Questions tagged [enriched-category-theory]

For questions about a category where the Hom-spaces have additional structure. Should probably be used with the general (category-theory) tag.

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If Rel is enriched over suplattices what is Span enriched over?

In Rel the category of relations between sets you can "and", "or" and do several other operations over relations in nice ways. I don't fully get the details here but we can say the ...
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Internal language of an enriched category

I think I need a gadget along the lines of an operad enriched over the category of bounded lattices. But I'm having trouble thinking through what features enrichment would correspond to in an internal ...
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Why is a closed monidal category enriched over itself?

Let $M$ be a left-closed monidal category (Assume more such as symmetry if needed). Let $i$ be the unity object of $M$. Let $\alpha,\lambda,\rho$ be the associator, left unitor, right unitor of $M$, ...
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Recovering composition in an enriched category

Let $(\mathcal{V},\otimes,e)$ be a closed symmetric monoidal category and $\underline{\mathcal{A}}$ a tensored $\mathcal{V}$-category. We will write $\mathcal{A}$ for the underlying category of $\...
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Are the cylinders in the sketch for the tensor product of two $\mathcal F$-theories colimiting cylinders?

In section 6.5 of Basic Concepts of Enriched Category Theory Kelly writes If $\lambda : F \to \mathcal A(T-,M)$ is a cylinder in $\mathcal A$, where $F : \mathcal L^{op} \to \mathcal V$ and $T : \...
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(Co)products and exponentials for Lawvere metric spaces

A Lawvere metric space can be thought of as a $\textbf{Cost}$-enriched category where $\textbf{Cost}=([0,\infty],\geq,0,+)$ is a symmetric monoidal preorder (see Chapter 2 of Fong and Spivak for ...
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Commutative diagram of internal hom functors

Let $V$ be a symmetric closed monoidal category and denote with $[-,=]:V^{°} \otimes V \rightarrow V$ its internal hom. Let $j_x : 1 \rightarrow [x,x]$ be the adjoint to the left-unitor $\lambda_x : ...
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Is there a notion of "contravariant enriched functors"?

Is there a concept of "contravariant enriched functors"? I'll add some context. Let $C$ and $D$ be categories enriched over a monoidal category $M$. An enriched functor from $C$ to $D$ is a ...
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In a closed monoidal category $\mathcal{V}$ is the adjunction $\mathcal{V}$-natural?

Let $\mathcal{V}$ be a closed symmetric monoidal category. By definition we have a natural isomorphism of hom-sets $\text{Hom}_{\mathcal{V}}(X\otimes Y,Z) \cong \text{Hom}_{\mathcal{V}}(X,\mathcal{V}(...
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Enriched functor categories and the embedding $\mathcal{B} \to \mathcal{B}^I$

Let $\mathcal{A}$ be an enriched category, tensored and cotensored over a closed symmetric (co)complete monoidal category $\mathcal{V}.$ Let $I$ be a small enriched category, and consider the category ...
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What is a coproduct in an enriched category?

Does anybody know how to define coproducts in enriched categories ? For example I was wondering if they could be thought of as some kind of weighted colimits. I am particularly interested in if/how ...
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Tensor product of adjoint maps

Let $V$ be a symmetric monoidal category and $D$ a monoidal category. Assume that $D$ is $V$-enriched with an action $V*D\to D$ satisfying the usual axioms and hom-objects $\underline{D}(X,Y)\in V$ ...
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Equivalent definitions of weighted limits

Consider a symmetric monoidal closed, complete and cocomplete category $\mathcal{V}$. Let $\mathcal{A,C}$ be $\mathcal{V}$-categories and $\mathcal{D}:\mathcal{A} \rightarrow \mathcal{C}$, $\mathcal{W}...
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How is the morphism of composition in the enriched category of modules constructed?

In April, I asked the question of how the structure of the enriched category is introduced into the category $_AV$ of modules over a given monoid $A$ in a closed monoidal category $V$: if we consider ...
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Different definitions of profunctors

In nlab a profunctor from category $\mathbb{C}$ to category $\mathbb{D}$ is defined to be a functor $\mathbb{D}^{\text{op}} \times \mathbb{C} \overset{F}{\rightarrow} \textbf{Set}.$ However in Fong ...
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Examples of Weighted (co)limits in Algebra

When reading about weighted (co)limits in enriched category theory, most examples are given in the context of $\mathsf{Cat}$-enriched categories (ie. 2-limits in 2-category theory) or $\mathsf{sSet}$-...
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What is an enriched initial object / terminal object?

