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2
votes
1answer
194 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 ...
0
votes
1answer
185 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
171 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
0answers
160 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
157 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
173 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
471 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
190 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 ...
0
votes
1answer
283 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
276 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 ...
6
votes
0answers
234 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
142 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 ...
4
votes
1answer
239 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, ...
28
votes
0answers
633 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 ...
11
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0answers
474 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
217 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 ...
12
votes
2answers
683 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
346 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, ...
12
votes
0answers
512 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
283 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
299 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 ...
