# Questions tagged [enriched-category-theory]

The enriched-category-theory tag has no usage guidance.

**16**

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### A multicategory is a … with one object?

We all know that
A monoidal category is a bicategory with one object.
How do we fill in the blank in the following sentence?
A multicategory is a ... with one object.
The answer is fairly ...

**14**

votes

**2**answers

322 views

### Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?

It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid ...

**11**

votes

**1**answer

203 views

### Definitions of enriched monoidal category

This question is about two definitions of enriched monoidal categories I have:
Let $\mathcal{V}$ be a symmetric monoidal closed category.
The first definition: a $\mathcal{V}$-enriched category $\...

**8**

votes

**0**answers

212 views

### In what context can enriched category theory be done?

There are many possible situations one can do enriched category theory. See https://ncatlab.org/nlab/show/category+of+V-enriched+categories#possible_contexts for a list.
My question is what ...

**3**

votes

**1**answer

116 views

### Is the category of enriched operads (co)complete?

Let $V$ be a symmetric monoidal category
which is complete and cocomplete.
Is the category of small symmetric colored $V$-enriched operads complete and cocomplete?
If $V$ is presentable,
is it ...

**5**

votes

**2**answers

195 views

### Enrichments vs Internal homs

Consider the definition of existence internal homs for a general monoidal category category $\cal{C}$, mainly the existence of an adjoint for the functor
$$
X \otimes -: \cal{C} \to \cal{C},
$$
for ...

**8**

votes

**1**answer

143 views

### Are sifted (2,1)-colimits of fully faithful functors again fully faithful? (And a de-categorified variant)

1) Suppose that I have a sifted diagram of categories $\mathcal{C}_i$, another of the same shape $\mathcal{D}_i$, and that I have a system $F_i:\mathcal{C}_i\to\mathcal{D}_i$ commuting with the ...

**7**

votes

**1**answer

164 views

### Can an enriched functor be expressed as a colimit of representable functors?

Suppose that $\mathcal C$ is an ordinary category and $F:\mathcal C^{op}\longrightarrow Set$ a functor. Then, we can form the category $\mathcal C/F$ as follows : each object is a morphism of functors ...

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vote

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66 views

### Pointwise convergence in Lawvere metric spaces

In the formalism of Lawvere metric spaces, we have that the distance in the hom-space $[X,Y]$ is given by:
$$
d(f,g) = \sup_{x\in X} d(f(x),g(x)) .
$$
Therefore, a sequence of functions $f_n:X\to Y$ ...

**3**

votes

**1**answer

130 views

### Accessible categories in enriched category theory

I study some definitions of accessible category (see 1) and the applications of that notions; my question: exist a notion of accessible category in therm of enriched category theory? (in case of exist,...

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134 views

### Topological categories as enriched category

Lurie define in HTT(Def. 1.1.1.6) a topological category as a enriched category over compactly generated (and weakly Hausdorff) topological spaces, but usually we define a topological category as ...

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45 views

### Is composition in a simplicially enriched category always determined by a compatible simplicial tensoring (if such exists)?

Let $C$ be a simplicially enriched category, i.e., there are
a collection of objects $ob C$,
a simplicial set $map_C(X,Y)$ for $X,Y \in ob C$,
composition maps $map_C(Y,Z) \times map_C(X,Y) \to map (...

**10**

votes

**1**answer

211 views

### The category theory of Span-enriched categories / 2-Segal spaces

The category $\mathsf{Span}$ of spans of sets is symmetric monoidal closed under $\times$ (the cartesian product from $\mathsf{Set}$, which is not the categorical product in $\mathsf{Span}$), ...

**3**

votes

**1**answer

117 views

### Kan extension of conservative functors

Suppose the right Kan extension $\text{Ran}_F G$ of a conservative functor $F$ along a conservative functor $G$ exists (with the category $\text{dom} F=\text{dom} G$ not necessarily small).
Is it ...

**3**

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**0**answers

145 views

### dg-categories and fully faithful functor

dg: is for differential graded
Suppose that $F: C\rightarrow D$ is a dg-functor between small dg-categories such that:
F: Objects of $C$ $\rightarrow$ Objects of $D$ is injective.
$Hom_{C}(a,b)\...

**4**

votes

**1**answer

219 views

### Simplicial mapping spaces, stable $\infty$-categories, and triangles

Let $C$ be a stable $\infty$-category (presentable, if you like) and let $map(-,-)$ denote the simplicial mapping space. If $X \to Y \to Z$ is a fiber sequence, and $W$ is an object, when is $map(W,X) ...