I am sorry if this is a trivial matter, but I was unable to find a reference: Let $\cal{V}$ be a Benabou-cosmos. What is the definition of an initial object in a $\cal V$-category $\cal C$? From the ...
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Why is a monad on self C equivalent to a strong monad on C

In Chapter 12 of Call-by-Push-Value, Levy states that a strong monad on a Cartesian category $\mathcal C$ is equivalent to a monad in the 2-category of $[\mathcal C^{\mathit{op}},Sets]$-enriched ...
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How to explain Category is Set-category, enriched in category: Set?

In book: https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf chapter: "4.4.4 Categories enriched in a symmetric monoidal category" declared: categories should really be called Set-...
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Category with special structure on its objects

My question stems from a categorial investigation and precise classification of $\mathtt{FinVect}_\mathbb{k}$, i.e. the category of finite-dimensional vector spaces over a field $\mathbb{k}$, in the ...
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Is the category of modules over an abstract monoid enriched?

If $A$ is an algebra over $\mathbb C$, or, in other words, a monoid in the closed monoidal category $_{\mathbb C}\operatorname{Vect}$ of all vector spaces over $\mathbb C$, then, clearly, the category ...
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Explicit description of pullback of $(2,1)$-categories.

Let us consider the ordinary category of $(2,1)$-categories. Its objects are groupoid enriched categories, and its morphisms are 2-functors. Is there an explicit way to define the objects, morphisms ...
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Two rings with the same multiplicative structure but non-isomorphic underlying Abelian groups

I am giving a series of lectures where I introduce some undergraduates to basic ideas from category theory. One of the things I would like to show them is how category theory could be used to make ...
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Relations between many category theory structures - introductory texts and maps

There are many kinds of category theory structures. What are some good texts and good ways to remember the relations between them? For example, can there be a web of embedding relations between these ...
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Name for a certain type of derived category similar to $(\mathbf{C}\downarrow\mathbf{C})$

$\newcommand{Hom}{\operatorname{Hom}}$ Consider a category $\mathbf{C}$, and draw a graph according to the following rules: Place all objects onto the graph Between every two objects $A,B\in\...
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11 votes
1 answer
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"Numerical results" from category theory

I want to keep a mini class on category theory, and I would be very satisfied to have a class of examples in which some theorems - or just the reformulation in categorical terms - yields numerical ...
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1 answer
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Category of vector spaces is monoidal closed

How to show that the category $\textbf{Vect}_k$ of vector spaces with ground field $k$ is monoidal closed? I am aware that the right adjoint to the functor $-\otimes Y$ should be $[Y,-]: X\mapsto \...
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Enriched category has to be locally small

I am reading material on enriched categories. The requirements for the category over which one enriches (be it $\textbf{Grp}$, $\textbf{Vect}_k$ are very clear, that they need to be monoidal and so). ...
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2 answers
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Categorical characterisation of fields and groups

Given a monoidal category $\mathcal{V}$, is there a categorical characterisation of an object in $\mathcal{V}$ such that it is a group if $\mathcal{V} = Set$ and a field if $\mathcal{V} = Ab$? Or does ...
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Equivalence between lax monoidal functors and monoids in the functor category.

I'm trying to go through the details of Proposition 3.4 of: https://ncatlab.org/nlab/show/Day+convolution For whatever reason, I don't see how to translate the conditions of a lax monoidal functor ...
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Drinfeld center as lax end

The Bernstein center of a category and the Drinfeld center of a monoidal category are very similar notions: While the first one is defined as the natural endomorphisms of the identity functor on the ...
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2 answers
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Completeness of the category of enriched categories

In the nLab entry on strict n-categories, one reads: For $V$ any complete and cocomplete closed monoidal category, also $VCat$ (the category of V-enriched categories) has these same properties. Is ...
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Are 1-bounded ultrametric spaces and continuous maps cartesian closed?

