2021/06/21 by Brandenburg, Martin
#18C05 #18C10 #18C15 #18C30 (Primary) #18C40 #18F70 (Secondary) #54A05 #54A20 #54B30 #Category Theory (math.CT) #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2106.11115
For a (possibly large) realized limit sketch S such that every S-model is small in a suitable sense we show that the category of cocontinuous functors Mod(S) → C into a cocomplete category C is equivalent to the category ModC(Sop) of C-valued Sop-models. From this result we deduce universal properties of several examples of cocomplete categories appearing in practice. It can be applied in particular to infinitary Lawvere theories, generalizing the well-known case of finitary Lawvere theories. We also look at a large limit sketch that models Top, study the corresponding notion of an internal net-based topological space object, and deduce from our main result that cocontinuous functors Top → C into a cocomplete category C correspond to net-based cotopological space objects internal to C. Finally, we describe a limit sketch that models Topop and deduce from our main result that continuous functors Top → C into a complete category C correspond to frame-based topological space objects internal to C. Thus, we characterize Top both as a cocomplete and as a complete category. Thereby we get two new conceptual proofs of Isbell's classification of cocontinuous functors Top → Top in terms of topological topologies.