Mac Lane's inclusions

Mac Lane's inclusions

[Colin McLarty]

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.

RE: Mac Lane's inclusions

[Colin McLarty]

Robert W. McGrail wrote:

>Correct me if I am wrong, but it seems to me that every \tau-category is a 
>category with inclusions.  Moreover, I recall a result by Freyd that every 
>(sufficiently small? cartesian?) category is equivalent to a \tau-category. 
> The proof does not use choice.

        There is certainly a similarity. Tau categories are cartesian
categories (categories with all finite limits) which not only have selected
monics but also selected jointly-monic n-tuples for all finite n; and all
these things compose. Freyd shows every cartesian category is equivalent to
a tau category.

        My construction seems very like Peter's but shortened by working
only on monics and not jointly monic lists, and never using limits.  

best, Colin

RE: Mac Lane's inclusions

[Robert W. McGrail]

Colin,

Correct me if I am wrong, but it seems to me that every \tau-category is a 
category with inclusions.  Moreover, I recall a result by Freyd that every 
(sufficiently small? cartesian?) category is equivalent to a \tau-category. 
 The proof does not use choice.

In any event, see Categories, Allegories by Freyd and Scedrov for 
\tau-categories.  Hope this helps.

Bob McGrail

On Thursday, July 15, 1999 12:32 PM, Colin McLarty [SMTP:cxm7@po.cwru.edu] 
wrote:
> 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.
>
>