Splitting epis by wishful thinking

Splitting epis by wishful thinking

[Andrej Bauer]

On Mathoverflow there is a discussion (see
http://mathoverflow.net/questions/117921/relative-consistency-of-etcs-over-the-theory-of-a-well-pointed-topos-with-nno)
which got me thinking.

Is there a construction which "freely" splits all epis in a category
C? Something like: we add sections to every epi and wish we are done?

The question is somewhat lose, but I think it is clear nontheless.

With kind regards,

Andrej


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Re: Splitting epis by wishful thinking

[Richard Garner]

Hi Andrej,

This isn't exactly what you want, but it's along the right lines. Given a
small category with cokernel pairs, one can construct another category with
cokernel pairs in which all epis have been "freely" split. By this, I mean
that a chosen section has been freely added to every epi in the category,
even the ones that already had a section; thus the construction is not
idempotent. Basically one uses the small object argument.

Consider the category K of small categories with cokernel pairs, and
functors preserving such. Let C be the free category with cokernel pairs
containing an epi e: it can be obtained by first forming the free category
with cokernel pairs on an arrow f, and then coinverting the codiagonal of
f. Let D be the free category with cokernel pairs containing a
section-retraction pair (i,p). There is an obvious map C --> D in K which
sends e to p. Now some E in K satisfies the axiom of choice if and only if
it is projective (has the weak right lifting property) with respect to this
map C --> D.

K is locally finitely presentable, and so the map C --> D generates via the
small object argument a weak factorisation system (L,R) on it. By the above
argument, the fibrant objects for (L,R) are those small categories with
cokernel pairs satisfying the axiom of choice. If one uses the algebraic
version of the small object argument, the fibrant replacement for this
w.f.s. is a monad, S, say. The action of this monad on objects freely
adjoins sections for all epis; its algebras are precisely the small
categories with cokernel pairs with a chosen section for each epi.

One can ask what happens if one drops the assumption of cokernel pairs.
Consider the category Cat_ac, whose objects are small categories in which
every epi comes equipped with a chosen section. There is an obvious
forgetful functor Cat_ac ---> Cat, and a more precise formulation of your
original question would be to ask if this functor has a left adjoint. This
is unclear to me; at the moment I feel like it probably doesn't. What does
seem clear is that, if it does have a left adjoint, then it can't possibly
be monadic, so whatever construction one gives won't be entirely honest or
straightforward.

Richard


On 3 January 2013 23:36, Andrej Bauer <andrej.bauer@andrej.com> wrote:

> On Mathoverflow there is a discussion (see
>
> http://mathoverflow.net/questions/117921/relative-consistency-of-etcs-over-the-theory-of-a-well-pointed-topos-with-nno
> )
> which got me thinking.
>
> Is there a construction which "freely" splits all epis in a category
> C? Something like: we add sections to every epi and wish we are done?
>
> The question is somewhat lose, but I think it is clear nontheless.
>
> With kind regards,
>
> Andrej
>

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Re: Splitting epis by wishful thinking

[Steve Vickers]

Preliminary thoughts:

1. Certainly for any category C and any set of morphisms in it you can freely freely adjoin splittings for all those morphisms. In particular you might start with the set of all epis in C. (They're going to become epi anyway.)

This construction is just universal algebra, but the universal algebra dooen't answer questions like (a) Is the functor from C faithful? (b) Does the new category inherit nice properties of C? (c) Do all epis split in the new category?

2. Suppose you are willing to restrict to categories with finite colimits. Then epis can be characterized equationally, so there is a cartesian theory of finite colimit categories in which every epi has a chosen splitting. Then the forgetful functor from such categories to finite colimit categories has a  left adjoint. You can extend this to deal with categories with additional structure on top of the finite colimits, so long as the additional structure can be expressed as cartesian theory.

Happy New Year,

Steve.





On 3 Jan 2013, at 12:36, Andrej Bauer <andrej.bauer@andrej.com> wrote:

> On Mathoverflow there is a discussion (see
> http://mathoverflow.net/questions/117921/relative-consistency-of-etcs-over-the-theory-of-a-well-pointed-topos-with-nno)
> which got me thinking.
> 
> Is there a construction which "freely" splits all epis in a category
> C? Something like: we add sections to every epi and wish we are done?
> 
> The question is somewhat lose, but I think it is clear nontheless.
> 
> With kind regards,
> 
> Andrej
> 

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Re: Splitting epis by wishful thinking

[Ohad Kammar]

Dear Andrej,

The following formulation of your question has negative answers:

Let Cat be the category of small categories and functors between them,
and SCat be its full sub-category of small categories in which all
epis split. Let U : SCat -> Cat be the full inclusion.

U does not have a left adjoint, nor a right adjoint.

Proof:

Let C be the following category:
It has 3 objects:
0 - an initial object
A and B.

Apart from the identities and the initial maps, C has the following 3 morphisms:
a parallel pair f, g : A -> B
an endomorphism h : B -> B

The non-trivial compositions are given by:

x o h = x, for x = f, g, and h.

