Subject: categories: Mac Lane's inclusions

From: [email-masked] (Colin McLarty)
Date: Thu, 15 Jul 1999 13:32:06 -0400 (EDT)

This note gives an alternative characterization of "inclusions" in Mac
Lane's sense; and proves every category with selected monics is equivalent
to one with "inclusions". Some of it may have been known before and if so I
would appreciate references.

	In a circular letter Saunders Mac Lane has suggested paying attention to
"inclusions" in categorical set theory.  The main questions about this have
general categorical answers as follows. A category has "selected monics" if
for each object C, each equivalence class of monics to C has one selected
representative. The selected monics are called "inclusions" if whenever
C'>->C and C">->C' are selected, the composite C">->C is also selected.

	The following alternative characterization motivates the construction in
the next theorem but is not actually used to prove it. Given monics i and j
to a single object, with i<j, the monic h such that jh=i will be called the
"transition monic" from i to j. (For e-mail convenience I write i<j to mean
i is included in or equal to j.) 

Fact: In any category with selected monics, the selected monics are
inclusions if and only if: for all selected monics i and j with i<j, the
transition monic h is also selected.

Proof: First suppose i:I>->C and j:J>->C are selected, and selected monics
compose. The transition h must have some selected equivalent k:K>->J, and so
jk is selected and equivalent to i, which implies jk=i and k=h so h is
selected. Conversely suppose h:I>->J and j:J>->C are selected and transition
monics between selected monics are selected. Then jh:I>->C has some selected
equivalent m and for some iso g we have jhg=m. Thus hg is transition monic
between the selected m and j, and so hg is selected. But hg is also
equivalent to the selected h so hg=h and g is an identity. Thus jh=m is
selected. 

	In any topos with selected pullbacks we have selected monics, since for any
equivalence class of monics we can take the selected pullback of "true"
along the characteristic arrow. These need not be inclusions. However, we
have:  

Theorem: Any category A with selected monics is equivalent to a category AI
with inclusions.
Proof: 	Let the objects of AI be the selected monics C>->B of A.  An AI
arrow from C'>->B' to C>->B is simply an A arrow C'-->C. That is, AI arrows
ignore the monics and just look at the domains. Obviously AI is equivalent
to A and an arrow is monic in AI iff it is monic in A. As the selected
monics to an AI  object C>->B we take those AI monics h

                               h:C'>-------->C
                                 v           v
                                 |           |
                                 |           |
                                 v           v
                                 B    =      B

which lie over the same B and make the triangle commute in A. Clearly these
compose, and each monic to C>->B in AI is equivalent to exactly one of these.

        In particular, given any axiomatic theory of a topos with selected
pullbacks: if the axioms are preserved by equivalence, then we can
consistently add the assumption of inclusions. We can assume selected monics
compose. Of course it remains to actually use this assumption to secure the
advantages that Saunders sees for it.