
element of U. The word 'isomorphic' is underlined.

Let us consider (C2) first. The axioms of a universe only allow one to
construct sets in the universe from other sets in the universe. Since
U is not in U, there is no way that any definition/construction
involving it can produce a set in U. Certainly one needs to involve U
to be able to define the Hom sets of Func(C, U-Ens).

But the Hom sets of Func(C, U-Ens) are isomorphic to elements of U,
i.e. are U-small. This is because U-Ens is a U-category, i.e. the Hom
sets are U-small.

One can make the same point about the set of objects of Func(C,
U-Ens).  One needs U to define it, so it is certainly not an element
of U. It is not even isomorphic to an element of U, because this would
imply that the cardinality of U is strictly less than U. In (C1), it
is not exactly this that is asked, but rather that the set of objects
of Func(C, U-Ens) is a subset of U. This is impossible for the same
kind of reasons: one will need to consider for instance a subset of
the set C x U to define it, and there is no way that show that this is
a subset of U (it is perfectly possible if instead of U we have some
element u of U, because then the product of C and u belongs to U).

Hence Func(C, U-Ens) is, as SGA claims, the prototypical example to
illustrate why the definition of a U-category is exactly the way it
is.

Best wishes,
Richard


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