# Questions tagged [enriched-category-theory]

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### decomposition of hom-functors in a self-enriched category

Let $\mathbb{C}$ be a self-enriched category (such as Set). The Functor $\mathbb{C}(X, \mathbb{C}(Y,\_))$ is the same than the composition of functors $\mathbb{C}(X,\_) \circ \mathbb{C}(Y,\_)$. In a ...
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### Recognising categories of enriched categories

Given a monoidal category $\mathcal V$ one has the category $\mathcal V{-}\mathbf {Cat}$ of (small) $\mathcal V$-enriched categories and $\mathcal V$-enriched functors (and $\mathcal V$-enriched ...
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### Do we have to redo category theory when learning about enriched categories? [closed]

I'd like to know whether there exists a common language that encompasses both categories and enriched categories, so that results pertaining to either may be proven in a uniform way. I'd prefer it if ...
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### Enriched category over non-monoidal category

I'm finally learning homological algebra, and the notion of an enriched category seems to be the right setting in which to define the $\mathbf{Ext}$ functor. Yet, the definitions of an enriched ...
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### A linear category is a Vect-module

I would like to know how to show that any linear category is a $\mathrm{Vec}$-module. Here $\mathrm{Vect}$ denotes a category of finite dimensional vector spaces. More general statement can be found ...
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### k-linear category

In the tasks I have it says "Let $k$ be a field. Show that the structure of a $k$-linear category on a category $\mathcal{C}$ is equivalent to $\mathcal{C}$ being a module category (see Module ...
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### Notation: why are categories enriched over $\mathcal V$?

In most references about enriched categories, $(\mathcal V, \otimes)$ is supposed to be a monoidal category and then $\mathcal V$-enriched categories are defined. Why is the letter $\mathcal V$ used ...
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### Do 2-categorical left adjoints preserve $\mathcal{V}$-colimits?

Let $\mathcal{C}$ be a category enriched in $\mathcal{V}$, and $\mathcal{D}$ be a category enriched in $\mathcal{W}$. It is well known that $\mathcal{V}Cat$, the category of $\mathcal{V}$-enriched ...
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### Some questions about Enrichement Definition in Category Theory

Here is the definition of enrichment captured from Borceux. My questions: It seems to me we cannot define enrichment over any monoidal category, because: First, take the 3rd requirement, the ...
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### DG cofibrant replacement functor

This problem is one occurred in Bertrand Toën’s Lectures on DG-categories Prop 4.3.4. Let $M$ be a cofibrantly generated $C(k)-$model category. Then it is automatically a DG category and its ...
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### What is the right notion of separator in a 2-category?

A separator, or separating family, in a category is a full subcategory $\mathcal{S} \hookrightarrow \mathcal{E}$ of a category $\mathcal{E}$ which satisfies the following: For any parallel pair of ...
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### Examples of weakly dualizable objects in a non-closed monoidal category.

The following is a straightforward generalization of the notion of dualizable object in a symmetric monoidal category given in Duality, Trace and Transfer by Albrecht Dold and Dieter Puppe to non-...
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### Reference Request. Using algebraic geometry to study categories enriched over rings.

EDIT: As mentioned below in the comments, take subcategory of $\mathbf{Vec}_k$ consisting of endomorphisms. Then $\text{End}_k(V)$ carries a natural ring structure. My question is in multiple ...
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### What is a Presheaf-category enriched pullback?

I have a question about presheaf-enriched categories, like sSet for example that I think is pretty basic, but I don't know how to go about. So I have a category $C$, like $\Delta^\text{op}$, that is ...
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### Monoidal Category (Coherence Conditions)

In the definition of a monoidal category below, can someone please explain the idea behind the coherence conditions, especially the pentagon diagram's construction. Why do we need four elements A,B,C, ...
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### Division (or internal hom) notation in monoidal categories

At the end of section A.1.4 of the book "higher topos theory," there is a formula $X\otimes (C\otimes D)\simeq (X\otimes C)\otimes D$ which means the action property of tensoring in enriched ...
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### Definiton of the beta in Lurie's HTT

In the section §A.3.1 of Lurie's "Higher Topos Theory", the map $\beta_{X,S} \colon S \otimes FX \to F(S \otimes X)$ is defined without assuming $F$ has the structure of $\mathbf{S}$-enriched ...
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### Definition of enriched category as lax-monoidal functor

I am striking against the following definition/characterization (from the nlab) $\newcommand{\id}{\text{id}} \newcommand{\comp}{\text{comp}}$ of enriched category ... an alternative way of viewing ...
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### Is there a known connection between graded categories and enriched categories?

