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Here is a reformulation/generalisation of G. Stefanich's counterexample, showing that sheaf-locality can fail very dramatically once we leave the realm of locally finitely presentable categories. More …

Let $\mathcal{A}$ be a locally small category with colimits of small filtered diagrams. For the purposes of this question, an $\mathcal{A}$-presheaf on a topological space $X$ is a functor $\Omega (X) …

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. …

Let $C$ be a small (1-)category. There is always a poset $D$ and a functor $p : D \to C$ such that: $p$ is surjective on objects, i.e. for every $c$ in $C$ there is a $d$ in $D$ such that $p (d) = c$ …

I think for your specific problem it suffices to add a compatibility condition between the locally internal category $\mathcal{C}$ and the NNO. First, let me describe the case where $\mathcal{C}$ is e …

Terence Tao's cheap non-standard analysis can be interpreted as happening in a certain elementary topos, which I define here. Amusingly, this construction can itself be interpreted as happening in the …

To answer your question directly, WISC does not imply the existence of subobject classifiers. Notice that when there are only trivial covers, WISC is trivially satisfied, so it suffices to find a cate …

This is ultimately the same construction as the one Simon Henry describes, but you might like the different perspective. Definition. Let $A$ be a commutative rig and let $L$ be a distributive lattice. …

They are (it is?) the same theorem, but emphasising different aspects. We can exploit the object classifier to get from the formulation in terms of (pseudo)colimits to the "elementary" formulation in …

There's a long story that can be told here but I will try to be brief. In one sense, the answer is yes – you can certainly define cohomology and homotopy groups and so on for pretoposes and have them …

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 …

There is a precise, almost literal, sense in which $f^* : \textbf{Sh} (Y) \to \textbf{Sh} (X)$ generalises the inverse image as defined in elementary set theory. Observe that open subspaces $V \subset …

There is a notion of the étale homotopy type of a (Grothendieck) topos, going back to Artin and Mazur (I think). However, in classic "French" fashion they turned a theorem (in one setting) into a defi …

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 …

There are several theorems in category-theoretic logic which say something like, "any proposition in X logic that is provable in topos logic assuming (the law of excluded middle and) the axiom of choi …