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In a topos, the question is not whether a sentence is true or false, but where it is true, because toposes--at least the geometric ones you're talking about--are generalized spaces. There can be limit …

For each groupoid $G$ there is a monad such that the cloven fibred groupoids over $G$ are algebras for this monad. I am looking for an infinite dimensional counterpart: a monad on simplicial sets whos …

A generic monomorphism is a monomorphism of which every monomorphisms in the same category is a pullback; it is like a subobject classifier without uniqueness. I am looking for a locally Cartesian clo …

I don't know any examples with $\Omega$, but $x\mapsto 2^{2^x}$ has an initial algebra in the effective topos. The object $2^{2^x}$ is a quotient of a subobject of the natural number object for any ob …

Question 1] Morphisms $\lambda:X\times Y\to\Omega$ correspond to subobjects $L\subseteq X\times Y$. The conditions ed says that the projection $\pi_0:L\to X$ is a (regular) epimorphism, and uv says th …