Questions tagged [enriched-category-theory]

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Applications of Day convolution for monoidal bicategories

Day convolution is a very powerful tool to build monoidal structures on categories of functors from a pro/monoidal $\mathcal{V}$-category. For instance, it is used in stable homotopy theory to ...
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Is the tensor product of chain complexes a Day convolution?

Recently, Jade Master asked whether the tensor product of chain complexes could be viewed as a special case of Day convolution. Noting that chain complexes may be viewed as $\mathsf{Ab}$-functors from ...
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Kan extensions between categories of monoid objects

Let $K\colon\mathcal{A}\longrightarrow\mathcal{B}$ be a functor between $\mathcal{V}$-enriched categories. Endowing $\mathcal{A}$ and $\mathcal{B}$ with promonoidal structures, we obtain induced ...
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1answer
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Literature on linear categories

I am trying to understand Deligne's 'Categories Tensorielles', and therefore I need some knowledge on linear categories. Looking at Wikipedia and nLab, I found some definitions and explanations, but I ...
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Classification of absolute 2-limits?

Let $\mathcal V$ be a good enriching category. Recall that an enriched limit weight $\phi: D \to \mathcal V$ is called absolute if $\phi$-weighted limits are preserved by any $\mathcal V$-enriched ...
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1answer
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Is there such a thing as a weighted Kan extension?

The title pretty much sums it up. More in detail. Let $C$, $D$ and $E$ be categories, let $F:C\to D$ and $G:C\to E$ be functors, and let $P:C^{op}\to \mathrm{Set}$ be a presheaf. The colimit of $F$ ...
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2answers
232 views

The symmetric monoidal closed structure on the category of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors

In 6.5 of the book by Kelly, Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, No. 10, 2005. the author claims that the $2$-category $\mathsf{Cat}_{\...
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1answer
300 views

(Co)tensoring of enriched slice categories

In an answer to this question: Enriched slice categories, a description of the enrichment of the slice category in an enriched category is given. I'm interested in going a bit further. If we assume ...
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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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3answers
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The homotopy category of the category of enriched categories

We know that if $\mathcal C$ is a combinatorial monoidal model category such that all objects are cofibrant and the class of weak equivalences is stable under filtered colimits, then $\mathsf{Cat}_{\...
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Are (complete) 2-Segal spaces the same as Span-enriched infinity categories?

The question is basically in the title. More generally, I would like to know if this, or any reasonable variant of it, is true. Or perhaps, to understand better the gap between 2-Segal spaces and Span-...
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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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1answer
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By general reasons, $i_A \colon \mathbb{D}\text{-}\mathrm{cont}[A,\mathbf{Set}] \to [A,\mathbf{Set}]$ has a left adjoint

In Centazzo and Vitale's A Duality Relative to a Limit Doctrine (TAC, 2002, abstract), early on, they make the above claim and cite Kelly's Basic Concepts in Enriched Category Theory (TAC reprints). I ...
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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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1answer
247 views

Tannaka duality for closed monoidal categories

I asked this some time ago at mathstackexchange, and people there explained to me the mathematical part of what I was asking, but the question about references remains open. In my impression, people ...
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2answers
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Multiplication and division by a morphism under the “inner composition” in closed monoidal categories

I asked this a week ago at math.stackexchange, without success, so I hope it will be appropriate here. Let ${\mathcal C}$ be a symmetric closed monoidal category, and let me denote the internal hom-...
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1answer
144 views

Weak enrichment and bicategories

I'm trying to find examples where the following perspective on bicategories is developed. We can define a 2-category as being enriched in Cat, where Cat is treated as a monoidal category using the ...
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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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1answer
112 views

Why is the category of all small $\mathbf{S}$-enriched categories locally presentable?

