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.
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
"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
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
compose, and each monic to C>->B in AI is equivalent to exactly one of
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: the transition these these.
In particular, given any axiomatic theory of a topos with
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
selected the
advantages that Saunders sees for it.
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
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Robert W. McGrail