Re: Category without objects

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

It is difficult to understand "without objects"  without any definition of "object". Remember that , already before the 21st century, modern mathematics had begun to overcome medieval metaphysics. In fact ,in the late 1950s, Alexander Grothendieck had made explicit the definition of "subobject", which seems relevant here, as does his powerful legacy of relativization in several senses. Now we understand that a category C in a category U  is a truncated simplicial object C0->...->C3 satisfying certain limit conditions. We are free to call C0 'objects" and C1 "maps" and since C0->C1 is a subobject of C1, we could also say that objects "are" maps,but "mimicked by" seems \x10unnecessary (as well as undefined).
(Recall that it is actions of such a C in a topos U that form the topos enveloping, as a full subtopos of sheaves, the typical U-topos E->U).
To give a category "with objects" i\x10n a serious sense would seem to be giving MORE than ju\x10st a category, for example an interpretation as structuresC-> B^A, the (functor category also emphasized by Grothendieck)of structures of shape A in background B. (Where perhaps B is equipped with an internal embedding in U itself)
The case of no structure and featureless background ( which seems to be the   default setting of modern mathematics despite the preference of MacLane'sdear teacher for a vonNeuman-like setting) means in particular that the C0 in a category there consists of "lauter Einsen" in the sense of Cantor.
Those featureless elements X of C0 do obtain a structure by virtue of C1,C2 because taking the latter into account we can see the inside of  X as the "comma" category C/X involving (not only the subobjects of X and their inclusions, but also singular figures and reparameterizations) as very extensively utilized by Grothendieck .
Bill
> Date: Sat, 7 Mar 2015 14:36:36 +0000
> From: ronnie.profbrown@btinternet.com
> To: Uwe.Wolter@ii.uib.no; p.l.lumsdaine@gmail.com
> CC: categories@mta.ca
> Subject: categories: Re: Category without objects
> 
> I remember Henry Whitehead said that he was very impressed by the axioms
> for a category in the Eilenberg-Mac Lane paper.
> 
> A curiosity about the definition is that groupoids were defined by
> Brandy in 1926, and this definition was used by the Chicago school of
> algebra and applied to ring theory.  Bill Cockcroft told me that the
> groupoid notion was an influence.  In 1985 I asked Eilenberg about this,
> and said no, since if it had been, they would have used it as an
> example! I forgot to ask Mac Lane!
> 
> Ronnie Brown
> 

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