more on inclusion maps

more on inclusion maps

[Peter Freyd]

One doesn't need all the tau-category stuff just for finding an
equivalent category with inclusion maps. There's a first-order
construction but first an easier way to see how to do it. Given a
category  *A*  define  *B*  to be the full subcategory of
representable contravariant set-valued functors and all subfunctors
thereof that are isomorphic to representable functors. The Yoneda
embedding of  *A*  into  *B*  is clearly an equivalence functor. By
saying that  T  is a subfunctor of  (-,A)  I mean that  TX  is a
subset of  (X,A)  for each  X  and that the inclusions form a natural
transformation, which transformations are to be, of course, the
chosen notion of inclusion map in  *B*. Done..

If one objects that this works only for small  *A*  use this first-
order construction: take the objects of  *B*  to be the monics of  *A*.
The forgetful functor from  *B*  to  *A*  carries  [A >-> B]  in  *B*
to  A  in  *A*.  Take *B*  to be the "inflation" that results from
this onto forgetful functor, that is, the maps from  [A'>-> B']  to
[A >-> B]  are named by maps from  A'  to  A. Such a map is said to be
an inclusion map iff  A' >-> B' = A' -> A >-> B  (in particular,
A' -> A  is monic and  B' = B).

Given  [A'>-> B'], [A>-> B]  and an arbitrary monic  A' >-> A  note
that the identity map on  A'  names an isomorphism from  [A' >-> B']
to  [A'>-> A >-> B]  such that the given monic
[A' >-> B'] -> [A >-> B]  is equal to
[A' >-> B'] -> [A' >-> A >-> B] -> [A >-> B].

Define  *A* => *B*  to be the functor that sends each object in  *A*
to the object in  *B*  named by its identity map and define
*B* => *A*  to be the obvious forgetful functor. Then
*A* => *B* => *A*  is the identity functor on  *A*  and there's a
natural equivalence from  *B* => *A* => *B*  to the identity functor
on  *B*. We don't need the axiom of choice to construct that natural
equivalence; take each of its component isomorphisms as the one map
that fits and that's named by an identity map. (The first would
require the axiom of choice. The second construction is isomorphic to
the category whose objects are ordered pairs, where the first
coordinate is a subfunctor of a representable and the second is a
specified isomorphism to a representable.)










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,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 automorphism 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 those
two equivalence classes, 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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