# Questions tagged [enriched-category-theory]

The enriched-category-theory tag has no usage guidance.

119
questions

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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 ...

3
votes

0
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### Reference request for facts about bi(co)descent objects

I know the following facts are true, but I struggle to find adequate references for them:
Let $T$ be a pseudo-monad in a bicategory $\mathcal{C}$, and let $A$, $B$ be pseudo-algebras for $T$. Then, ...

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0
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53
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### Tensored and cotensored simplicial comma category

To transfer a tensored and cotensored simplicially enriched structure from a category $\mathcal{C}$ to $(\mathcal{C}\downarrow Z)$, we define $(X\to Z)\otimes K$ by the composite $(X\otimes K \to X \...

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### Constructing lax limits from lax limits

Let $K$ be a 2-category. It's well-known that if $K$ has all PIE limits, then $K$ also has all lax limits. But I don't know a general "limit-decomposition" result which works "...

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### Examples of nonpointwise Kan extensions that "play a mathematical role"

Most Kan extensions arising in nature are pointwise, and this observation prompts Kelly to write [1]:
Our present choice of nomenclature is based on
our failure to find a single instance where a [...

3
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0
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### Enriched categories over a semi-monoidal category

Let $\mathcal{V}$ be a semi-monoidal category, meaning it satisfies the axioms of a monoidal category except missing a unit and the unit axiom. One could then still go about defining a $\mathcal{V}$-...

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443
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### What are abelian categories enriched over themselves?

As far as I understand, an arbitrary abelian category is not enriched over itself, for example, $\mathrm{ChainComplex}(\mathrm{Ab})$ is, right? On the other hand, the categories $\mathrm{Mod}(R)$ (in ...

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### $V$-cat and $V$-graph: coequalizers in the category of enriched functors

This question is regarding the 1974 JPAA paper $V$-cat and $V$-graph by Harvey Wolff.
To be precise, I don't understand a certain step in the proof of Corollary 2.9, which (the corollary) is crucial ...

8
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1
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### What is the right notion of a functor from an internal topological category to topological spaces?

Let $\mathcal C=(\mathcal O, \mathcal M)$ be a category internal to topological spaces. Thus $\mathcal O$ and $\mathcal M$ are topological spaces: the space of objects and the space of morphisms ...

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135
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### When this coend is invariant up to homotopy?

It is a follow-up of my question Calculation of the homotopy colimit of a diagram of spaces which was badly formulated.
Consider a fixed diagram $D:I^{op}\to {\rm Top}$ where ${\rm Top}$ is
a ...

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83
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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 ...

2
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### Determining enriched limit-preserving functors by their global sections

Let $\mathcal{C}$ be a small 1-category and let $\mathcal{M}$ be a category enriched over the presheaf category $\widehat{\mathcal{C}}$ which is complete as $\widehat{\mathcal{C}}$ enriched category. ...

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### Are weighted limits terminal in a category of cones?

Consider a Benabou-cosmos $(\mathcal{V},\otimes,J)$, $\mathcal{V}$-categories $\mathcal{I},\mathcal{C}$ and $\mathcal{V}$-functors $\mathcal{W}:\mathcal{I} \rightarrow \mathcal{V}$ and $\mathcal{D}:\...

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### Tannaka without Yoneda?

I am studying enriched categories, and as I wrote in my previous question How is the morphism of composition in the enriched category of modules constructed?, this is very difficult because there are ...

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2
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### How is the morphism of composition in the enriched category of modules constructed?

I asked this a week ago at MSE, but without success.
I am studying enriched categories and I have a feeling that I am doing something wrong because all the way each step, each elementary proposition, ...

3
votes

1
answer

210
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### When are enriched categories equivalent?

$F : \mathbf{MonCat} \to \mathbf{2Cat}$ is the 2-functor for change of enrichment. What is the maximal subcategory of $\mathbf{MonCat}$ whose arrows $b : V \to W$ each induce an equivalence of ...

5
votes

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### Does an Ab-enriched category have a unique Ab-enrichment?

I know that the group structure on Hom sets can be recovered from biproducts if they exit. Indeed, if $f, g : A \to B$ are two maps then there is a uniquely defined map $f \oplus g : A \oplus A \to B \...

