Questions tagged [enriched-category-theory]
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Tensored and cotensored simplicial comma category
To transfer a tensored and cotensored simplicially enriched structure from a category $\mathcal{C}$ to $(\mathcal{C}\downarrow Z)$, we define $(X\to Z)\otimes K$ by the composite $(X\otimes K \to X \...
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Constructing lax limits from lax limits
Let $K$ be a 2-category. It's well-known that if $K$ has all PIE limits, then $K$ also has all lax limits. But I don't know a general "limit-decomposition" result which works "...
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Examples of nonpointwise Kan extensions that "play a mathematical role"
Most Kan extensions arising in nature are pointwise, and this observation prompts Kelly to write [1]:
Our present choice of nomenclature is based on
our failure to find a single instance where a [...
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Enriched categories over a semi-monoidal category
Let $\mathcal{V}$ be a semi-monoidal category, meaning it satisfies the axioms of a monoidal category except missing a unit and the unit axiom. One could then still go about defining a $\mathcal{V}$-...
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What are abelian categories enriched over themselves?
As far as I understand, an arbitrary abelian category is not enriched over itself, for example, $\mathrm{ChainComplex}(\mathrm{Ab})$ is, right? On the other hand, the categories $\mathrm{Mod}(R)$ (in ...
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$V$-cat and $V$-graph: coequalizers in the category of enriched functors
This question is regarding the 1974 JPAA paper $V$-cat and $V$-graph by Harvey Wolff.
To be precise, I don't understand a certain step in the proof of Corollary 2.9, which (the corollary) is crucial ...
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What is the right notion of a functor from an internal topological category to topological spaces?
Let $\mathcal C=(\mathcal O, \mathcal M)$ be a category internal to topological spaces. Thus $\mathcal O$ and $\mathcal M$ are topological spaces: the space of objects and the space of morphisms ...
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When this coend is invariant up to homotopy?
It is a follow-up of my question Calculation of the homotopy colimit of a diagram of spaces which was badly formulated.
Consider a fixed diagram $D:I^{op}\to {\rm Top}$ where ${\rm Top}$ is
a ...
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Universal property of the V-Mat construction
Internal categories and enriched categories can both be realised as monads in certain bicategories. If $\mathcal E$ is a category with pullbacks, then a monad in $\mathbf{Span}(\mathcal E)$ is a ...
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Determining enriched limit-preserving functors by their global sections
Let $\mathcal{C}$ be a small 1-category and let $\mathcal{M}$ be a category enriched over the presheaf category $\widehat{\mathcal{C}}$ which is complete as $\widehat{\mathcal{C}}$ enriched category. ...
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Are weighted limits terminal in a category of cones?
Consider a Benabou-cosmos $(\mathcal{V},\otimes,J)$, $\mathcal{V}$-categories $\mathcal{I},\mathcal{C}$ and $\mathcal{V}$-functors $\mathcal{W}:\mathcal{I} \rightarrow \mathcal{V}$ and $\mathcal{D}:\...
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Tannaka without Yoneda?
I am studying enriched categories, and as I wrote in my previous question How is the morphism of composition in the enriched category of modules constructed?, this is very difficult because there are ...
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How is the morphism of composition in the enriched category of modules constructed?
I asked this a week ago at MSE, but without success.
I am studying enriched categories and I have a feeling that I am doing something wrong because all the way each step, each elementary proposition, ...
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When are enriched categories equivalent?
$F : \mathbf{MonCat} \to \mathbf{2Cat}$ is the 2-functor for change of enrichment. What is the maximal subcategory of $\mathbf{MonCat}$ whose arrows $b : V \to W$ each induce an equivalence of ...
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Does an Ab-enriched category have a unique Ab-enrichment?
I know that the group structure on Hom sets can be recovered from biproducts if they exit. Indeed, if $f, g : A \to B$ are two maps then there is a uniquely defined map $f \oplus g : A \oplus A \to B \...
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