# Questions tagged [internalization]

The internalization tag has no usage guidance.

13
questions

**4**

votes

**1**answer

168 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 ...

**3**

votes

**0**answers

121 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 ...

**15**

votes

**1**answer

329 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
$$\mathbf{Set}^{\mathrm{op}} \to \mathbf{CABA}, ~ X \mapsto (P(X),\bigcup,\bigcap).$$
...

**3**

votes

**1**answer

146 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 ...

**3**

votes

**0**answers

93 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 ...

**3**

votes

**0**answers

139 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}:\...

**6**

votes

**0**answers

202 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 ...

**29**

votes

**7**answers

4k 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 ...

**23**

votes

**1**answer

646 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 ...

**7**

votes

**3**answers

653 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 ...

**13**

votes

**1**answer

474 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)...

**6**

votes

**2**answers

584 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 ...

**12**

votes

**1**answer

778 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 ...