# Questions tagged [internalization]

The internalization tag has no usage guidance.

7
questions

**4**

votes

**0**answers

127 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

3k 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

607 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 ...

**6**

votes

**3**answers

568 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

410 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

572 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

694 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 ...