Some questions about different notions of "theory"

[Mike Stay <metaweta@gmail.com>]

1) Finitary monads correspond to Lawvere theories.  Is there a name
for those monads that correspond to toposes?

2) In topos theory is there any analogous result to Lawvere's theorem
that the opposite of the category of free finitely generated gadgets
is equivalent to the Lawvere theory of gadgets?  Something like "the
opposite of the category of fooable gadgets is equivalent to the topos
of gadgets"?

3) nLab says a sketch is a small category T equipped with subsets
(L,C) of its limit cones and colimit cocones.  A model of a sketch is
a Set-valued functor preserving the specified limits and colimits.  Is
preserving limits and colimits like a ring homomorphism?  Preserving
both limits and colimits sounds like it ought to involve profunctors,
but maybe I'm level slipping.

-- 
Mike Stay - metaweta@gmail.com
http://www.cs.auckland.ac.nz/~mike
http://reperiendi.wordpress.com


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