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1
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203
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### Structural properties of $\mathcal{V}$-$\mathsf{Cat}$

In this question $(\mathcal{V}, \otimes, e)$ is a (bi)complete symmetric monoidal category.
We have an adjunction $$\mathscr{l}: \mathsf{Cat} \leftrightarrows \mathcal{V}\text{-}\mathsf{Cat} :(-)_0,$$ ...

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3
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532
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### Relation between Ind-completion and "additive"-ind-completion

Suppose that $\mathcal{C}$ is a skeletally small additive category.
To enlarge $\mathcal{C}$ and produce a bigger category whose "small" objects can be identified with those in $\mathcal{C}$,...

3
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88
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### Density with respect to a family of diagrams, versus a class of weights

In Theorem 5.19 of Kelly's Basic Concepts of Enriched Category Theory, it is proven that a fully faithful functor $K \colon \mathcal A \to \mathcal C$ is dense if and only if $\mathcal C$ is the ...

11
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1
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488
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### Yoneda Lemma for monoidal functors

Let $(\mathcal V,\otimes,I)$ be a closed symmetric monoidal category, and let $\mathcal C$ be a $\mathcal V$-enriched category. The (weak) enriched Yoneda Lemma gives us a nice description of the set $...

6
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291
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### Examples of (co)ends

I am reading texts about (co)ends, and everywhere I see a lack of examples. I am not an expert in this area, and without examples it is difficult for me to use my intuition to grasp the idea. MacLane ...

3
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148
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### Explicit description of a pullback of $(2,1)$-categories

In the 1-category of 2-categories, with objects being categories enriched over Cat, and morphisms being 2-functors, is there an explicit way to describe a pullback of two functors $G:E\to D$ and $F:C\...

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145
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### Is there a nice way to define discrete enriched categories?

In the setting of classical category theory, one defines the discrete category associated to a set $X$ as the category $X_\mathsf{disc}$ having the elements of $X$ as its objects and only identities ...

4
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1
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### How to compute Homotopy Pullback

What on Earth is a homotopy pullback of
$$A \rightarrow B \leftarrow C \ \ \ \ \ ???$$
Here $A,B,C$ are elements of a category ${\mathcal V}$ enriched in topological spaces (any convenient category ...

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0
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136
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### Reference request: a class of matrices leading to interesting metric geometry

For $0 \le A \in GL(n,\mathbb{R})$, let $Aw = \Delta(A)$, where $\Delta$ denotes the map taking a matrix to a vector of its diagonal entries and/or forming a diagonal matrix from a vector, according ...

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1
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575
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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, ...

2
votes

1
answer

95
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### "Lie theory" for anchored bundles and reflexive graphs

Perhaps Lie theory is not the correct term, but I'm thinking of the intermediate result in the Lie groupoid to Lie algebroid correspondence. Given a Lie groupoid $G$ over $M$, we may construct the Lie ...

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96
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### Degree shift of multilinear maps

Let $V$ be a graded vector space over $\mathbb{k}$ and $V[1]$ its odd degree shift.
Given $k$, $l\in \mathbb{N}_0$, is there a natural way to define the following map,
$$
\psi: \hom_{\mathbb{k}}(V^{\...

7
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1
answer

274
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### Enrichment as extra structure on a category

We will suppose, for the sake of simplicity, that everything is happening within a fixed 'metacategory' $\textbf{SET}$ of sets and functions. So, from now on, a 'category' just means a category object ...

3
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1
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### Morphisms of $\infty$-groupoids

As far as I understand, there are several ways of defining $\infty$-categories. One of them is to think of $\infty$-cateogries as $top$-enriched categories. Hence we can think of $\infty$-groupoids as ...

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338
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### Theory of weak enrichment in higher categories

Has there been work towards a general theory of weak enrichment in higher categories? To be more pointed, has there been any work towards trying to make sense of statements such as
There is a (weak) $...

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2
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### Enriched vs ordinary filtered colimits

Filtered categories can be defined as those categories $\mathbf{C}$ such that $\mathbf{C}$-indexed colimits in $\mathrm{Set}$ commute with finite limits.
Similarly, for categories enriched in $\mathbf{...

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votes

1
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### Weighted Co/ends?

