Re: Question on choosing subobjects consistently

[John Kennison <JKennison@clarku.edu>]

Peter just said it better, but to clarify what I said: First, it is possible that Grothendieck intended the following: as usual, say that two monos are equivalent if they have the same codomain with an isomorphism between their domains which commutes with the two monos.  Choose exactly one mono from  each equivalence class and call it a canonical mono. Then define a subobject as a canonical mono and a subobject of a subobject as the canonical mono  equivalent to the composition.

Secondly, it is not always possible to define a class of canonical monos (as above) which is closed under composition. Consider the category with 3 objects, A,B,C, in which every hom set. Hom(X,Y) is a subset of the group Z_2  with two elements. Let Hom(A,A) =Hom(A,B)=Hom(A,C)=Hom(B,C)=Z_2. Let Hom(B,A) and Hom(C,B) be empty and let Hom(B,B) and Hom(C,C) each consist of just the identity of Z_2. All compositions will be given by the group operation. Then there are two non-equivalent monos from B to C, so they must both be canonical. The two monos from A to B are equivalent, so only one of them can be canonical. But the compositions of this canonical mono with each of the two canonical monos from B to C give us two canonical monos from A to C but they are equivalent. 

________________________________________
From: Peter Freyd [pjf@seas.upenn.edu]
Sent: Saturday, August 28, 2010 8:24 AM
To: categories@mta.ca
Subject: categories: Re: Question on choosing subobjects consistently

Mike asks how to pick what I'll call "inclusion maps" that is a choice
of monics so that each subobject is named by a unique inclusion map
and such that the inclusion maps are closed under composition. I don't
know what Grothendieck had in mind, but it's an easy application of
the stuff about tau-categories in Cats & Alligators copied from my
1975 "pamphlet." The setting there is cats with finite limits but of
course you could always work in the full subcategory of representable
functors in the category of set-valued contraviariant functors.

A tau-structure gives you not only (transitive) canonical subobjects
but associative canonical products and all sorts of goodies. But alas
it delivers an equivalent category, not necessarily the category you
started with. I don't know how to do it in an arbitrary category.

But neither does anyone else, not even Grothendieck:

Consider the subcategory of the category of sets with three objects,
A,B,C  each of which is a two-element set. There well be a total of
ten maps, three of which, of course, are identity maps. The other
seven maps will be one-to-one and onto but only one of them an
isomorphism (as defined, of course, in the subcategory). The one
non-trivial isomorphism will be on  A  (necessarily, the twist map).
The hom-sets  (B,B)  and  (C,C)  each have a single element
(necessarily their identity maps).  The hom-sets  (B,A), (C,A,)  and
(C,B)  are empty. The hom-sets  (A,A), (A,B), (A,C)  and  (B,C)  each
have two maps, to wit, the two possible one-to-one onto maps. Note
that all maps are monic.  A has no proper subobjects. B  has only one
(the two monics from  A  to  B  name the same subject of  B). C  has
three proper subobjects one of which is named by each of the two maps
from  A  to  C. Each of the other two subobjects of C are named by one
of the two maps from  B  to  C  (since  B  has no non-trivial
automorphisms they necessarily name different subobjects). Hence, of
the four proper subobjects in this category, two of them have just one
name, the other two each have two names. No matter how one chooses a
choice of "inclusion map" in the latter cases, that is no matter how
one chooses a map  A -> B  and a map  A -> C, when you compose the
chosen  A -> B  with each of the two maps from  B  to  C  you will get
both names for the single proper subobject named by each of the two
maps from  A  to  C.  They can't both be the chosen map in  (A,C).


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