The tag has no usage guidance.

learn more… | top users | synonyms

1
vote
1answer
112 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 $\...
13
votes
1answer
267 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
0answers
70 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) ...
19
votes
1answer
678 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 ...
2
votes
2answers
215 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?
8
votes
2answers
311 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 ...
4
votes
1answer
130 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] \...
2
votes
0answers
113 views

How to compute (co)limits of enriched categories?

I've asked this question on math.stackexchange some time ago (http://math.stackexchange.com/questions/1380176/how-to-compute-colimits-of-enriched-categories) and I received no complete answers, so I'm ...
1
vote
1answer
103 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
1answer
276 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 ...
1
vote
2answers
167 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
1answer
253 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-...
0
votes
1answer
214 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 ...
3
votes
0answers
205 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 ...
3
votes
1answer
220 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 ...
1
vote
0answers
174 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
1answer
199 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 ...
11
votes
2answers
526 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 ...
0
votes
0answers
195 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 ...
1
vote
1answer
386 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
votes
1answer
296 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,...
6
votes
0answers
252 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 ...
2
votes
1answer
145 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_{\...
4
votes
1answer
268 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 $(\mathcal{V},\otimes, I)$....
34
votes
0answers
745 views

Enriched Categories: Ideals/Submodules and algebraic geometry

While working through Atiyah/MacDonald for my final exams I realized the following: The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...
13
votes
0answers
625 views

Can the similarity between the Riesz representation theorem and the Yoneda embedding lemma be given a formal undergirding?

For example, by viewing Hilbert spaces as enriched categories in some fashion? (I suppose the same idea of considering the inner product of a Hilbert space as a generalized Hom-set has also been ...
2
votes
0answers
225 views

Name for enrichment with Hom(1,-) a full functor?

Let C be a V-enriched category and 1 be a terminal object of C. V is not necessarily a closed category, and C does not necessarily have an internal hom (nor is C even necessarily a monoidal category)....
13
votes
2answers
771 views

Model category structure on categories enriched over quasi-coherent sheaves

Gonçalo Tabuada has shown that there is a Quillen model category structure on the category of small dg-categories, i.e. the category of small categories enriched over chain complexes (for a fixed ...
2
votes
2answers
371 views

When does the 2-category V-Cat have pseudo-pullbacks?

Edit: rewritten the question (edit:again), as I realised I wanted weak pseudo-pullbacks, not comma objects. Consider the 2-category $V$-$Cat$ of $V$-enriched categories. An example is $Cat$ itself, ...
13
votes
0answers
609 views

Is there some way to see a Hilbert space as a C-enriched category?

The inner product of vectors in a Hilbert space has many properties in common with a hom functor. I know that one can make a projectivized Hilbert space into a metric space with the Fubini-Study ...
2
votes
4answers
326 views

Is there a sensible way to enrich over SymMonCat such that id_X is not the monoidal unit?

SymMonCat is the cartesian 2-category of symmetric monoidal categories, braided monoidal functors, and monoidal natural transformations. The terminal symmetric monoidal category 1 has one object $I$ ...
5
votes
2answers
335 views

Pointer to literature on double enrichment and functors among enriching categories?

I'm currently working with the following two situations: $\mathbb A$ is a monoidal category, $\mathbb B$ is an $\mathbb A$-enriched monoidal category, and $\mathbb C$ is a $\mathbb B$-enriched ...