RE: Terminology of locally small categories without replacement

["F. William Lawvere" <wlawvere@hotmail.com>]

Dear Colin,I think your stronger definition is the correct one, by analogy withother categories (where properness and other manifestations ofintensive and extensive objective quantities come up).  If your definition of 'category' itself is equivalent to 'internal category  C3=>C2->C1 x C1 in the category of classes ', then your notion seems to be a case of the condition (on E ->B) that the pullback of any small S ->B is small.The AXIOM of  replacement would perhaps be the extra condition on the universe that the pullbacks of S = 1 suffice to test the above, a condition that is perhaps appropriate for abstract constant discrete sets but not for cohesive variable ones.It would not be the 'scheme' of replacement that is relevant here since the category of objective classes (not their sometimes representing  subjective formulas) is directly under consideration.I presume that you are here trying to extend the Bernays-Mac Lane framework.It is not clear what  would result if we alternatively considered that a category C itself is just a formula, i.e objectively, a subset naturally defined in every model. Bill
> Date: Wed, 1 Dec 2010 17:00:51 -0500
> Subject: categories: Terminology of locally small categories without replacement
> From: colin.mclarty@case.edu
> To: categories@mta.ca
> 
> Locally small categories are always defined as categories such that:
> 
> LS) for any objects A,B there is a set of all arrows A-->B.
> 
> When the base set theory includes the axiom scheme of replacement that
> is equivalent to a prima facie stronger property:
> 
> ??) for any set of objects there is a set of all arrows between them.
> 
> These two are not equivalent in the absence of the axiom scheme of
> replacement.  There the second is much stronger, but it remains
> important.  Is there a good term for it?
> 
> thanks, Colin
> 

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