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Kleene defined a continuous realizability over Baire space, i.e. $\mathbb N^{\mathbb N}$ with the product topology. In this model $\forall x\exists y\phi(x,y)$ is valid, if there is a continuous funct …

I have a couple of conjectures on recursive functions, that I feel must have been proved or refuted by someone else, but I don't know where to look. In short: 1. The primitive recursive functions for …

Despite having left academia, I still work on this problem from time to time, and I (hope I) recently worked out a combinatorial proof for the base case of extension along a horn inclusion: https://g …

The answer is essentially already in the comments. Commutativity erases a monoid's sense of order, so the free commutative monoid monad is the 'bag monad' even if the free monoid monad is the 'list mo …

The haskell monad you mention models the process of drawing from a sequence at random. It is a state monad--with underlying map of objects $X \mapsto (S\times X)^S$--where the state includes the remai …

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 …

A suplattice automatically has all meets, and a suplattice where meets distribute over all joins is a locale. The category of partial (i.e. nonreflexive) equivalence relations and functional relations …

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 …

The pullback $\forall$ and $\exists$ are adjoint to the inverse image map of a morphism, which sends a subobject to its pullback along the morphism. I don't know what a pushout of a subobject is, thou …

I think the answer is no, because being an natural number object is a universal property and being a model of Nelson arithmetic is not. As long as a category is Heyting (is a regular category where t …

You can find it as an example of a monoidal category in Tom Leinster's "Higher Operads, Higher Categories", which contains loads of coherence proofs for higher categories.

Small (co)complete categories are posets by a theorem of Freyd. If $C$ has all small coproducts and its class of morphisms $C_1$ is small, then $C(x,y)^{C_1}\simeq C(\coprod_{f\in C_1} x, y)\subseteq …

The internal language of a locally Cartesian closed category is a dependent type theory. A category $\mathcal C$ is locally Cartesian closed, if all of its slices $\mathcal C/X$ are Cartesian closed. …

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 …

As Freyd and Scedrov show in "Categories, Allegories", you can think of a category as some kind of partial monoid. Such a monoid has a set of partial units that take over the role of objects in standa …