Questions tagged [internal-categories]

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14
votes
1answer
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, ...
12
votes
1answer
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 $\mathbb{D}\to\mathbb{C}$. ...
10
votes
1answer
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 ...
8
votes
1answer
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
3answers
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 ...
7
votes
1answer
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-...
6
votes
2answers
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 ...
6
votes
1answer
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 ...
6
votes
2answers
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 ...
6
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0answers
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
0answers
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 ...
5
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0answers
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 ...
5
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0answers
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 ...
4
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0answers
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 ...
4
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0answers
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 ...
3
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1answer
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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0answers
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
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0answers
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
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1answer
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. ...
2
votes
1answer
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 ...
1
vote
1answer
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: $...
1
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0answers
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 ...