Questions tagged [enriched-category-theory]
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147
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Enriched vs ordinary filtered colimits
Filtered categories can be defined as those categories $\mathbf{C}$ such that $\mathbf{C}$-indexed colimits in $\mathrm{Set}$ commute with finite limits.
Similarly, for categories enriched in $\mathbf{...
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Weighted Co/ends?
Recall: Limits
Recall that the limit of a functor $D\colon\mathcal{I}\to\mathcal{C}$ is, if it exists, the pair $(\mathrm{lim}(D),\pi)$ with
$\lim(D)$ an object of $\mathcal{C}$, and
$\pi\colon\...
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Kan liftings and projective varieties
Regard the following two bicategories:
$\operatorname{dg-\mathcal{B}imod}$, with objects dg categories, and morphisms categories from $C$ to $D$ being the categories of $C$-$D$-bimodules. Composition ...
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Is monadicity preserved by the underlying functor?
Let $\mathcal{V}$ be a monoidal closed (complete, cocomplete, reasonable...) category.
Let $\mathsf{T}$ be an enriched monad over $\mathcal{V}$. The forgetful functor $\mathsf{U}: \mathsf{Alg}(\...
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Is the tensor product of chain complexes a Day convolution?
Recently, Jade Master asked whether the tensor product of chain complexes could be viewed as a special case of Day convolution. Noting that chain complexes may be viewed as $\mathsf{Ab}$-functors from ...
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Literature on linear categories
I am trying to understand Deligne's 'Categories Tensorielles', and therefore I need some knowledge on linear categories. Looking at Wikipedia and nLab, I found some definitions and explanations, but I ...
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Classification of absolute 2-limits?
Let $\mathcal V$ be a good enriching category. Recall that an enriched limit weight $\phi: D \to \mathcal V$ is called absolute if $\phi$-weighted limits are preserved by any $\mathcal V$-enriched ...
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Is there such a thing as a weighted Kan extension?
The title pretty much sums it up.
More in detail. Let $C$, $D$ and $E$ be categories, let $F:C\to D$ and $G:C\to E$ be functors, and let $P:C^{op}\to \mathrm{Set}$ be a presheaf. The colimit of $F$ ...
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The symmetric monoidal closed structure on the category of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors
In 6.5 of the book by Kelly,
Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, No. 10, 2005.
the author claims that the $2$-category $\mathsf{Cat}_{\...
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(Co)tensoring of enriched slice categories
In an answer to this question: Enriched slice categories, a description of the enrichment of the slice category in an enriched category is given. I'm interested in going a bit further. If we assume ...
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Monoidal V-categories, and monoids
I am guessing that the definition of monoidal V-category is a V-category $\mathbf{A}$ together with a V-functor $(\boxtimes) \colon \mathbf{A} \times \mathbf{A} \to \mathbf{A}$ and a functor $i \colon ...
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The homotopy category of the category of enriched categories
We know that if $\mathcal C$ is a combinatorial monoidal model category such that all objects are cofibrant and the class of weak equivalences is stable under filtered colimits, then $\mathsf{Cat}_{\...
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Are (complete) 2-Segal spaces the same as Span-enriched infinity categories?
The question is basically in the title. More generally, I would like to know if this, or any reasonable variant of it, is true. Or perhaps, to understand better the gap between 2-Segal spaces and Span-...
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Do adjoints of enriched functors preserve the enriched structure?
Is there is any reason in general for adjoints of enriched functors to preserve the enriched structure of categories?
The specific example I'm thinking of is the following:
Fix a commutative ring $R$...
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By general reasons, $i_A \colon \mathbb{D}\text{-}\mathrm{cont}[A,\mathbf{Set}] \to [A,\mathbf{Set}]$ has a left adjoint
In Centazzo and Vitale's A Duality Relative to a Limit Doctrine (TAC, 2002, abstract), early on, they make the above claim and cite Kelly's Basic Concepts in Enriched Category Theory (TAC reprints). I ...
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