Questions tagged [internalization]

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30 votes
7 answers
5k views

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 ...
seldon's user avatar
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24 votes
1 answer
726 views

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 ...
Peter LeFanu Lumsdaine's user avatar
22 votes
2 answers
2k views

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 ...
user1005113's user avatar
17 votes
1 answer
918 views

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 ...
Dmitri Pavlov's user avatar
16 votes
2 answers
1k views

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 ...
14 votes
1 answer
553 views

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)...
Mike Shulman's user avatar
14 votes
1 answer
449 views

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 $$\mathbf{Set}^{\mathrm{op}} \to \mathbf{CABA}, ~ X \mapsto (P(X),\bigcup,\bigcap).$$ ...
Martin Brandenburg's user avatar
8 votes
3 answers
804 views

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 ...
Pedro's user avatar
  • 733
8 votes
1 answer
403 views

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 ...
Alec Rhea's user avatar
  • 8,977
6 votes
2 answers
605 views

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 ...
Eduardo J. Dubuc's user avatar
5 votes
1 answer
244 views

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 ...
Emily's user avatar
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4 votes
1 answer
236 views

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 ...
Alec Rhea's user avatar
  • 8,977
3 votes
1 answer
287 views

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 $...
Antoine Labelle's user avatar
3 votes
0 answers
184 views

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 ...
Emily's user avatar
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3 votes
0 answers
111 views

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 ...
Alec Rhea's user avatar
  • 8,977
3 votes
0 answers
153 views

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}:\...
Alec Rhea's user avatar
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