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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.

If the right adjoints are also a right inverses, than such functors are called (Grothendieck) fibrations. If the right adjoint are full and faithful, than such functors are called Street fibrations.

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 …

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 …

I need a name for a regular category where the inverse image maps have right adjoints. If $\mathcal C$ is a regular category, then the poset of subobjects $\mathsf{Sub}(X)$ of any object $X$ is a sem …

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 …

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 …

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 …

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 …

A category is small if its objects and morphisms form a set rather then some other kind of class. Smallness is therefore relative to the model of set theory we are working in and the whole notion was …

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 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. …

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 …

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 …