Re: Reflexive coequalizers

Re: Reflexive coequalizers

[Jiri Adamek]

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On Mon, 9 Oct 2006, Richard Garner wrote:
>
> I have a proof that the indiscrete category functor
> Set -> Cat preserves reflexive coequalizers which,
> although straightfoward, uses the explicit
> description of colimits in Cat. Is this necessary,
> or can I deduce the result from general
> principles?
>
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This is a nice example of an algebraically exact functor:
for varieties Alg T, where T is an algebraic theory (and
Alg T is the category of all finite-product preserving
functors in [T, Set]), all theory morphisms F: T -> S
induce functors Alg F: Alg S -> Alg T given by precomposing
with F; they are called algebraically exact. These are precisely
the right adjoints between varieties which preserve sifted
colimits- and reflexive coequalizers are special sifted colimits.
This all is a part of the duality between varieties and algebraic
theories (described by F.W.Lawvere, J. Rosicky and myself, Algebra
Universalis 49 (2003), 1-45).

Consider the "obvious" algebraic theory S of Set, the dual of
finite sets, and the "obvious" theory C of Cat, the dual of
finitely presentable cats. The functor F: C -> S which forgets
morphisms induces the indiscrete category functor as Alg F.

reflexive coequalizers

[Jiri Adamek]

The indiscrete-category functor I: Set -> Cat is not algebraically
exact as I claimed in my posting of October 9. But I is
a full codomain restriction of one: as in that posting, let F
be the forgetful functor the Gabriel-Ulmer theory T
of categories to the theory of sets. Then Alg F is an algebraically
exact functor from Set to Alg T, and the Yoneda embedding
Y: Cat -> Alg T is fully faithful (since the dual of T is dense in Cat).
It is easy to see that Alg F is naturally isomorphic to Y.I , thus,
I preserves sifted colimits.

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alternative e-mail address (in case reply key does not work):
J.Adamek@tu-bs.de
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Re: Reflexive coequalizers

[Richard Garner]

Ah yes, I see now. I had observed that the
underlying diagram of indiscrete graphs was still a
coequalizer -- which unfortunately is to truncate
one's simplicial sets a little too much! Many
thanks for the enlightenment.

Richard


--On 09 October 2006 20:40 George Janelidze wrote:

> Dear Richard,
>
> If we were asking the same question about the category SimplSet of
> simplicial sets instead of Cat, the answer would be obvious since:
>
> (*) The functor Set ---> Set sending X to X^n preserves reflexive
> coequalizers for each natural n. This (very simple) fact should be
> considered as well known since it is involved in one of several well-known
> proofs of monadicity of varieties of universal algebras over Set.
>
> (**) Since SimplSet is a Set-valued functor category, all colimits in it
> reduce to colimits in Set.
>
> For those who are familiar with the standard adjunction between SimplSet and
> Cat, the result for SimplSet will immediately imply the result for Cat
> (since the fundamental category functor being the left adjoint in that
> adjunction preserves all colimits and obviously sends "indiscrete simplicial
> sets" to indiscrete categories).
>
> One could also use various (not too much) truncated simplicial sets instead
> of the simplicial sets of course (e.g. what I once called "precategories").
>
> I mean, I do not remember any reference, but I would not need the explicit
> description of colimits in Cat.
>
> George Janelidze
>
> ----- Original Message -----
> From: "Richard Garner" <rhgg2@hermes.cam.ac.uk>
> To: <categories@mta.ca>
> Sent: Monday, October 09, 2006 12:14 PM
> Subject: categories: Reflexive coequalizers
>
>
>>
>> Dear categorists,
>>
>> I have a proof that the indiscrete category functor
>> Set -> Cat preserves reflexive coequalizers which,
>> although straightfoward, uses the explicit
>> description of colimits in Cat. Is this necessary,
>> or can I deduce the result from general
>> principles?
>>
>> [Also, I'm sure this result must appear somewhere
>> but I can't find a reference for it. If anyone
>> knows of one, I'd be grateful.]
>>
>> Many thanks,
>>
>> Richard
>>
>>
>>
>
>

Re: Reflexive coequalizers

[George Janelidze]

Dear Richard,

If we were asking the same question about the category SimplSet of
simplicial sets instead of Cat, the answer would be obvious since:

(*) The functor Set ---> Set sending X to X^n preserves reflexive
coequalizers for each natural n. This (very simple) fact should be
considered as well known since it is involved in one of several well-known
proofs of monadicity of varieties of universal algebras over Set.

(**) Since SimplSet is a Set-valued functor category, all colimits in it
reduce to colimits in Set.

For those who are familiar with the standard adjunction between SimplSet and
Cat, the result for SimplSet will immediately imply the result for Cat
(since the fundamental category functor being the left adjoint in that
adjunction preserves all colimits and obviously sends "indiscrete simplicial
sets" to indiscrete categories).

One could also use various (not too much) truncated simplicial sets instead
of the simplicial sets of course (e.g. what I once called "precategories").

I mean, I do not remember any reference, but I would not need the explicit
description of colimits in Cat.

George Janelidze

----- Original Message -----
From: "Richard Garner" <rhgg2@hermes.cam.ac.uk>
To: <categories@mta.ca>
Sent: Monday, October 09, 2006 12:14 PM
Subject: categories: Reflexive coequalizers


>
> Dear categorists,
>
> I have a proof that the indiscrete category functor
> Set -> Cat preserves reflexive coequalizers which,
> although straightfoward, uses the explicit
> description of colimits in Cat. Is this necessary,
> or can I deduce the result from general
> principles?
>
> [Also, I'm sure this result must appear somewhere
> but I can't find a reference for it. If anyone
> knows of one, I'd be grateful.]
>
> Many thanks,
>
> Richard
>
>
>

Re: Reflexive coequalizers

[Prof. Peter Johnstone]

On Mon, 9 Oct 2006, Richard Garner wrote:

>
> Dear categorists,
>
> I have a proof that the indiscrete category functor
> Set -> Cat preserves reflexive coequalizers which,
> although straightfoward, uses the explicit
> description of colimits in Cat. Is this necessary,
> or can I deduce the result from general
> principles?
>
It certainly follows from the fact that reflexive coequalizers
commute with finite products in Set (or in any cartesian closed
category). This is a result that some people attribute to me,
since the first place it was explicitly written down seems to
have been my PhD thesis, though I'm sure it was known well
before that.

Peter Johnstone

Reflexive coequalizers

[Richard Garner]

Dear categorists,

I have a proof that the indiscrete category functor
Set -> Cat preserves reflexive coequalizers which,
although straightfoward, uses the explicit
description of colimits in Cat. Is this necessary,
or can I deduce the result from general
principles?

[Also, I'm sure this result must appear somewhere
but I can't find a reference for it. If anyone
knows of one, I'd be grateful.]

Many thanks,

Richard