Questions tagged [enriched-category-theory]
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147
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Free cocompletion of a 2-category under pseudo colimits, lax colimits, and colax colimits
Let $\mathscr K$ be a small 2-category. It follows from $\mathrm{Cat}$-enriched category theory that the free cocompletion of $\mathscr K$ under strict 2-colimits of 2-functors is given by the 2-...
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Enriched tensor product of chain complexes
Question (idea): Is there a notion of tensor product of chain complexes in a $\mathcal{V}$-enriched monoidal category $\mathcal{C}$, for $\mathcal{V}$ a linear symmetric monoidal category?
Let me ...
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Automorphism groups for simple objects in abelian linear categories
Let $\mathcal{A}$ be an abelian category that is also $k$-linear, where $k$ is some algebraically closed field. Let $X$ be a simple object in $\mathcal{A}$. What can we say about $\mathrm{Aut}(X)$? I ...
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Is there a correction to the failure of geometric morphisms to preserve internal homs?
Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have ...
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Interpreting a diagram in Borceux-Quinteiro's paper on enriched sheaves
I am somewhat new to working with enriched categories, and have a question about how to interpret Definition 1.2 in Borceux-Quinteiro's paper A theory of enriched sheaves.
The authors consider a ...
13
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Large V-categories admitting the construction of V-presheaves
By a result of Foltz, and Freyd and Street, a category $C$ is essentially small (i.e. equivalent to a small category) if and only if both $C$ and $[C^{\text{op}}, \mathrm{Set}]$ are locally small. I ...
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Self-enrichment for a closed monoidal bicategory
First, there are two possible generalization of the notion of closed category, vertical and horizontal.
I'm interested in the vertical one, something saying, I guess, that a monoidal bicategory $\...
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Day convolution and sheafification
$\DeclareMathOperator\Psh{Psh}\DeclareMathOperator\Sh{Sh}\newcommand\copower{\mathrm{copower}}$I was looking through Bodil Biering's thesis On the Logic of Bunched Implications - and its relation to ...
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Can every weighted colimit in a $\mathbf{Pos}$-enriched category be rephrased as a conical colimit?
For ordinary category theory, we have the following fact.
A weighted colimit of a functor can always be equivalently expressed as a colimit of a different functor.
Specifically, the weighted colimit ...
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Tensor product of objectwise weak homotopy equivalences of $\mathcal{M}$-spaces
I consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category $...
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Are $\mathscr{V}$-modules uniquely (nicely) enrichable?
$\require{AMScd}\newcommand{\V}{\mathscr{V}}\newcommand{\M}{\mathcal{M}}\newcommand{\hom}{\operatorname{hom}}\newcommand{\op}{{^\mathsf{op}}}$Fix a closed symmetric monoidal category $(\V;\otimes;\...
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Enriched cofibrant replacement in spectrally enriched categories
If $\mathcal{V}$ is a monoidal model category with all objects cofibrant, Theorem 13.5.2. of Categorical Homotopy Theory will guarantee that the functorial cofibrant replacement of a $\mathcal{V}$-...
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Free enriched monoidal categories
Suppose $(\mathcal{V},\otimes,1)$ is a symmetric monoidal category and $\mathbb{C}$ is a $\mathcal{V}$-category. I will deliberately avoid usual powerful assumptions (eg completeness/cocompleteness) ...
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Understanding this (standard?) notion of enriched product category
$\newcommand{\V}{\mathscr{V}}\newcommand{\A}{\mathcal{A}}\newcommand{\B}{\mathcal{B}}\newcommand{\C}{\mathcal{C}}$Fix a closed symmetric monoidal category $\V$, writing the product as $\otimes$, the ...
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Preservation of lax limits in categories of functors and lax natural transformations
Let $f:\mathbb{C} \to \mathbb{D}$ be a functor of 2-categories and let $\operatorname{Fun}(\mathbb{C},\operatorname{Cat})^{\operatorname{lax}}$ denote the 2-category of functors and lax natural ...
