# Questions tagged [internalization]

The internalization tag has no usage guidance.

The internalization tag has no usage guidance.

16
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An undirected graph is sometimes defined as a pair of sets $V$ and $E$ (vertices and oriented edges), together with two maps $i,f: E\to V$ (sending a directed edge its initial/final vertex) and a map $...

16
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2
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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 ...

20
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2
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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 ...

4
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1
answer

194
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It is well known that the notion of an internal category can be generalized to categories without pullbacks by considering cotensors of comodules in a monoidal category. I'm curious about the other ...

3
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Recall that
The biinitial monoidal category with a monoid is given by the augmented simplex category together with the monoid $([0],\sigma^{0}_{0},\delta^{0}_{0})$ there.
The biinitial symmetric ...

14
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1
answer

355
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The category of complete atomic boolean algebras $\mathbf{CABA}$ is equivalent to $\mathbf{Set}^{\mathrm{op}}$ via
$$\mathbf{Set}^{\mathrm{op}} \to \mathbf{CABA}, ~ X \mapsto (P(X),\bigcup,\bigcap).$$
...

4
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1
answer

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In enriched category theory over a base monoidal category $(\mathcal{V},\otimes_{\mathcal{V}},\mathbf{1}_{\mathcal{V}})$, one can consider $\mathcal{V}$ itself as a $\mathcal{V}$-enriched category ...

3
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100
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Has there been any work done on internal categories inside internal categories?
I'm familiar with $n$-fold categories, but I don't want an internal category inside the category of internal categories ...

3
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0
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146
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In any topos $\mathcal{S}$, we have the ability to speak about power objects $\mathcal{P}(X)$ of objects $X\in\mathcal{S}$. We can then define an internal closure operator as an arrow ${\sf cl}:\...

8
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1
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Has the notion of an internal $2$-category been studied, or more generally an internal $n$-category? Do we have any examples of naturally occurring internal $2$-categories/$n$-categories?
The ...

30
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7
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The internal hom (or exponential object) is basically a reification of the 'external' hom. It can be defined in any cartesian (or even monoidal, more on this later) category as the right adjoint of ...

23
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1
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Lemma D4.5.3 of Johnstone’s Sketches of an Elephant states:
Lemma. For an object $A$
of a topos $\newcommand{\E}{\mathcal{E}}\E$, the following are equivalent:
$A$ is internally ...

8
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3
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nLab defines a strict 2-groups in many different but equivalent ways, among them:
an internal group object in Cat,
an internal group object in Grpd
Also, it is known that strict 2-groups may be ...

13
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1
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502
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An object $X$ of a category $C$ is said to be projective if the hom-functor $C(X,-)$ preserves epimorphisms (or, in general, some restricted class of epimorphisms such as the regular or effective ones)...

6
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I am not an expert on elementary topos (meaning by this to work with the internal language in a Grothendiek topos) and I rather be told than struggle with the following:
Consider an elementary topos ...

14
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1
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834
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The category of internal locales in the Grothendieck topos of sheaves on a locale X
is equivalent to the slice category over X.
In other words, internal locales over X are precisely morphisms of ...