The notion of an enriched category and that of a graded category are both similar in the sense that they both endow the usual morphisms of a category with additional structure. A natural question is ...
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### Does Yoneda embedding reflect equivalent categories?

Let $\mathsf{Cat}$ denote the category of small categories. For categories $\mathcal A$ and $\mathcal B$ in $\mathsf{Cat}$, let $[\mathcal A,\mathcal B]$ denote the category whose objects are functors ...
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### The one-arrow category as a weighted limit in Cat

Many categories can be defined as weighted limits or colimits in the 2-category of categories Cat. For example the category 1 (one object with its identity) is the terminal object of Cat, the category ...
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I'm studying the notion of a a $\mathcal{V}$ category $\underline{\mathscr{A}}$ which is powered or compowered over $\mathcal{V}$. I'm having trouble finding a proof that powering/copowering gives a $\... 1answer 172 views ### Interchange in 2-category. Consider a (strict) 2-category$\textbf{A}$, with vertical composition denoted$\circ$, and horizontal composition denoted$\ast$. Let$f,g: A \to B$be$1$-cells, and$\alpha: g \to f$a$2$-cell. If ... 1answer 159 views ### Reference request: Categories enriched over$\textbf{FinLat}$Let$\textbf{FinLat}$be the category of finite lattices with$0$, regarded as a monoidal category by the tensor product of semilattices. It is known that the tensor product of two finite lattices ... 2answers 190 views ### Familiar categorical limits viewed elegantly as weighted limits? Are there any limits in ordinary category theory that are more elegantly seen as weighted limits? In$\mathsf{Set}$-enriched category theory, one can say that the limit of$\mathbf{J} \xrightarrow{D} ...
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A pre-additive category with a single object $\bullet$ is simply a ring $R = \mathrm{Hom}(\bullet,\bullet)$: pre-additivity makes this Hom-space an abelian group and with bilinear composition, i.e. a ...
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### Delooping of a ring?

I'm no expert on category theory, so the definition of delooping in the nlab article is a bit over my head. However, I do understand the practical idea that we can think of a group $G$ as a one-object ...
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### Graphic intuition for generalizing to weighted limits

One of the ways to define a limit of a functor $F:\mathsf C\longrightarrow\mathsf D$ is a representation of $\mathsf{Nat}(\Delta-,F)$. Along the journey of generalization to the enriched setting, one ...
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### Is $Set$ a 2-category?

I've read that $Rel$ (the category of sets and relations) is a 2-category by considering 2-morphisms to be inclusion of relations. Is $Set$ also a 2-category by considering 2-morphisms to be inclusion ...
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### Weighted limits in the $Cat$-category of categories

What is a weighted limit in the $Cat$-category of categories, functors and natural transformations? I can find the general definition of a weighted limit for enriched categories in Kelly's book or ...
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### The category of elements, enrichment, and weighted limits

Every so often, when reading notes online or skimming through books, the category of elements and the Grothendieck construction pop up. I don't know anything about the Grothendieck construction, and I ...
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### Relation between strong profunctors and enriched profunctors

We know that in a closed monoidal category $V$, the two statements are equivalent: $T$ is a strong endofunctor $T$ is an (enriched) endofunctor in $V$ enriched over itself. I'd like to know if ...
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### How to compute (co)limits of enriched categories?

Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$. This ...
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It is well known that if $\mathcal{C}$ is a category with finite biproducts, then we can define a binary operation "$+$" on every set of morphisms $Hom_{\mathcal{C}}(X,Y)$ using the diagonal $\Delta_{... 1answer 414 views ### Yoneda lemma for enriched categories Let$\mathcal{M}$be a monoidal category. Is there a generalization of the Yoneda lemma to categories enriched over$\mathcal{M}$? In the specific case I need,$\mathcal{M}$would be the category$k\...
Let $K$ be a field, regarded as a monoidal category in the following way: Objects are elements $x\in K$; Morphisms between $x\to y$ are only identities; The (strictly symmetric) monoidal structure \$x\...