If you have done any algebraic geometry, then you will have come across descriptions of “covering families” of various types, usually described as a collection of maps with a) common codomain, b) some nice property, such that c) they are jointly surjective. For instance, an Zariski/étale/smooth/flat covering family of a scheme is a collection of open embedding/étale/smooth/flat maps such that is an epimorphism. (one might need more adjectives over mere flatness, but these can be safely added with no change of anything below.) One can play similar games in other categories; there is a notion of cover of manifolds that is a surjective submersion (or, in principle, a jointly surjective family of submersions), or a cover of a topological space that is an open surjection. That surjectivity is here is good and right: this makes the covers/covering families into subcanonical pretopologies, in that in the case of the single covering maps, these are regular epimorphisms, and really do capture the notion of “being a cover” (in the covering family version, an analogous property holds).
But what is common to the examples above (and many others) is that the other property also has nice features. Submersions, open maps, open embeddings, étale maps, smooth maps etc all contain the isomorphisms, are closed under composition, and are closed under pullback. As such these classes of maps, not assuming surjectivity, already form Grothendieck pretopologies, though they are far from being subcanonical. Further, the categories of schemes, manifolds and topological spaces exhibit a property called extensivity. As a result, coproducts behave as you expect disjoint unions to, rather than, say, direct sums of vector spaces (which are also coproducts). Moreover, the coproduct inclusions form a covering family in each of the cases given (so we have a superextensive pretopology). This implies that one can turn each of these pretopologies into a pretopology where every covering family consists of a single map (a singleton pretopology). This process preserves the property of being subcanonical, and so I will assume that we are working with a singleton subcanonical pretopology.
Another feature that these examples display is that given, for instance, a submersion , then given any surjective submersion , the resulting fold map is a surjective submersion. The same is true for open maps, for open embeddings, for étale maps and so on. There is a recent PhD thesis by Giorgi Arabidze (not this Giorgi Arabidze!), that deals with internal groupoids in categories equipped with an abstraction of this type of structure, though I have a slightly different take. He starts from an arbitrary singleton pretopology (though calls it a “stronger pretopology”), whose maps are called partial covers, then singles out the subclass of partial covers that are also regular epimorphisms, and calls those covers. The covers then form a subcanonical singleton pretopology. I’m interested in starting from the latter, and seeing what sort of larger classes of maps I can take that have the fold map property above.
As an example, whenever a subcanonical superextensive pretopology is given by specifying that a covering family is a collection of jointly-surjective -maps, for a class of maps that form a pretopology on their own, then this is the sort of setup I’d like to abstract, but merely starting from a subcanonical singleton pretopology. So here is a first attempt at this. Nothing deep, just wanting to see if it seems like something that’s out there (and the name is totally provisional).
Definition: Let be a subcanonical singleton pretopology on an extensive category . Let be another singleton pretopology such that
- Given any in and in , the fold map is in
- is local for the extensive pretopology.
We then call a class of precovers for .
By “local for the extensive topology” I mean that if I have a map in with codomain a coproduct, if its corestriction to each summand is in , then it is itself in . This implies that the coproduct of maps in is again in . What I am interested in is the largest possible class of precovers for a given pretopology as in the definition. As noted in the title, these maps are not quite covers, but can be ‘completed’ to a cover. Another way to think about this might be something like finding a class of maps in which is a kind of ‘ideal’ for the operation of forming the fold map in a suitable slice category.
As an example that is somewhat contrasting to the above more geometric ones, consider an extensive category with all finite limits (a ‘lextensive’ category; a topos is an example), a subcanonical singleton pretopology , and consider the subcanonical singleton pretopology given by maps that admit -local sections. Then the class of all maps in form a class of precovers. For an more specific example of interest, take a pretopos and the singleton pretopology consisting of the regular epimorphisms (this, together with the extensive pretopology generate the coherent pretopology). What are the maps that admit -local sections? And then what is the largest class of precovers, is it all maps again?