**4**

votes

**1**answer

130 views

### Is there a monoidal category that coclassifies enriched category structures for a given set?

Let $S$ be a set. Is there a monoidal category $TS$ that we can construct from $S$ such that monoidal functors $F: TS \to M$ (up to monoidal natural isomorphism) correspond to $M$-enriched categories ...

**7**

votes

**1**answer

71 views

### generalized elements in monodial categories

In a category $\mathcal{C}$, a generalized element of an object $A$ means a morphism to $A$. It follows from Yoneda lemma that the object $A$ is determined by the collection of generalized of elements ...

**2**

votes

**4**answers

379 views

### Examples of enriched categories which are (co)powered or (co)tensored

For $\mathsf{V}$ a closed monoidal category, it is canonically powered (or cotensored) and copowered (or tensored) over itself with respect to the internal hom and tensor product.
Likewise, any (co)...

**2**

votes

**0**answers

83 views

### Classification of unitary pointed monoidal category

I wonder if the following classification results are true (and are there any references):
Unitary pointed monoidal categories (the fusion rule of the objects is given by a finite group $G$) are ...

**5**

votes

**1**answer

212 views

### Does every enriched functor preserve tensors?

Let $\cal{P}$ be a $k$-linear semisimple abelian rigid monoidal category with finite dimensional (over $k$) Hom-spaces (for a field $k$).
By a tensored $\cal{P}$-category we mean a $\cal{P}$-category ...

**4**

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67 views

### To what kind of generalized Lawvere theory does the “free cartesian closed category” 2-monad on $\mbox{Cat}_g$ correspond?

Thinking of Cat as a mere 1-category, there is a 1-monad $\Lambda$ for the free cartesian closed category on a category. To every category X it assigns the category $\Lambda(X)$ whose objects are ...

**2**

votes

**1**answer

273 views

### Colimits in the category of simplicial categories

A simplicial category is a category enriched over the monoidal category of simplicial sets (morphism sets are now simplicial sets), and the collection of all such categories forms a category itself (...

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94 views

### Posets as (0,1)-categories

I am reading on the nLab that a poset can be seen as a (0,1)-category.
I was assuming all along that an ($n$,$r$)-category were a category where all morphisms of order larger than $n$ are trivial. ...

**1**

vote

**1**answer

210 views

### Comma category as weighted limit

Let $F : C \to E$ and $G : D \to E$ be functors. Consider the comma category $(F \downarrow G)$ with its projections $\pi_1 : (F \downarrow G) \to C$ and $\pi_2 : (F \downarrow G) \to D$. Using the ...

**1**

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**1**answer

124 views

### Colocal Objects in Enriched Bousfield Colocalizations

Let $C$ be a $V$-model category, and $\mathcal{K}$ a set of objects of $C$.
Let me denote (derived) simplicial homotopy function complexes by $\text{Dmap}$
and derived $V$-function complexes by $\text{...

**1**

vote

**1**answer

271 views

### Yoneda extension of a faithful functor is faithful

Let $F: \mathcal C \to \mathcal D$ be a functor with $\mathcal D$ cocomplete, and let $\mathscr P \mathcal C$ be the free cocompletion of $\mathcal C$ (i.e., the category of small presheaves on $\...

**15**

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**1**answer

363 views

### Why is every object cofibrant in an excellent model category?

In Appendix A.3 of the book higher topos theory appears the notion of an excellent model category (see Definition A.3.2.16). The main feature of this notion is that when $\mathbf{S}$ is an excellent ...

**3**

votes

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80 views

### Reference for generalized ind-completions?

I am wondering whether any enriched versions of ind- and pro- completions have been studied? I can not find any literature on them, even though I believe people (most likely the Australian school) ...

**21**

votes

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1k views

### How to stop worrying about enriched categories?

Recently I realized that ordinary category theory is not a suitable language for a big portion of the math I'm having a hard time with these days. One thing in common to all my examples is that they ...

**3**

votes

**2**answers

394 views

### Question about Enriched Categories and Functors

How would one describe the process of enriching a category C over some monoidal category D? Is there some functor between them that adds structure to the hom-sets?

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553 views

### The category of elements, enrichment, and weighted limits

This is a crosspost of this MSE question.
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 ...

**5**

votes

**1**answer

199 views

### Enriched Cauchy completions and underlying categories

The ordinary Cauchy completion $\overline{C}$ of a small category $C$ satisfies a number of conditions: Every idempotent in $\overline{C}$ splits, there's an equivalence of categories $[C^{op}, Set] \...

**4**

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376 views

### How to compute (co)limits of enriched categories?

I've asked this question on math.stackexchange some time ago (https://math.stackexchange.com/questions/1380176/how-to-compute-colimits-of-enriched-categories) and I received no complete answers, so I'...

