It is well-known that in a 1-pretopos,
For a similar statement in an -pretopos for , the natural first guess is to replace “monic” by “ff” and “regular epic” by “eso.” However, there are (at least) two reasonable replacements for “epic” and “regular monic:”
Since esos in Cat are cofaithful and liberal but not co-ff, it wouldn’t work to replace “epic” by “co-ff.” Unfortunately, both ideas fail in Cat, where equifiers and inverters are not just ff but are closed under retracts. Likewise, the map from a category to its Cauchy completion is full, faithful, cofaithful, and liberal, but generally not an equivalence.
There are two ways to deal with this problem. One is to restrict to smaller values of , for which there are no nontrivial retracts. The other is to go back and change “ff” to “ff and closed under retracts” and change “eso” to “surjective up to retracts.”
In a (2,1)-exact positive coherent 2-category, every ff with groupoidal codomain is an equifier (in fact, an identifier of an involution).
Recall that an identifier of a 2-cell in a 2-category is the equifier of and . By an involution we mean an (invertible) 2-cell that is its own inverse.
Let be ff with groupoidal, and consider first the (2,1)-congruence given by , where one copy of gives the identities, and the composition treats the other copy of as an involution. Its quotient is the copower of by the “walking involution.”
Now consider the following equivalence relation on in . We have
and we define the relation to be given by
Since is a 1-pretopos, this relation has a quotient, say . It is easy to verify that is then a (2,1)-congruence on with . (This depends on being groupoidal; otherwise it would be a homwise-discrete category but not necessarily a congruence.) Let be the quotient in of this (2,1)-congruence. Then we have a 2-fork such that and is an involution of .
We claim that is an identifier of . By construction of , we have , so since is ff, it suffices to show that for any with , factors through . But since with is the kernel of , the assumption implies that in fact . If we write for the two inclusions, this means that and become equal in , and therefore factor through the kernel pair of , namely . But this is evidently tantamount to saying that factors through .
In a (2,1)-pretopos, 1. every ff is an equifier, 1. every cofaithful ff is an equivalence, and 1. every cofaithful morphism is eso.
Theorem 1 shows the first statement. Then any ff is an equifier of , so in particular ; but if is also cofaithful, this implies , and thus their equifier is an equivalence. Finally, if is just cofaithful, we factor it as where is ff and is eso; but then is also cofaithful, hence an equivalence, and so , like , is eso.
In a (1,2)-exact positive coherent 2-category, every ff with posetal codomain is an inverter.
Let be ff, and consider the 2-congruence on defined as follows. We have
(adding subscripts to distinguish the two copies of ) and the congruence is given by
Here is defined to be the ff image of the “composition” morphism ; in other words it is “the object of arrows in which factor through some element of .” The composition is easy to define making this into a 2-congruence, and if is posetal, then it is a (1,2)-congruence.
Let be the quotient of this congruence. Analogously to the proof of Theorem 1, the fork defining this quotient gives a 2-cell such that is an isomorphism. Thus, to show that is the inverter of , it suffices to show that for any with invertible, factors through . Now is induced by the fork defining together with the composite . If is invertible, then its inverse is given by some map , which must lie over the diagonal .
Now, pulling back the eso along we obtain an eso with a morphism such that the identity 2-cell of is the composite of 2-cells ; in other words, is a retract of . But since is posetal, this means , and then the fact that is eso implies that factors through , as desired.
In a (1,2)-pretopos, 1. every ff is an inverter, 1. every liberal ff is an equivalence, and 1. every liberal morphism is eso.
Now, in any regular 2-category, in addition to the (eso,ff) factorization system we also have a Cauchy factorization system consisting of the cso (Cauchy surjective) and rff (ff and retract-closed) morphisms. Moreover, every cso is cofaithful and liberal. In “Modulated bicategories” by Carboni, Johnson, Street, and Verity (CJSV), it is shown that the liberal functors in Cat are precisely the Cauchy surjective ones; we now show that the same is true in any 2-pretopos.
In a 2-pretopos, 1. every rff is an inverter, 1. every liberal rff is an equivalence, and 1. every liberal morphism is Cauchy surjective.
In particular, every liberal morphism is cofaithful.
The first statement is proven exactly like Theorem 2, except that at the last step we use the assumption that is retract-closed rather than the assumption that is posetal. The other statements follow as before, recalling that Cauchy surjective morphisms are cofaithful.
In (CJSV) a 2-category is defined to be co-conservational (liberational?) if
and faithfully co-conservational if moreover
Here a strong conservative morphism is one that is right orthogonal to all liberals. Theorem 3 shows that in a 2-pretopos, strong conservatives coincide with rffs, since liberals coincide with csos; thus any 2-pretopos satisfies the second condition to be co-conservational. It need not have finite colimits, but this can be remedied with some infinitary structure. The construction of copowers in a 2-pretopos can be used to show that rffs are preserved by copowers with . And we have also seen that since every liberal is cso, it is cofaithful.
However, even fails the final condition, as shown in (CJSV, Prop. 3.4). This can be remedied by passing to the 2-category of Cauchy-complete categories. I do not yet know whether a similar construction is possible in any 2-pretopos.