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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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0answers
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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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1answer
55 views

How does this statement reduces to the Yoneda lemma?

Context: I am reading the following page in the nLab, which is about simplicial presheaves, i.e Functors $\mathscr{C}^{\mathrm{op}} \to [\mathbf{\Delta}^{\mathrm{op}}, \mathbf{Set}]$. Equivalently, ...
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1answer
33 views

(co)Limits in $\infty$-categories.

It seems to me that the following must be true, and has been used in Lurie's HA, and in my question. The precise statement is: Let $p:K \rightarrow C$ be a small diagram in $C$. Here $C$ is an $\...
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0answers
10 views

How should one define the join of two simplicial categories?

This is in Cor. 4.2.1.4 of HTT. Given two simplicial categories, $A,B$, what is the definition for their join $$A \star B$$ which makes it still a simplicial category? One approach I could ...
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1answer
34 views

Internal hom: products and coproducts

It seems to me that Assuming the all smalllimits and colimits exist: Internal hom for a closed symmetric monoidal category satisfies: $$[\bigsqcup C_i, X] \cong \prod_i [C_i, X] $$ where $(...
2
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1answer
26 views

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 ...
2
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1answer
102 views

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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3answers
47 views

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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47 views

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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0answers
12 views

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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61 views

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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1answer
95 views

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 ...
2
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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 ...
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1answer
149 views

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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0answers
52 views

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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0answers
85 views

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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0answers
17 views

Colimit functor on an enriched category

Let $\mathscr{M}$ be a cocomplete category enriched over topological spaces, and $J$ be a small (ordinary) category toplogized with the discrete topology. Is it true that the functor $Fun(J,\mathscr{M}...
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35 views

$\mathbf{sSet}$-enriched Algebraic Theories

If $\mathcal{L}$ is the $\mathbf{sSet}$-enriched subcategory of $\mathbf{sSet}$ whose objects are finite coproducts of the terminal simplicial set $\Delta^0 = \Delta(-,[0]) = *$, identify the object $\...
3
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1answer
64 views

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 ...
3
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1answer
74 views

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 ...
2
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1answer
56 views

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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0answers
63 views

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-...
2
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1answer
104 views

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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1answer
91 views

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 ...
4
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1answer
71 views

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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0answers
41 views

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 ...
3
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1answer
155 views

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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2answers
186 views

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 ...
2
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0answers
94 views

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 ...
4
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1answer
151 views

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 ...
2
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1answer
146 views

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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1answer
303 views

Enriching an adjunction

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 $\...
2
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1answer
265 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 ...
5
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1answer
170 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 ...
2
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2answers
229 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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2answers
124 views

Single-object additive category

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 ...
2
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1answer
128 views

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 ...
2
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1answer
142 views

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 ...
2
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1answer
123 views

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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1answer
214 views

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 ...
2
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1answer
210 views

How to define a weighted cone?

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 =...
1
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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 ...
4
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2answers
542 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 ...
2
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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 ...
3
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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 $Ab$? ...
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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 ...
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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.
3
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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 "$\...
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0answers
155 views

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 ...