January 17, 2012 5 Comments
Similar to last time, I want to talk about something mildly pedantic, but again, very important. There’s something in-between large categories and small categories. We don’t call it a medium category (I suppose my letters to the AMS and MAA haven’t been that persuasive). The collection of homomorphisms between and we denoted as . This may be a set or a proper class. If it is a set, we’ll call it the “hom-set.” Easy enough. What if for every , is a hom-set? This is what we call a locally small category. Make sense? We may have way too many objects to have it actually be a small category, but there may be few enough maps between them to be locally small. All small categories are locally small. The obvious example of a locally small category which is not small is the category of sets with set maps as the morphisms. There is no “set of all sets,” but given any two sets, the collection of maps between them is a set. Check your set theory axioms.
I’d also like to point out in slightly further detail the mistake I made last time (and have since edited). The definitions of large versus small categories are accurate. Most examples that we know of are large categories. But many of these examples are at least locally small. It seems that smallness is a very restrictive property. Too restrictive to say anything about the categories we care about. On the other hand largeness is just too wide open to say anything interesting. Being locally small is in some sense a balance, and it turns out to have some interesting properties.