In Lurie's Higher Topos Theory Proposition A.3.2.4, the author used Proposition A.2.6.15 to prove that for any combinatorial monoidal model category $\mathbf{S}$ with all objects cofibrant and weak ...
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54 views

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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1answer
135 views

Isomorphisms in enriched categories

Let $(M,\otimes,1)$ be closed monoidal category and $C$ an $M$-enriched category. Assume we have $C$-objects $X$ and $X'$ and a morphism $f:1\to C(X,X')$ in $M$. We call $f$ an isomorphism if there is ...
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1answer
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Two monoidal structures and copowering

Let $(\mathbf{M},\otimes,1)$ be a closed monoidal category and $(\mathbf{C},\oplus,0)$ an $\mathbf{M}$-enriched monoidal category. Furthermore, assume that we have a copowering $\odot:\mathbf{M}\times\...
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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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1answer
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Simplicially enriched cartesian closed categories

In this question I asked whether for a complete and cocomplete cartesian closed category $V$, there can be a complete and cocomplete $V$-category $C$ (with powers and copowers) whose underlying ...
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3answers
670 views

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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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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2answers
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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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2answers
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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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1answer
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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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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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1answer
138 views

Is the category of enriched operads (co)complete?

Let $V$ be a symmetric monoidal category which is complete and cocomplete. Is the category of small symmetric colored $V$-enriched operads complete and cocomplete? If $V$ is presentable, is it ...
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2answers
320 views

Enrichments vs Internal homs

Consider the definition of existence internal homs for a general monoidal category category $\cal{C}$, mainly the existence of an adjoint for the functor $$ X \otimes -: \cal{C} \to \cal{C}, $$ for ...
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1answer
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Are sifted (2,1)-colimits of fully faithful functors again fully faithful? (And a de-categorified variant)

1) Suppose that I have a sifted diagram of categories $\mathcal{C}_i$, another of the same shape $\mathcal{D}_i$, and that I have a system $F_i:\mathcal{C}_i\to\mathcal{D}_i$ commuting with the ...
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1answer
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Can an enriched functor be expressed as a colimit of representable functors?

Suppose that $\mathcal C$ is an ordinary category and $F:\mathcal C^{op}\longrightarrow Set$ a functor. Then, we can form the category $\mathcal C/F$ as follows : each object is a morphism of functors ...
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75 views

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 $f_n:X\to Y$ ...
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1answer
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Accessible categories in enriched category theory

I study some definitions of accessible category (see 1) and the applications of that notions; my question: exist a notion of accessible category in therm of enriched category theory? (in case of exist,...
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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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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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1answer
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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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1answer
134 views

Kan extension of conservative functors

Suppose the right Kan extension $\text{Ran}_F G$ of a conservative functor $F$ along a conservative functor $G$ exists (with the category $\text{dom} F=\text{dom} G$ not necessarily small). Is it ...
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dg-categories and fully faithful functor

dg: is for differential graded Suppose that $F: C\rightarrow D$ is a dg-functor between small dg-categories such that: F: Objects of $C$ $\rightarrow$ Objects of $D$ is injective. $Hom_{C}(a,b)\...
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1answer
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Simplicial mapping spaces, stable $\infty$-categories, and triangles

Let $C$ be a stable $\infty$-category (presentable, if you like) and let $map(-,-)$ denote the simplicial mapping space. If $X \to Y \to Z$ is a fiber sequence, and $W$ is an object, when is $map(W,X) ...
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1answer
143 views

Is there a monoidal category that coclassifies enriched category structures for a given set?

Let $S$ be a set. Is there a monoidal category $TS$ that we can construct from $S$ such that monoidal functors $F: TS \to M$ (up to monoidal natural isomorphism) correspond to $M$-enriched categories ...
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1answer
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generalized elements in monodial categories

In a category $\mathcal{C}$, a generalized element of an object $A$ means a morphism to $A$. It follows from Yoneda lemma that the object $A$ is determined by the collection of generalized of elements ...
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572 views

Examples of enriched categories which are (co)powered or (co)tensored

For $\mathsf{V}$ a closed monoidal category, it is canonically powered (or cotensored) and copowered (or tensored) over itself with respect to the internal hom and tensor product. Likewise, any (co)...
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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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1answer
235 views

Does every enriched functor preserve tensors?

Let $\cal{P}$ be a $k$-linear semisimple abelian rigid monoidal category with finite dimensional (over $k$) Hom-spaces (for a field $k$). By a tensored $\cal{P}$-category we mean a $\cal{P}$-category ...
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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 ...
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1answer
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Colimits in the category of simplicial categories

A simplicial category is a category enriched over the monoidal category of simplicial sets (morphism sets are now simplicial sets), and the collection of all such categories forms a category itself (...
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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. ...