We’re rewarding the question askers & reputations are being recalculated! Read more.

Questions tagged [enriched-category-theory]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0
votes
1answer
17 views

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 ...
1
vote
0answers
39 views

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 ...
2
votes
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 $\...
2
votes
1answer
53 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, ...
0
votes
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 ...
0
votes
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
votes
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
votes
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 $X,Y$ ...
2
votes
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 ...
0
votes
0answers
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 ...
1
vote
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 ...
2
votes
0answers
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 ...
0
votes
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 ...
0
votes
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 ...
3
votes
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 ...
3
votes
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}...
0
votes
0answers
34 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
votes
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
votes
1answer
73 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
votes
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 ...
2
votes
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 ...
1
vote
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-...
3
votes
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
votes
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, ...
0
votes
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
votes
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 ...
1
vote
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 ...
2
votes
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
votes
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 ...
1
vote
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
votes
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 ...
6
votes
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
votes
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 ...
2
votes
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} ...
1
vote
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 ...
5
votes
1answer
169 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
votes
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
votes
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
votes
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 ...
0
votes
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
votes
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 =...
2
votes
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 ...
1
vote
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 ...
2
votes
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 ...
4
votes
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 ...
3
votes
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 $Ab$? ...
0
votes
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 ...
1
vote
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
votes
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 "$\...
2
votes
0answers
362 views

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 ...