# Questions tagged [enriched-category-theory]

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47 questions
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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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### 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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### 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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### Underlying functor of tensor product in a closed and symmetric monoidal category.

I will follow, for terminology and notation, G. M. Kelly, Basic Concepts of Enriched Category Theory. For sake of a self-contained exposition, I will try to write here all the needed concepts. Let ...
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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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### 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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### 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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### 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 ...
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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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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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### 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 ...
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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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### 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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### 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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### 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.
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### 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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### 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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### 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 ...
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 ...