Questions tagged [enriched-category-theory]
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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 ...
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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 ...
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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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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 ...
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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 ...
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A multicategory is a ... with one object?
We all know that
A monoidal category is a bicategory with one object.
How do we fill in the blank in the following sentence?
A multicategory is a ... with one object.
The answer is fairly ...
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Relationship between enriched, internal, and fibered categories
In this question, let $(\mathcal{V}, \otimes, [-,-], e)$ be a nice enough symmetric monoidal closed bicomplete category.
The usual set-based Category theory has been generalized in many directions, ...
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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 ...
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Enriched cartesian closed categories
Let $V$ be a complete and cocomplete cartesian closed category. Feel free to assume more about $V$ if necessary; in my application $V$ is simplicial sets, so it is a presheaf topos and hence has all ...
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Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?
It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid ...
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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 ...
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Definitions of enriched monoidal category
This question is about two definitions of enriched monoidal categories I have:
Let $\mathcal{V}$ be a symmetric monoidal closed category.
The first definition: a $\mathcal{V}$-enriched category $\...
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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 ...
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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 anything ...
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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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