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Let $G$ and $G'$ be quivers. If their path categories $Path[G]$ and $Path[G']$ are isomorphic, does is follow that $G$ is isomorphic to $G'$?

The notion of a group object $G$ in a category with finite products can either be defined with a few commutative diagrams or via requiring that each hom set $\hom(X,G)$ is a group. There is a theorem …

Let $U$ be the forgetful functor from categories to quivers. Then the left adjoint $F$ of $U$ is the functor sending a quiver to its path category. It's a fact that $F$ is injective up to isomorphism, …

This proof of Lawvere's fixed point theorem suggests (since it uses $\lambda$ notation) that it is written in the internal language of cartesian closed categories (which is the $\lambda$-calculus, as …

Is there a geometric theory $T$ and a Grothendieck topos $\mathcal E$ such that (2) holds but (1) doesn't: $\mathcal E$ 2-represents the 2-functor $$\mathbf{GrothTop}\to\mathbf{Cat}$$ which sends a G …

This question is about Joyal and Tierney's famous An extension of the Galois theory of Grothendieck. One of the main results states (see the MathSciNet review by Peter Johnstone): Joyal and Tierney's …

Let $f\colon X\to Y$ be a continuous map. Then $f$ induces a geometric morphism $f^\ast\dashv f_\ast\colon \mathrm{Sh}(X)\leftrightarrows\mathrm{Sh}(Y)$, whose left adjoint is called inverse image and …

What are the major applications of the internal language of toposes? Here are a few applications I know: Mulvey's proof of the Serre–Swan theorem in which he interprets the intuitionistically valid r …