Questions tagged [internalization]
The internalization tag has no usage guidance.
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Can graphs of groups be thought of as "graph objects" in the category of groupoids?
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 $...
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Major applications of the internal language of toposes
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 ...
22
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2
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An extension of the Galois theory of Grothendieck
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 ...
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4
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Internal monoidal categories
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 ...
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What is the initial semiring category with a (commutative) semiring?
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 ...
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14
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What are internal complete atomic boolean algebras, intuitively?
The category of complete atomic boolean algebras $\mathbf{CABA}$ is equivalent to $\mathbf{Set}^{\mathrm{op}}$ via
$$\mathbf{Set}^{\mathrm{op}} \to \mathbf{CABA}, ~ X \mapsto (P(X),\bigcup,\bigcap).$$
...
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Internalising the base in internal category theory
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 ...
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Reclusive Categories
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 ...
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Internalizing 'topology on a set'
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}:\...
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Internal $2$-categories
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 ...
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Are there categories whose internal hom is somewhat 'exotic'?
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 ...
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Is Lemma D4.5.3 in the Elephant correct? (“In a topos, weakly projective implies internally projective.”)
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 ...
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strict 2-groups VS crossed modules
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 ...
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14
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Pullback-stability of internally projective objects
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)...
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On internal functions and arrows in a Topos
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 ...
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What are the smooth manifolds in the topos of sheaves on a smooth manifold?
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 ...
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