Questions tagged [internal-categories]
The internal-categories tag has no usage guidance.
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Example of a groupoid internal to the category of smooth manifolds that is not a Lie groupoid
This questions is about the distinction between:
Lie groupoids: we require source and target maps to be submersions. This implies that the domain of the composition map, $G_1 \;{}_s\!\times_t G_1$, ...
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Are (commutative) squares in some sense universal among edge-symmetric double categories?
Definitions:
Given a category $\mathcal{C}$, and a double category $\mathbb{D}$, i.e. a category internal to $Cat$ with an “objects category” $\mathcal{D}_0$ and a “morphisms category” $\mathcal{D}_1$,...
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Geometric realization of crossed square
Given a crossed square of groups, you can "totalize" it and get a 2 crossed module in the sense of Conduché "Modules croisés généralisés de longueur 2", then you can apply his ...
4
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The internalization hierarchy
For a complete category $\mathcal{C}$, we can consider the strict $2$-category ${\sf Cat}(\mathcal{C}$) of internal categories in $\mathcal{C}$. Similarly, for any continuous functor $F:\mathcal{C}\to\...
4
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What is a correct notion of an internal pseudofunctor?
Let $C$ be a category internal to a category $K$. It is well known (for example see Proposition 2.4 in the paper Higher Dimensional Algebra VI: Lie 2-Algebra by Baez and Crans https://digitalcommons....
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Condensed categories vs categories (co)tensored with condensed sets
I am not sure how to solve set-theoretic issues properly, so let me first ignore them.
There are two notions, probably closely related:
Condensed categories, i.e. condensed objects in the category of ...
12
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What is the right notion of a functor from an internal topological category to a topologically enriched category?
Let $\mathcal{C}$ be a category internal to (some convenient model for) topological spaces (which I will denote by $\mathsf{Top}$). In the question Greg Arone asks:
What is the correct notion of a ...
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Universal property of the V-Mat construction
Internal categories and enriched categories can both be realised as monads in certain bicategories. If $\mathcal E$ is a category with pullbacks, then a monad in $\mathbf{Span}(\mathcal E)$ is a ...
4
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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 ...
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Cotensor products (in monoidal categories) without regularity
In Internal Categories and Quantum Groups, Aguiar defines the cotensor product of two bicomodules as follows. Let
$(\mathcal{V},\otimes_{\mathcal{V}},\mathbf{1}_{\mathcal{V}})$ be a monoidal category;...
4
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Externalisation for non-Cartesian internal categories
In the context of category theory internal to a category $(\mathcal{E},\times,\mathbf{1}_{\mathcal{E}})$ with pullbacks and a terminal object, the process of externalisation builds an indexed ...
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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 ...
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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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'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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 ...
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683
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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 ...
8
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3
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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 ...
2
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1
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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: $...
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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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1
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735
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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. ...
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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
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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
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2
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605
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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
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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 ...
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J.W. Gray's monumental work notes on the formal theory of internal (2-)categories
In the book "Topos Theory" of Peter Johnstone (Topos Theory, LMS Monographs no. 10. Academic, 1977) one finds at page 41 in Chapter 2:
"For a detailed account of internal categories ...
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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 ...
8
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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-...
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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 ...