…is called an infinitary pretopos. Once you add the existence of a small set of generators then Giraud’s theorem applies and you know you have a Grothendieck topos. A pretopos can even have a subobject classifier, but lacking cartesian closedness fail to be a topos (I’ll say just ‘pretopos’ from now on, see the end for an explanation).

Bernie Sanders "I am once again asking you"captioned meme, with text "I am once again asking you to sau thje category of condensed sets is a pretopos" and included is a footnote to "Johnstone (1977), Definition 7.38". Added to the still image of Sanders is a carefully placed copy of Johnstone's 1977 book "Topos Theory" positioned to look like it's being held just below frame, with the title and author just visible.

The oldest reference I can find is in the introduction by Lawvere to Springer LNM 445, dating to probably 1973, and he claims the idea is in SGA4 Exposé VI (I can’t find the word there, but I can absolutely believe Grothendieck and Verdier knew of the concept EDIT: prétopos are defined in Exercice 3.11 a), see the comments). And there is a formal definition included in the Baby Elephant as in the above image, from 1977. For a modern take, Lurie has a lecture titled Pretopoi with notes on his website.

Sorry to be a curmudgeon, but it’s been more than five years, and a new (very nice!) thesis just dropped that referred to a category with all the non-size conditions from Giraud’s theorem, and apparently didn’t know there was a word for these. And with a really nice classical canonical example being the category of compact Hausdorff spaces, and now the much fancier category of condensed sets (as a special case of a colimit along inverse image functors of a large diagram of Grothendieck toposes), having a precise name for such categories is helpful (‘precise’ here being actually imprecise in the original version of this blog post, see the next paragraph).

ADDED: Both Mike Shulman in the comments below and an email correspondent have prompted me to tweak the post: a vanilla pretopos of course is defined in such a way that only finite disjoint coproducts are needed. In my brain there is a spectrum of pretoposes, i.e. \kappa-pretoposes, for \kappa a regular cardinal or the size of the universe of all sets, and one asks for appropriately universal <\kappa-sized colimits in the definition. And I am happy to leave the size parameter implicit, but this is not a universal practice, which is a mistake when trying to make a terminological point. A Grothendieck topos isn’t just (forgetting the generating set) a pretopos, but a cocomplete pretopos, and in fact an infinitary pretopos, so that its colimits of all sizes are well-behaved. Additionally, a Grothendieck topos has a geometric morphism to \mathbf{Set}, which is not specified by just saying ‘pretopos’. I have another short blog post in mind, more serious, about the family of generalisations of Grothendieck toposes that I like that include pretoposes, so more about this later. But this is relevant for the category of condensed sets, too: there is a global points functor, and sets embed as discrete condensed sets.