Let $X$ be a 1-bounded ultrametric space, i.e. a pair $(|X|,d_X)$ where $|X|$ is a set and $d_X:|X|\times |X|\to [0,1]$ is a distance such that $d_X(x, x'')\le \max\{d_X(x,x'),d_X(x',x'')\}$. The ...
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An elementary question about horizontal composition in (weak) 2-categories

Let: $\mathcal{K}$ be a (weak) 2-category $A$, $B$ and $C$ be objects of $\mathcal{K}$ $f_1, f_2 : A \to B$ and $g : B \to C$ be 1-cells $\alpha, \beta : f_1 \to f_2$ be 2-cells Assuming that $\...
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Tensoring for a left dualisable object is an absolute colimit

It's never too late to learn the following: Let $\mathcal C$ be a $\mathcal V$-category; then, if $A\in\mathcal V$ is a left dualizable object (this means that $A$ regarded as an object of the one-...
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Complex group representations as an enriched category?

In my lecture notes it says: ‘Complex representations of a given group G, together with intertwiners, form a category enriched over the complex numbers.’ Is it true that the category is enriched ...
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7 votes
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Does this condition imply isomorphism?

Let $k$ be a field and let $X$ and $Y$ be objects of a category $C$ enriched over $k$-vector spaces. Composition gives us a linear map $\hom(X,Y) \otimes \hom(Y,X) \to \hom(X,X) \oplus \hom(Y,Y)$, ...
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Semantic doubt about enrichment of categories and what it means to have 'group structure'

We define a category $C$ to be $Grp$-Enriched if for every $X$,$Y\in C$, we have that $C(X,Y)$ 'has group structure'/is a group. But what does that mean really? If I am given any finite set $A$, there ...
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1 vote
1 answer
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Cocomplete $R$-linear categories are tensored : adjoint functor theorem?

Let $B$ be an abelian category which is actually $Mod_R$-enriched for some ring $R$ (say unital commutative ring). For $b\in B$, we have a functor $\hom(b,-) : B\to Mod_R$ which preserves limits, so ...
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1 vote
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Enriched yoneda lemma proof

This is an exercise in Riehl's Categorical Homotopy Theory. Lemma 7.3.5 Give a small $V$-category $D$, and object $d \in D$, a $V$-functor $F:D \rightarrow V$, the canonical map is a $V$-natural ...
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(co)Limits in $\infty$-categories.

It seems to me that the following must be true, and has been used in Lurie's HA, and in my question. The precise statement is: Let $p:K \rightarrow C$ be a small diagram in $C$. Here $C$ is an $\...
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How does this statement reduces to the Yoneda lemma?

Context: I am reading the following page in the nLab, which is about simplicial presheaves, i.e Functors $\mathscr{C}^{\mathrm{op}} \to [\mathbf{\Delta}^{\mathrm{op}}, \mathbf{Set}]$. Equivalently, ...
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1 answer
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Internal hom: products and coproducts

It seems to me that Assuming the all smalllimits and colimits exist: Internal hom for a closed symmetric monoidal category satisfies: $$[\bigsqcup C_i, X] \cong \prod_i [C_i, X] $$ where $(...
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2 votes
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What is the hom space in over category of simplicial sets?

Let $Set_\Delta$ denote category of simplicial sets, which is enriched in $Set_\Delta$. Let $B$ be a simplicial set. We can form the over category. $(Set_\Delta)_{/B}$. Then is this also enriched ...
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3 votes
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Definition of simplicially enriched category

In Philip Hirschhorn's Model Category and Their Localizations, Def. 9.1.2, pg. 159, He defines a simplicially enriched category as A category $M$ together with Every two objects $X,Y$ ...
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2 votes
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A group that is also a category

I have a group with a category structure (i.e. category whose objects form a group), such that the left multiplication with any fixed element is a category automorphism. The same is true for right ...
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decomposition of hom-functors in a self-enriched category

Let $\mathbb{C}$ be a self-enriched category (such as Set). The Functor $\mathbb{C}(X, \mathbb{C}(Y,\_))$ is the same than the composition of functors $\mathbb{C}(X,\_) \circ \mathbb{C}(Y,\_)$. In a ...
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Recognising categories of enriched categories

Given a monoidal category $\mathcal V$ one has the category $\mathcal V{-}\mathbf {Cat}$ of (small) $\mathcal V$-enriched categories and $\mathcal V$-enriched functors (and $\mathcal V$-enriched ...
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Do we have to redo category theory when learning about enriched categories? [closed]

I'd like to know whether there exists a common language that encompasses both categories and enriched categories, so that results pertaining to either may be proven in a uniform way. I'd prefer it if ...
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1 vote
1 answer
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Enriched category over non-monoidal category

I'm finally learning homological algebra, and the notion of an enriched category seems to be the right setting in which to define the $\mathbf{Ext}$ functor. Yet, the definitions of an enriched ...
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