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# Questions tagged [enriched-category-theory]

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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 ...
72 views

### 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 ...
232 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 ...
206 views

### 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-...
134 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 ...
58 views

### 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 \...
71 views

### Presentation of enriched categories

For an ordinary category, it is clear to me what a representation is: We have a notion of: Free category over a quiver, Congruence relations (a family of equivalence relations on each $C(x,y)$ such ...
104 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 ...
49 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 ...
92 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 ...
168 views

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 $\odot:\mathbf{M}\times\... 0answers 47 views ### Enrichment of lax monoidal functors between closed monoidal categories Let$\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}$. Now ... 1answer 200 views ### Simplicially enriched cartesian closed categories In this question I asked whether for a complete and cocomplete cartesian closed category$V$, there can be a complete and cocomplete$V$-category$C$(with powers and copowers) whose underlying ... 3answers 558 views ### Enriched cartesian closed categories Let$V$be a complete and cocomplete cartesian closed category. Feel free to assume more about$V$if necessary; in my application$V$is simplicial sets, so it is a presheaf topos and hence has all ... 0answers 107 views ### Hopf monoid from comonoidal structures Let$\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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