Object of connected components for internal categories

Object of connected components for internal categories

[David Roberts]

Dear all,

I was wondering if there is any place in the literature that people
consider a hypothesis on a category E of the following form:

(*) For an internal category C in E, there is an object of connected
components pi_0(C).

This is equivalent to saying that s,t: Mor(C) ==> Obj(C) has a
coequaliser for any C.

One setup I know that is *sufficient* is if C is a sigma-pretopos (in
the language of the Elephant) aka an aleph_1-ary pretopos (in the
language of Shulman's exact completions article). This is a pretopos
that additionally has countable disjoint sums (rather than just
finite). This is sufficient because then we can take the underlying
internal graph of C, then form the corresponding relation, freely form
the associated symmetric, reflexive relation (this part so far only
requires the finitary structure) and then form the transitive closure
(this requires the countable sums).

However, given an internal category C I already have a transitive (and
reflexive) relation, namely the preorder reflection of C (which only
requires finitary operations). So while I don't necessarily expect any
other construction that doesn't require the countable sums, I was
wondering if people had considered this condition (*) directly and in
isolation. My best guess is it would be in some cohesion-type setup,
but I'm not interested in the rest of the cohesion axioms at this
time.

Many thanks,
David






David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com


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Re: Object of connected components for internal categories

[ptj]

The hypothesis that I used in both my topos theory books was that E has
coequalizers of reflexive pairs (which are moreover stable under pullback).
Apart from stability under pullback this is implied by David's hypothesis,
and it eems to be satisfied in all the cases of interest.

Peter Johnstone

On Jul 26 2020, David Roberts wrote:

>Dear all,
>
>I was wondering if there is any place in the literature that people
>consider a hypothesis on a category E of the following form:
>
>(*) For an internal category C in E, there is an object of connected
>components pi_0(C).
>
>This is equivalent to saying that s,t: Mor(C) ==> Obj(C) has a
>coequaliser for any C.
>
>One setup I know that is *sufficient* is if C is a sigma-pretopos (in
>the language of the Elephant) aka an aleph_1-ary pretopos (in the
>language of Shulman's exact completions article). This is a pretopos
>that additionally has countable disjoint sums (rather than just
>finite). This is sufficient because then we can take the underlying
>internal graph of C, then form the corresponding relation, freely form
>the associated symmetric, reflexive relation (this part so far only
>requires the finitary structure) and then form the transitive closure
>(this requires the countable sums).
>
>However, given an internal category C I already have a transitive (and
>reflexive) relation, namely the preorder reflection of C (which only
>requires finitary operations). So while I don't necessarily expect any
>other construction that doesn't require the countable sums, I was
>wondering if people had considered this condition (*) directly and in
>isolation. My best guess is it would be in some cohesion-type setup,
>but I'm not interested in the rest of the cohesion axioms at this
>time.
>
>Many thanks,
>David
>
>
>
>
>
>
>David Roberts
>Webpage: https://ncatlab.org/nlab/show/David+Roberts
>Blog: https://thehighergeometer.wordpress.com
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]