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)βC into a
cocomplete category C is equivalent to the category
ModCβ(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 which models Top, study the corresponding notion of an
internal net-based topological space object, and deduce from our main result
that cocontinuous functors TopβC into a cocomplete
category C correspond to net-based cotopological space objects
internal to C. Finally we describe a limit sketch which models
Topop and deduce from our main result that continuous
functors TopβC 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 TopβTop in terms of
topological topologies.Comment: 42 pages; comments are appreciated, in particular if detailed proofs
of the 'folklore results' already appear elsewher