Questions tagged [enriched-category-theory]
The enriched-category-theory tag has no usage guidance.
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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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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 ...
- 1,532
12
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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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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 ...
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99
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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 ...
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In what context can enriched category theory be done?
There are many possible situations one can do enriched category theory. See https://ncatlab.org/nlab/show/category+of+V-enriched+categories#possible_contexts for a list.
My question is what ...
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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 ...
- 1,532
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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 ...
- 2,003
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+50
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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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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0
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845
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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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Is composition in a simplicially enriched category always determined by a compatible simplicial tensoring (if such exists)?
Let $C$ be a simplicially enriched category, i.e., there are
a collection of objects $ob C$,
a simplicial set $map_C(X,Y)$ for $X,Y \in ob C$,
composition maps $map_C(Y,Z) \times map_C(X,Y) \to map (...
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Enriched Categories: Metric Spaces, Monoidal Endofunctors and Lipschitz-Continuous Maps.
In the introduction to the reprint of "Metric spaces, generalized logic and closed categories" Lawvere talks about the following situation:
Let $\mathbb R_+$ denote $\mathbb R_{\geq 0}^\infty$. Every ...
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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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$V$-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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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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Is there a nice way to define discrete enriched categories?
In the setting of classical category theory, one defines the discrete category associated to a set $X$ as the category $X_\mathsf{disc}$ having the elements of $X$ as its objects and only identities ...
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Pushforward of an internal category along a functor
Let $F:C\to D$ be a “nice” functor (for example, $H_*(-;\mathbb{Z}):\mathbf{Top}\to \mathbf{Ab}^{\mathbb{Z}}$). Now assume that we have a category $O$ internal to $C$. Is there a canonical way to ...
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Hopf monoid from comonoidal structures
Let $\mathcal{V}$ be a closed braided monoidal category and $\mathcal{V}-Cat$ the monoidal bicategory of small $\mathcal{V}$-enriched categories. Let $\mathcal{C}$ be a pseudo-comonoid in $\mathcal{V}-...
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How to compute (co)limits of enriched categories?
I've asked this question on math.stackexchange some time ago (https://math.stackexchange.com/questions/1380176/how-to-compute-colimits-of-enriched-categories) and I received no complete answers, so 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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75
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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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Density with respect to a family of diagrams, versus a class of weights
In Theorem 5.19 of Kelly's Basic Concepts of Enriched Category Theory, it is proven that a fully faithful functor $K \colon \mathcal A \to \mathcal C$ is dense if and only if $\mathcal C$ is the ...
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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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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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On cofibrations of simplicially enriched categories
Let $\mathbb{C}$ be an strict 2-category and denote by $C$ is underlying 1-category viewed as as a 2-category only having identity 2-cells.
We have a canonical inclusion functor ,
$$i: C \...
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Generating an enriched multicategory
Let $C$ be an $(M,\otimes,1)$-enriched category. I am looking for a reference for a notion of “generating the morphisms of $C$” (for ordinary categories, but also for multicategories, see below).
My ...
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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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Name for enrichment with Hom(1,-) a full functor?
Let C be a V-enriched category and 1 be a terminal object of C. V is not necessarily a closed category, and C does not necessarily have an internal hom (nor is C even necessarily a monoidal category)....
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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 $...
- 3,759
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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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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 ...
- 1,071
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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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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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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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Enrichment of lax monoidal functors between closed monoidal categories
Let $\mathscr C,\mathscr D$ be (right) closed monoidal categories. Then both of them can be considered as enriched over themselves via their internal homs, which I will denote by $\textbf{Maps}$. Now ...
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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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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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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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Reference request: a class of matrices leading to interesting metric geometry
For $0 \le A \in GL(n,\mathbb{R})$, let $Aw = \Delta(A)$, where $\Delta$ denotes the map taking a matrix to a vector of its diagonal entries and/or forming a diagonal matrix from a vector, according ...
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Degree shift of multilinear maps
Let $V$ be a graded vector space over $\mathbb{k}$ and $V[1]$ its odd degree shift.
Given $k$, $l\in \mathbb{N}_0$, is there a natural way to define the following map,
$$
\psi: \hom_{\mathbb{k}}(V^{\...
- 456
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Pointwise convergence in Lawvere metric spaces
In the formalism of Lawvere metric spaces, we have that the distance in the hom-space $[X,Y]$ is given by:
$$
d(f,g) = \sup_{x\in X} d(f(x),g(x)) .
$$
Therefore, a sequence of functions $f_n:X\to Y$ ...
- 2,129
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Topological categories as enriched category
Lurie define in HTT(Def. 1.1.1.6) a topological category as a enriched category over compactly generated (and weakly Hausdorff) topological spaces, but usually we define a topological category as ...
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Posets as (0,1)-categories
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. ...
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225
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Reference request: Grothendieck construction for $\mathbb V$-distributors?
I'm currently working with an analogue of the Grothendieck construction for enriched categories:
Given a distributor a.k.a. $\mathbb V$-functor $D:X^\mathrm{op}\otimes Y\to \mathbb V$ there is a ...
- 3,161
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63
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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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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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99
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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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how many ways can an algebra be a weighted colimit of free algebras?
For a given weight $W : \mathcal{S}^{op} \to \mathcal{V}$ and diagram $D : \mathcal{S} \to \mathcal{A}$, the weighted colimit is an object $W \cdot D$ together with an isomorphism
$$\mathcal{A}(W\cdot ...
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