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It is well-known that the category of local rings and ring homomorphisms admits an axiomatisation in coherent logic. Explicitly, it is the coherent theory over the signature $0, 1, -, +, \times$ with …

Let $\mathcal{A}$ be a category. There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf of sets" (i.e. …

Apparently, there is an abstract nonsense argument that shows $\mathbf{Sch}$ is concretisable. Here is a hands-on proof. We define $U_0 : \mathbf{Sch} \to \mathbf{Set}$ to be the functor that sends …

[Edit: Corrected some false claims and modified questions accordingly.] Let $\mathcal{S}$ be the cocomplete $(\infty, 1)$-category generated by a point. This is conventionally known as the $(\infty, 1 …

Consider the language of rigs (also called semirings): it has constants $0$ and $1$ and binary operations $+$ and $\times$. The theory of commutative rigs is generated by the usual axioms: $+$ is asso …

Here's a simple observation: the category of commutative monoids has an object that is both initial and terminal (just like the category of groups or abelian groups), so it cannot be cartesian closed. …

I think the language of classifying toposes is helpful in understanding this view of forcing. Let $P$ be a poset. The set theorists have the intuition that forcing over $P$ adjoins a generic filter of …

I offer the following summary/interpretation of Broodryk's results. In short, a category of algebras is coextensive if and only if there is a well behaved interpolation operation in the algebraic theo …

No. In fact any full subcategory of $\mathbf{Top}$ that contains all the discrete spaces cannot be monadic over $\mathbf{Set}$ unless it contains only discrete spaces. Indeed, for any such subcategory …

There is a Grothendieck topos $\textbf{Set}[\mathbb{O}]$ with the following universal property: for all Grothendieck toposes $\mathcal{E}$, the category $\textbf{Geom}(\mathcal{E}, \textbf{Set}[\mathb …

Starting from an ordinary 1-categorical point of view, there are various obvious candidate definitions for ‘finite homotopy type’: The homotopy type of a simplicial set that has only finitely many n …

Let $\mathcal{M}$ be the following category: The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels. The morphisms are functors that preserve the struct …

Here's a somewhat trivial one, but it is one that category theorists use all the time: Let us say that a functor $F : \mathcal{C} \to \mathcal{D}$ is a weak equivalence if it is fully faithful and …

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent …

I don't know if this counts as an application of the internal language or as an avoidance of it, but I think it is worth listing anyway. In the development of homological algebra and homotopy theory i …