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Stone's representation theorem gives you an isomorphism between every Boolean algebra and a field of sets. Viewed categorically, this is an isomorphism of categories, since isomorphism and equivalence …

This didn't fit in the comments, so I'm posting it as an answer. Ignoring size issues, we can define a category as a set of objects, together with a family of sets of morphisms, with one set for eac …

As it happens, I just saw a paper about this very subject today -- Martin Hyland's "Some Reaons for Generalizing Domain Theory", which is concerned with precisely the generalization you suggest, in or …

Hi Todd, I was hoping that someone more knowledgeable than me would answer this question, but since that hasn't happened yet I'll post a few comments. As you probably already know, your observation …

The theory of species is the full answer to your question -- but in this specific case all that is needed are some very basic properties of polynomial functors. Let's focus on you fixed-point equatio …

Freyd covers are a fundamental tool in the semantics of programming languages. Here, the technique is called "logical relations" or sometimes "Tait-Girard reducibility candidates". The general idea …

Here's a sequence of answers. The set of streams of elements of $A$ can be thought of as the final coalgebra $\nu F$ of the functor $F(X) = A \times X$. It is also the initial algebra of the functo …

For monoids (which are associative) the Krohn–Rhodes theorem gives a powerful decomposition result: every finite monoid is a quotient of a submonoid of an alternating wreath product of finite groups a …

It seems like many of the standard examples of monads in functional programming can be transported to linear logic to produce examples of non-monoidal functors. E.g., the linear state monad $T_S(A) …

The usual way of getting a category of metric spaces is to take metric spaces as objects, and the nonexpansive maps (ie, functions $f : A \to B$ such that $d_B(f(a), f(a')) \leq d_A(a, a')$) as morphi …

Yes, there is. See Peter Aczel's 1988 notes on models of non-well-founded set theory, here. The basic idea is that you can model sets as graphs (ie, as a collection of element sets together with a bin …

If $z$ is a constant, it's completely unproblematic, but it's troublesome if $z$ is a variable. Here's a simple example: suppose $A$ is a type operator of kind $\mathbb{N} \to \star$, defined as follo …

Lawvere theories generalize to higher-order logic in a straightforward way. A first-order hyperdoctrine is a functor $\mathcal{P} : C^{\mathrm{op}} \to \mathrm{Poset}$ where $C$ has products and is us …

Natural examples arise in proof theory: if you equate isomorphic propositions, then the semantics of proofs collapses -- all proofs of a given proposition end up having the same denotation. But if y …

This answer is largely a rendition of Sridhar's comment into lambda calculus. A strong monad $T$ has the following introduction and elimination rules in the lambda calculus. $$ \frac{\Gamma \vdash e …