# Questions tagged [enriched-category-theory]

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

137
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### Are $\mathscr{V}$-modules uniquely (nicely) enrichable?

$\require{AMScd}\newcommand{\V}{\mathscr{V}}\newcommand{\M}{\mathcal{M}}\newcommand{\hom}{\operatorname{hom}}\newcommand{\op}{{^\mathsf{op}}}$Fix a closed symmetric monoidal category $(\V;\otimes;\...

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### Enriched cofibrant replacement in spectrally enriched categories

If $\mathcal{V}$ is a monoidal model category with all objects cofibrant, Theorem 13.5.2. of Categorical Homotopy Theory will guarantee that the functorial cofibrant replacement of a $\mathcal{V}$-...

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1
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### Free enriched monoidal categories

Suppose $(\mathcal{V},\otimes,1)$ is a symmetric monoidal category and $\mathbb{C}$ is a $\mathcal{V}$-category. I will deliberately avoid usual powerful assumptions (eg completeness/cocompleteness) ...

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### Understanding this (standard?) notion of enriched product category

$\newcommand{\V}{\mathscr{V}}\newcommand{\A}{\mathcal{A}}\newcommand{\B}{\mathcal{B}}\newcommand{\C}{\mathcal{C}}$Fix a closed symmetric monoidal category $\V$, writing the product as $\otimes$, the ...

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### Preservation of lax limits in categories of functors and lax natural transformations

Let $f:\mathbb{C} \to \mathbb{D}$ be a functor of 2-categories and let $\operatorname{Fun}(\mathbb{C},\operatorname{Cat})^{\operatorname{lax}}$ denote the 2-category of functors and lax natural ...

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### Condition for an equivalence of functor categories to imply an equivalence of categories

Given small categories $\mathcal{C}$ and $\mathcal{D}$, we have that $[\mathcal{C}^\text{op},\textbf{Set}]\simeq[\mathcal{D}^\text{op},\textbf{Set}]$ if and only if the Cauchy-completions of $\...

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### Can we do away with cotensors when exploring the equivalence between closed $\mathscr{V}$-modules and strongly tensored $\mathscr{V}$-categories?

$\newcommand{\M}{\mathcal{M}}\newcommand{\ML}{\underline{\mathcal{M}}}\newcommand{\N}{\mathcal{N}}\newcommand{\NL}{\underline{\mathcal{N}}}\newcommand{\V}{\mathscr{V}}\newcommand{\VL}{\underline{\...

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813
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### Are algebroids "just matrices"?

$\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Mat{Mat}$This question was originally asked on MSE but may be better here.
Algebroids are particularly interesting structures: they are basically ...

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353
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### Why are enriched (co)ends defined like that?

I'm mainly following references such as Kelly, Loregian and the nLab, and it seems customary there to generalize (co)ends to the enriched context (over a symmetric monoidal category $\mathcal{V}$) by ...

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### Conditions for natural transformations of weights to induce adjunctions of weighted limits

Suppose we have:
-) A $2$-category $\mathsf{J}$
-) A natural transformation of $\alpha : M \Longrightarrow W : \mathsf{J} \longrightarrow \mathsf{Cat}$
-) A functor $X:\mathsf{J} \longrightarrow \...

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1
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### Why do we need enriched model categories?

As far as I understand, model categories mainly provide tools for studying the "homotopy theories" (that is, $\infty$-categories) that are ubiquitous in mathematics. From this point of view, ...

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### Finite groups acting on algebraic groups and representations

Let $H$ be a connected algebraic group over an algebraically closed field $k$, and $I$ a finite group which acts on $H$ through group scheme morphisms. Denote by $Rep(H)$ the category of finite ...

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422
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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 ...

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505
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### Can a category be enriched over abelian groups in more than one way?

An $\mathbf{Ab}$-category is a category enriched over the category of abelian groups. What is an example of a category that can be enriched over abelian groups in more than one way?
An abelian ...

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### Cocompleteness of enriched categories of algebras

A useful result due to Linton is that for a cocomplete category $C$ and monad $T$ on $C$, if the category of algebras $C^T$ admits reflexive coequalisers, then it is cocomplete (see here for a sketch ...

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### V-categories enriched in a monoidal V-category

In an email to the categories mailing list dated 21 August 2003, Street writes:
Max reminded me of his old result (not in the LaJolla Proceedings,
but known soon after) that a monoidal V-category is ...

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### Are homotopy colimits strict?

Let's say we are working with a fibrant simplicially enriched category $\mathbf{B}$ that has all limits and all homotopy limits, and let $\mathbf{A}$ be a full subcategory that is closed under weak ...

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1
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160
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### Strictification of $\mathcal{V}$-pseudofunctors

Let $\mathcal{B}$ be a bicategory. Section 4.10 of Gordon, Power and Street's paper "Coherence for Tricategories" states that there is a bicategory $\textbf{st}\mathcal{B}$ and a ...

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 ...

3
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1
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50
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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 on a bicategory $\mathcal{C}$, and let $A$, $B$ be pseudo-algebras for $T$. Then, ...

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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 [...

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1
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170
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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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### 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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268
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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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536
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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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141
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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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### 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 ...

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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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840
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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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### 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, ...

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271
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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 ...

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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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### 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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### 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 $\mathcal{C}$,...

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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 ...

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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 $...

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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 ...

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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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### 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 ...

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681
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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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### 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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774
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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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### "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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129
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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^{\...

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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 ...

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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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### 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) $...