Questions tagged [internal-categories]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
4 votes
0 answers
70 views

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\...
Alec Rhea's user avatar
  • 8,868
4 votes
1 answer
205 views

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....
Adittya Chaudhuri's user avatar
7 votes
0 answers
444 views

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 ...
Z. M's user avatar
  • 1,563
12 votes
1 answer
348 views

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 ...
Niall Taggart's user avatar
2 votes
0 answers
104 views

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 ...
varkor's user avatar
  • 7,797
4 votes
1 answer
223 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 ...
Alec Rhea's user avatar
  • 8,868
3 votes
1 answer
219 views

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;...
Emily's user avatar
  • 9,655
4 votes
0 answers
108 views

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 ...
Emily's user avatar
  • 9,655
4 votes
1 answer
223 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 ...
Emily's user avatar
  • 9,655
3 votes
0 answers
111 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 ...
Alec Rhea's user avatar
  • 8,868
4 votes
0 answers
102 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 ...
Alec Rhea's user avatar
  • 8,868
18 votes
2 answers
796 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, ...
Ivan Di Liberti's user avatar
7 votes
1 answer
236 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 ...
Alec Rhea's user avatar
  • 8,868
1 vote
0 answers
106 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 ...
pnips's user avatar
  • 11
8 votes
1 answer
383 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 ...
Alec Rhea's user avatar
  • 8,868
4 votes
0 answers
109 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 ...
FKranhold's user avatar
  • 1,623
6 votes
0 answers
125 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 $\...
Tim Campion's user avatar
  • 58.1k
5 votes
0 answers
154 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 ...
Damiano Mazza's user avatar
9 votes
1 answer
656 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 ...
Vladimir Sotirov's user avatar
8 votes
3 answers
783 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 ...
Pedro's user avatar
  • 723
2 votes
1 answer
270 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: $...
ABIM's user avatar
  • 5,001
3 votes
0 answers
114 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 $...
Maxime Lucas's user avatar
2 votes
1 answer
698 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. ...
Espen Nielsen's user avatar
11 votes
1 answer
1k 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 ...
David Roberts's user avatar
  • 33.1k
12 votes
1 answer
845 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 $\mathbb{D}\to\mathbb{C}$. ...
user35952's user avatar
  • 121
6 votes
2 answers
603 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 ...
Eduardo J. Dubuc's user avatar
5 votes
0 answers
171 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 ...
David Roberts's user avatar
  • 33.1k
8 votes
1 answer
734 views

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 ...
Buschi Sergio's user avatar
6 votes
2 answers
358 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 ...
Theo Johnson-Freyd's user avatar
8 votes
1 answer
589 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-...
David Roberts's user avatar
  • 33.1k
2 votes
1 answer
406 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 ...
David Roberts's user avatar
  • 33.1k