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# Questions tagged [enriched-category-theory]

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### Cocomplete $R$-linear categories are tensored : adjoint functor theorem?

Let $B$ be an abelian category which is actually $Mod_R$-enriched for some ring $R$ (say unital commutative ring). For $b\in B$, we have a functor $\hom(b,-) : B\to Mod_R$ which preserves limits, so ...
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### Enriched yoneda lemma proof

This is an exercise in Riehl's Categorical Homotopy Theory. Lemma 7.3.5 Give a small $V$-category $D$, and object $d \in D$, a $V$-functor $F:D \rightarrow V$, the canonical map is a $V$-natural ...
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### What is the hom space in over category of simplicial sets?

Let $Set_\Delta$ denote category of simplicial sets, which is enriched in $Set_\Delta$. Let $B$ be a simplicial set. We can form the over category. $(Set_\Delta)_{/B}$. Then is this also enriched ...
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### Definition of simplicially enriched category

In Philip Hirschhorn's Model Category and Their Localizations, Def. 9.1.2, pg. 159, He defines a simplicially enriched category as A category $M$ together with Every two objects $X,Y$ ...
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### A group that is also a category

I have a group with a category structure (i.e. category whose objects form a group), such that the left multiplication with any fixed element is a category automorphism. The same is true for right ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $\... 1answer 264 views ### Interchange in 2-category. Consider a (strict) 2-category$\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 ... 2answers 227 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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### 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 ...
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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 ...
Let $F : I \to C$ be a diagram in $C$, and $N$ an object of $C$. A cone from $N$ to $F$ is a family of morphism $P_X : N \to F(X)$ such that for every morphism $f : i1 \to i2$ in $I$, $F(f) \circ P_X =... 1answer 116 views ### A linear category is a Vect-module I would like to know how to show that any linear category is a$\mathrm{Vec}$-module. Here$\mathrm{Vect}$denotes a category of finite dimensional vector spaces. More general statement can be found ... 1answer 51 views ### On the existence of finite tensors/cotensors Suppose that we are in an ordinary ($\mathbf{Set}$-enriched) category$\mathcal{C}$. Is there a criterion that ensures the existence of finite tensors/finite cotensors? Does it suffice to be finitely ... 1answer 102 views ### A question about colimits in enriched categories I am just starting to learn about enriched categories, so excuse me if I am asking something trivial. Suppose$\mathcal{C}$is a$\mathcal{V}$-enriched category$\mathcal{C}$, with$\mathcal{V}$very ... 2answers 539 views ### Flat Modules are Filtered Colimits of Free Modules A result by Wraith and Blass states that every flat module is a filtered colimit of free modules (see nLab, Thm 1). I am wondering if this is simply a corollary of Yoneda's density theorem which ... 0answers 40 views ### On the definition of 2-rigs I am reading the nLab entry on 2-rigs. In its list of definitions, it says that a 2-rig category can be defined as a$Ab$-enriched category which is enriched monoidal. Why is the enrichment in$Ab$? ... 1answer 163 views ### Homotopy category of a simplicial category In many places (for example here) I've seen the following definition: For a simplicial category$\mathcal{C}$, it's homotopy category is defined to be the category$Ho(\mathcal{C})$with the same ... 0answers 102 views ### projective model structure on presheaves , hom-functors are always cofibrant Why hom-functors are always cofibrant in the projective model structure in$[\cal T,\cal V]$? The claim is here on page 5. 1answer 164 views ###$\mathcal{V}$-naturality in enriched category theory Let$\mathcal{V}$be a monoidal category, in section 1.2 of "Basic concepts of enriched category theory" (http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf) Max Kelly introduces the terms "$\...
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 ...