Note that C is in SCat, as it has no trivial epimorphisms: the
non-trivial arrow into A equalises f and g, which are different, and
the non-trivial arrows into B equalise h and id.

Let F : C -> C be the endofunctor that swaps f with g.

If U had a left adjoint, then SCat was a full reflective subcategory
of Cat, it was complete. Consider the equaliser of F, Id : C->C.
Whatever it is in SCat, this coequaliser cannot be preserved by U, as
the equaliser in Cat is given by dropping f and g. This resulting
subcategory has an epi, the initial arrow into A, that doesn't split.

Similarly, if U had a right adjoint, then SCat was cocomplete. Then
the coequaliser of F, Id : C->C in Cat would be C with g dropped
(=identified with f).  But then the initial map into A becomes epi,
without a section. Thus this colimit is not in SCat, and U doesn't
preserve it.

The same construction actually lies within the category Cat_ac that
Richard described (with specified sections), as C has no non-trivial
sections. Thus the same proof applies to his formulation, and his
forgetful functor also isn't a right nor a left adjoint.

Ohad.

On 4 January 2013 04:04, Richard Garner <richard.garner@mq.edu.au> wrote:
> Hi Andrej,
>
> This isn't exactly what you want, but it's along the right lines. Given a
> small category with cokernel pairs, one can construct another category with
> cokernel pairs in which all epis have been "freely" split. By this, I mean
> that a chosen section has been freely added to every epi in the category,
> even the ones that already had a section; thus the construction is not
> idempotent. Basically one uses the small object argument.
>
> Consider the category K of small categories with cokernel pairs, and
> functors preserving such. Let C be the free category with cokernel pairs
> containing an epi e: it can be obtained by first forming the free category
> with cokernel pairs on an arrow f, and then coinverting the codiagonal of
> f. Let D be the free category with cokernel pairs containing a
> section-retraction pair (i,p). There is an obvious map C --> D in K which
> sends e to p. Now some E in K satisfies the axiom of choice if and only if
> it is projective (has the weak right lifting property) with respect to this
> map C --> D.
>
> K is locally finitely presentable, and so the map C --> D generates via the
> small object argument a weak factorisation system (L,R) on it. By the above
> argument, the fibrant objects for (L,R) are those small categories with
> cokernel pairs satisfying the axiom of choice. If one uses the algebraic
> version of the small object argument, the fibrant replacement for this
> w.f.s. is a monad, S, say. The action of this monad on objects freely
> adjoins sections for all epis; its algebras are precisely the small
> categories with cokernel pairs with a chosen section for each epi.
>
> One can ask what happens if one drops the assumption of cokernel pairs.
> Consider the category Cat_ac, whose objects are small categories in which
> every epi comes equipped with a chosen section. There is an obvious
> forgetful functor Cat_ac ---> Cat, and a more precise formulation of your
> original question would be to ask if this functor has a left adjoint. This
> is unclear to me; at the moment I feel like it probably doesn't. What does
> seem clear is that, if it does have a left adjoint, then it can't possibly
> be monadic, so whatever construction one gives won't be entirely honest or
> straightforward.
>
> Richard
>
>
> On 3 January 2013 23:36, Andrej Bauer <andrej.bauer@andrej.com> wrote:
>
>> On Mathoverflow there is a discussion (see
>>
>> http://mathoverflow.net/questions/117921/relative-consistency-of-etcs-over-the-theory-of-a-well-pointed-topos-with-nno
>> )
>> which got me thinking.
>>
>> Is there a construction which "freely" splits all epis in a category
>> C? Something like: we add sections to every epi and wish we are done?
>>
>> The question is somewhat lose, but I think it is clear nontheless.
>>
>> With kind regards,
>>
>> Andrej
>>
>

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Scotland, with registration number SC005336.



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Re: Splitting epis by wishful thinking

[Fred E.J. Linton]

On Thu, 03 Jan 2013 08:55:07 PM EST Andrej Bauer <andrej.bauer@andrej.com>
asked

> Is there a construction which "freely" splits all epis in a category ... ?

To the responses already received I thought it perhaps worth adding the idea
of 
freely (or generically) splitting everything, after a fashion I first heard
described by
Bill Lawvere -- that is, freely adjoining, for each map e (epi or not), a  map
f with
  efe = e (and perhaps, if you like, fef = f) .

Note that, with e epi, efe = e will entail ef = id, i.e., f will be a  section
for e.
Likewise, for e mono, efe = e will entail fe = id, i.e., f will be a
retraction for e.
(In these two cases, of course, it will follow that fef = f. In general, tho',
... .)

One should, of course, ask oneself whether one should really be wanting
sections for 
quite all epimorphisms -- in the category [R] of unital rings R, for example,
should one 
really ever want the trivializing homomorphisms !: R --> 1 to the terminal
ring to split,
anywhere?

Cheers, -- Fred 





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