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Condition for an equivalence of functor categories to imply an equivalence of categories
Given small categories $\mathcal{C}$ and $\mathcal{D}$, we have that $[\mathcal{C}^\text{op},\textbf{Set}]\simeq[\mathcal{D}^\text{op},\textbf{Set}]$ if and only if the Cauchy-completions of $\...
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Can we do away with cotensors when exploring the equivalence between closed $\mathscr{V}$-modules and strongly tensored $\mathscr{V}$-categories?
$\newcommand{\M}{\mathcal{M}}\newcommand{\ML}{\underline{\mathcal{M}}}\newcommand{\N}{\mathcal{N}}\newcommand{\NL}{\underline{\mathcal{N}}}\newcommand{\V}{\mathscr{V}}\newcommand{\VL}{\underline{\...
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Are algebroids "just matrices"?
$\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Mat{Mat}$This question was originally asked on MSE but may be better here.
Algebroids are particularly interesting structures: they are basically ...
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Why are enriched (co)ends defined like that?
I'm mainly following references such as Kelly, Loregian and the nLab, and it seems customary there to generalize (co)ends to the enriched context (over a symmetric monoidal category $\mathcal{V}$) by ...
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Conditions for natural transformations of weights to induce adjunctions of weighted limits
Suppose we have:
-) A $2$-category $\mathsf{J}$
-) A natural transformation of $\alpha : M \Longrightarrow W : \mathsf{J} \longrightarrow \mathsf{Cat}$
-) A functor $X:\mathsf{J} \longrightarrow \...
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Why do we need enriched model categories?
As far as I understand, model categories mainly provide tools for studying the "homotopy theories" (that is, $\infty$-categories) that are ubiquitous in mathematics. From this point of view, ...
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Finite groups acting on algebraic groups and representations
Let $H$ be a connected algebraic group over an algebraically closed field $k$, and $I$ a finite group which acts on $H$ through group scheme morphisms. Denote by $Rep(H)$ the category of finite ...
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Condensed categories vs categories (co)tensored with condensed sets
I am not sure how to solve set-theoretic issues properly, so let me first ignore them.
There are two notions, probably closely related:
Condensed categories, i.e. condensed objects in the category of ...
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527
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Can a category be enriched over abelian groups in more than one way?
An $\mathbf{Ab}$-category is a category enriched over the category of abelian groups. What is an example of a category that can be enriched over abelian groups in more than one way?
An abelian ...
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Cocompleteness of enriched categories of algebras
A useful result due to Linton is that for a cocomplete category $C$ and monad $T$ on $C$, if the category of algebras $C^T$ admits reflexive coequalisers, then it is cocomplete (see here for a sketch ...
- 9,521
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V-categories enriched in a monoidal V-category
In an email to the categories mailing list dated 21 August 2003, Street writes:
Max reminded me of his old result (not in the LaJolla Proceedings,
but known soon after) that a monoidal V-category is ...
- 9,521
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Are homotopy colimits strict?
Let's say we are working with a fibrant simplicially enriched category $\mathbf{B}$ that has all limits and all homotopy limits, and let $\mathbf{A}$ be a full subcategory that is closed under weak ...
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Strictification of $\mathcal{V}$-pseudofunctors
Let $\mathcal{B}$ be a bicategory. Section 4.10 of Gordon, Power and Street's paper "Coherence for Tricategories" states that there is a bicategory $\textbf{st}\mathcal{B}$ and a ...
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What is the right notion of a functor from an internal topological category to a topologically enriched category?
Let $\mathcal{C}$ be a category internal to (some convenient model for) topological spaces (which I will denote by $\mathsf{Top}$). In the question Greg Arone asks:
What is the correct notion of a ...
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Reference request for facts about bi(co)descent objects
I know the following facts are true, but I struggle to find adequate references for them:
Let $T$ be a pseudo-monad on a bicategory $\mathcal{C}$, and let $A$, $B$ be pseudo-algebras for $T$. Then, ...
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