Recall: Limits
Recall that the limit of a functor $D\colon\mathcal{I}\to\mathcal{C}$ is, if it exists, the pair $(\mathrm{lim}(D),\pi)$ with
$\lim(D)$ an object of $\mathcal{C}$, and
$\pi\colon\...

3
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0
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### Kan liftings and projective varieties

Regard the following two bicategories:
$\operatorname{dg-\mathcal{B}imod}$, with objects dg categories, and morphisms categories from $C$ to $D$ being the categories of $C$-$D$-bimodules. Composition ...

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### Is monadicity preserved by the underlying functor?

Let $\mathcal{V}$ be a monoidal closed (complete, cocomplete, reasonable...) category.
Let $\mathsf{T}$ be an enriched monad over $\mathcal{V}$. The forgetful functor $\mathsf{U}: \mathsf{Alg}(\...

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2
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### Is the tensor product of chain complexes a Day convolution?

Recently, Jade Master asked whether the tensor product of chain complexes could be viewed as a special case of Day convolution. Noting that chain complexes may be viewed as $\mathsf{Ab}$-functors from ...

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1
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### Literature on linear categories

I am trying to understand Deligne's 'Categories Tensorielles', and therefore I need some knowledge on linear categories. Looking at Wikipedia and nLab, I found some definitions and explanations, but I ...

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### Classification of absolute 2-limits?

Let $\mathcal V$ be a good enriching category. Recall that an enriched limit weight $\phi: D \to \mathcal V$ is called absolute if $\phi$-weighted limits are preserved by any $\mathcal V$-enriched ...

6
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1
answer

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### Is there such a thing as a weighted Kan extension?

The title pretty much sums it up.
More in detail. Let $C$, $D$ and $E$ be categories, let $F:C\to D$ and $G:C\to E$ be functors, and let $P:C^{op}\to \mathrm{Set}$ be a presheaf. The colimit of $F$ ...

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### The symmetric monoidal closed structure on the category of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors

In 6.5 of the book by Kelly,
Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, No. 10, 2005.
the author claims that the $2$-category $\mathsf{Cat}_{\...

6
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1
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349
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### (Co)tensoring of enriched slice categories

In an answer to this question: Enriched slice categories, a description of the enrichment of the slice category in an enriched category is given. I'm interested in going a bit further. If we assume ...

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### Monoidal V-categories, and monoids

I am guessing that the definition of monoidal V-category is a V-category $\mathbf{A}$ together with a V-functor $(\boxtimes) \colon \mathbf{A} \times \mathbf{A} \to \mathbf{A}$ and a functor $i \colon ...

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### The homotopy category of the category of enriched categories

We know that if $\mathcal C$ is a combinatorial monoidal model category such that all objects are cofibrant and the class of weak equivalences is stable under filtered colimits, then $\mathsf{Cat}_{\...

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### Are (complete) 2-Segal spaces the same as Span-enriched infinity categories?

The question is basically in the title. More generally, I would like to know if this, or any reasonable variant of it, is true. Or perhaps, to understand better the gap between 2-Segal spaces and Span-...

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### Do adjoints of enriched functors preserve the enriched structure?

Is there is any reason in general for adjoints of enriched functors to preserve the enriched structure of categories?
The specific example I'm thinking of is the following:
Fix a commutative ring $R$...

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1
answer

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### By general reasons, $i_A \colon \mathbb{D}\text{-}\mathrm{cont}[A,\mathbf{Set}] \to [A,\mathbf{Set}]$ has a left adjoint

In Centazzo and Vitale's A Duality Relative to a Limit Doctrine (TAC, 2002, abstract), early on, they make the above claim and cite Kelly's Basic Concepts in Enriched Category Theory (TAC reprints). I ...

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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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### Tannaka duality for closed monoidal categories

I asked this some time ago at mathstackexchange, and people there explained to me the mathematical part of what I was asking, but the question about references remains open. In my impression, people ...

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### Multiplication and division by a morphism under the “inner composition” in closed monoidal categories

I asked this a week ago at math.stackexchange, without success, so I hope it will be appropriate here.
Let ${\mathcal C}$ be a symmetric closed monoidal category, and let me denote the internal hom-...