# Questions tagged [enriched-category-theory]

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### Applications of Day convolution for monoidal bicategories

Day convolution is a very powerful tool to build monoidal structures on categories of functors from a pro/monoidal $\mathcal{V}$-category. For instance, it is used in stable homotopy theory to ...
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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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### Kan extensions between categories of monoid objects

Let $K\colon\mathcal{A}\longrightarrow\mathcal{B}$ be a functor between $\mathcal{V}$-enriched categories. Endowing $\mathcal{A}$ and $\mathcal{B}$ with promonoidal structures, we obtain induced ...
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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 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 $\mathsf{Span}$), ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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)...
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### 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 ...
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### 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 ...
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### 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 ...
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. ...