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Assume a metatheory that supports lambda-abstraction, and an object language that is merely first-order. Now let $\varphi$ denote a formula in the object language with one free variable $x$. Then we c …

Write $\mathbf{Cat}$ for the world of categories. Then $\mathbf{Cat}$ has: objects (namely cateories) arrows (namely, functors) proarrows (namely, bimodules) squares (namely, functors between pairs …

Define the notion of a "foo digraph" recursively as follows. If we take any finite number of directed path graphs each of which has at least $2$ vertexes, and glue them at the start and end vertexes …

Let $M$ denote a monoid and suppose we're given a function $[-] : M \rightarrow M$ satisfying $[a[b]c] = [abc].$ Then: Proposition 0. $[-]$ is idempotent. Proof. Take $a=c=1$). Proposition 1 …

When I'm explaining things involving partial functions, I usually end up stumbling over my words, like so: "Suppose $f : A \rightarrow B$ is a function, uhh, sorry I mean a partial function, and suppo …

Definition. Call an object $X$ of a category $\mathbf{C}$ nearly initial iff firstly, it is weakly initial, and secondly, for all objects $Y$ and all morphisms $f,g : X \rightarrow Y$, there exists …

I am also interested in terminology for some or all of the following concepts: The construction $N \mapsto N^{sym}$ that takes commutative monoids to symmetric multicategories. …

I think we all occasionally come across terminology that we'd like to see supplanted (e.g. by something more systematic). … Highly-entrenched = The literature at the time had the old terminology written all over it. Succeeded = The new terminology is now used almost exclusively in research papers. …

Suppose we're in a bicartesian closed category. Then given a morphism $$f : X \rightarrow Y_1 + \ldots + Y_n$$ and a test object $T$, we get a corresponding morphism $$T^f : X \times [Y_1,T] \times \l …

Are there some accepted definitions or terminology that could get an interested reader started in learning about such things? …