4 research outputs found
Large limit sketches and topological space objects
For a (possibly large) realized limit sketch such that every
-model is small in a suitable sense we show that the category of
cocontinuous functors into a
cocomplete category is equivalent to the category
of -valued
-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 , study the corresponding notion of an
internal net-based topological space object, and deduce from our main result
that cocontinuous functors into a cocomplete
category correspond to net-based cotopological space objects
internal to . Finally we describe a limit sketch which models
and deduce from our main result that continuous
functors into a complete category
correspond to frame-based topological space objects internal to .
Thus, we characterize both as a cocomplete and as a complete
category. Thereby we get two new conceptual proofs of Isbell's classification
of cocontinuous functors in terms of
topological topologies.Comment: 42 pages; comments are appreciated, in particular if detailed proofs
of the 'folklore results' already appear elsewher