**1**

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**1**answer

176 views

### How to define the internal hom between presheaves valued in cotensored categories?

First let $\mathcal{V}$ be a closed symmetric monoidal category
and $\mathcal{M}$ be a category enriched over $\mathcal{V}$. Moreover we assume $\mathcal{M}$ is cotensored, or powered over $\mathcal{...

**4**

votes

**1**answer

605 views

### Properties of loop space functor from homotopy types to group objects in homotopy types

I am trying to understand some properties of categories enriched in homotopy types, and the following question has become important:
When we take the loop-space of a (connected) homotopy type, we get ...

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185 views

### Completion under weighted limits/colimits

Is there any further reference besides "Basic Concepts of Enriched Categories" (Kelly) for completion under T-(weighted) limits/colimits?
(in which T is a set of weights)
Thank you in advance

**2**

votes

**1**answer

298 views

### When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories?

Let $\mathrm{Quillen}$ be the model category of simplicial sets with the Quillen model structure, and $\mathrm{Joyal}$ the model category of simplicial sets with the Joyal model structure.
As is well-...

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votes

**1**answer

220 views

### equivalence in simplicial category

Let $(\mathcal{C},W)$ be a category with weak equivalences. One can build from $(\mathcal{C},W)$ its hammock localization $L^{H}(\mathcal{C},W)$ which is a simplicial category $\textit{ie}$ a category ...

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212 views

### Ends and Coends - Analogues for higher arity - Horn Filling

Consider the setting of categories enriched over a suitable monoidal category $\mathbb V$.
We define $$\mathrm{Dist}(X,Y):=\mathbb V−\mathrm{Cat}(X^ \mathrm{op}⊗Y,\mathbb V).$$
Recall the definition ...

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votes

**1**answer

298 views

### About a Double-pseudo-category generalization of the module bicategory construction

To a category with finite limits $\mathscr{C}$, it is associated the bicategory of its spans $Span(\mathscr{C})$. Furthermore the bicategory of (bi)modules (and monoids) on $Span(\mathscr{C})$ is ...

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194 views

### Reference request: Grothendieck construction for $\mathbb V$-distributors?

I'm currently working with an analogue of the Grothendieck construction for enriched categories:
Given a distributor a.k.a. $\mathbb V$-functor $D:X^\mathrm{op}\otimes Y\to \mathbb V$ there is a ...

**5**

votes

**1**answer

237 views

### Reference request: (co)limits in Eilenberg--Moore (V-)categories

The following result seems to be well known:
If T is a (V-)monad on a (V-)category C, then the forgetful functor $U^T \colon C^T \to C$ creates
any limits that exist in C, and
any ...

**12**

votes

**2**answers

601 views

### What are the higher morphisms between enriched higher categories?

This question is about $n$-categories, or perhaps $(\infty,n)$-categories, or ... My guess is that the answer will not depend sensitively on the model of higher categories, so rather than have me ...

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201 views

### how many ways can an algebra be a weighted colimit of free algebras?

For a given weight $W : \mathcal{S}^{op} \to \mathcal{V}$ and diagram $D : \mathcal{S} \to \mathcal{A}$, the weighted colimit is an object $W \cdot D$ together with an isomorphism
$$\mathcal{A}(W\cdot ...

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**1**answer

451 views

### Cotensor vs exponential objects.

Under which conditions can we say that the cotensor objects in a (closed) V-category are the exponential objects? It is just when V=Set?

**4**

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**1**answer

318 views

### A Reference on Multicategories with “Internal Hom”

The multicategory of Waldhausen categories is "enriched over itself": the Hom-set of $k$-exact functors can be given a Waldhausen category structure by letting the morphisms be natural transformations,...

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297 views

### Enriched Categories: Metric Spaces, Monoidal Endofunctors and Lipschitz-Continuous Maps.

In the introduction to the reprint of "Metric spaces, generalized logic and closed categories" Lawvere talks about the following situation:
Let $\mathbb R_+$ denote $\mathbb R_{\geq 0}^\infty$. Every ...

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**1**answer

161 views

### Reference Request(Enriched Categories): Metric on Lipschitz Continuous Functions

If we consider metric spaces to be categories enriched over $\mathbb R_{\geq 0}$, the object corresponding to presheaves should be lipschitz-continuous functions $\operatorname{Lip^ 1}(M, \mathbb R_{\...

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**1**answer

320 views

### Definition of enriched caterories or internal homs without using monoidal categories.

I know this question may seem nonsensical at first but let me exlain what i have in mind:
In enriched category theory we define categories enriched over a monoidal category $(\mathcal{V},\otimes, I)$....