# Questions tagged [internal-categories]

The internal-categories tag has no usage guidance.

22
questions

**3**

votes

**1**answer

132 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**

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**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 ...

**4**

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**0**answers

88 views

### 'The' object of composable triples in an internal category

In any category $\mathcal{C}$ with pullbacks, we can define an internal category $\mathscr{C}$ in $\mathcal{C}$ as an object ${\bf Ob}_\mathscr{C}$ of objects and an object ${\bf Hom}_\mathscr{C}$ of ...

**14**

votes

**1**answer

369 views

### Relationship between enriched, internal, and fibered categories

In this question, let $(\mathcal{V}, \otimes, [-,-], e)$ be a nice enough symmetric monoidal closed bicomplete category.
The usual set-based Category theory has been generalized in many directions, ...

**6**

votes

**1**answer

142 views

### Reference on internal categories and externalization

I'm looking for a reference on internal categories and externalization of internally defined notions.
The nlab has a stub on externalization (more details are available under small fibration) and the ...

**1**

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**0**answers

101 views

### Aggregations (e.g., cardinality, indexed sums/products) internal to a syntactic category?

Note: Expanded and rephrased, per Todd's question below.
Suppose that we have a set-valued functor $S:\mathcal{C}\to\mathbf{Sets}$, and an arrow $p:Y\to X$ such that $S(p)$ has finite fibers.
From ...

**5**

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**0**answers

180 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 ...

**4**

votes

**0**answers

86 views

### Pushforward of an internal category along a functor

Let $F:C\to D$ be a “nice” functor (for example, $H_*(-;\mathbb{Z}):\mathbf{Top}\to \mathbf{Ab}^{\mathbb{Z}}$). Now assume that we have a category $O$ internal to $C$. Is there a canonical way to ...

**6**

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96 views

### Internal van Kampen colimits

Let $C$ be a category with pullbacks. Recall that a colimit in $C$ is van Kampen if it is preserved by $C/(-): C^{op} \to CAT$, $c \mapsto C/c$.
We can $C$-internalize everything in sight:
Let $\...

**5**

votes

**0**answers

116 views

### Cartesian liftings in double categories

The question: I wonder whether the following definition, or something similar, has appeared somewhere (see below for motivations). Any reference or pointer is welcome!
(In what follows, I denote ...

**8**

votes

**1**answer

482 views

### Yoneda Lemma for internal presheaves

I'm looking for a reference explaining under what conditions the internal Yoneda lemma holds; in particular, I am wondering if it is known what properties of the ($2$-)category of categories are ...

**7**

votes

**3**answers

624 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 ...

**1**

vote

**1**answer

203 views

### Creating Duals in A Category

Before stating my question I would like to provide afew motivating examples:
Examples:
In the category of Finitely-generated-projective $R$-modules, we have that:
$M^{\vee}:=Hom_R(M,R)$ satisfies: $...

**3**

votes

**0**answers

108 views

### Colimits of n-fold categories

An $n$-fold category is an internal category in the category of $(n-1)$-fold categories (and a $0$-fold category is just a Set).
General results about internal categories assure that the category of $...

**2**

votes

**1**answer

523 views

### Classifying space of a colimit of topological categories

Say I have a diagram $D:I\rightarrow\text{Cat}(\text{Top})$ of categories internal to compactly generated topological spaces. This induces a diagram $BD:I\rightarrow \text{Top}$ of classifying spaces. ...

**10**

votes

**1**answer

894 views

### Existence of internal toposes/inner models in a topos

It has been known for some time that one can define a topos as a model of a (finitary) essentially algebraic theory (or in other words, can be defined internal to any category with finite limits). In ...

**12**

votes

**1**answer

738 views

### What is the difference between internal presheaves and presheaves on a total space?

Suppose that $\mathbb{C}$ is a category with finite limits and that $\mathcal{D}$ is a category internal to $\mathbb{C}$. We can also represent $\mathcal{D}$ as a fibration $\mathbb{D}\to\mathbb{C}$.
...

**6**

votes

**2**answers

579 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 ...

**5**

votes

**0**answers

156 views

### Maximal algebraic sub-groupoids

By a theorem of Ehresmann, topological and Lie categories (by which I mean categories internal to $Top$ and $Diff$ respectively, with the condition that the source and target in the latter case are ...

**6**

votes

**2**answers

309 views

### What condition on a “bibundle between categories” generalizes “right-principal bibundle between groupoids”?

My question is long on background and motivation, and almost but not quite answered over at the nLab. I'll write up a bunch before asking my question (feel free to skip to the end or look at the ...

**7**

votes

**1**answer

492 views

### Who first came up with the idea of essential/Morita equivalence of internal groupoids/categories?

The idea that stacks can be identified with groupoids internal to the base site $S$ up to what is variously called essential/Morita equivalence is well known. The basic idea is that one takes the 2-...

**2**

votes

**1**answer

374 views

### internal version of a flat functor?

I'm working out of Sheaves in geometry and logic, for reference.
There is a characterisation of flat functors $A:C \to Set$ as those such that the Grothendieck construction $\int_C A$ is a filtering ...