How to compute (co)limits of enriched categories?

I’ve asked this question on math.stackexchange some time ago ( and I received no complete answers, so I’m posting it here. Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories enriched over $\mathscr{V}$. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$. This leads to some questions. 1)When … Read more

Are there any dominant pivotal functors such that the regular representation is not mapped onto a multiple of the regular representation?

This question is related to Pivotal functors of that are substantially different from finite group homomorphisms. A tensor functor F:C→D is called dominant (sometimes called “surjective”) if for any Y:D, there is an X:C such that Y is a subobject of FX. It is known (“On fusion categories” by Pavel Etingof, Dmitri Nikshych, and Viktor … Read more

Nichols Algebras as Braided Hopf Algebras

Given a Hopf algebra H and a Yetter–Drinfeld module V over H, it is well-known that V has an induced braided vector space structure, and so, one can consider it’s Nichols algebra which is a braided Hopf algebra in the braided monoidal category of Yetter–Drinfeld modules over H. The notion, however, of a Nichols algebra … Read more

“Monomorphicness” of a natural morphism induced by a monoidal functor

Let (C,⊗,IC) and (D,⊗,ID) be left-closed monoidal categories with internal-hom denoted by [X,Y] and F:C→D be a monoidal functor. It is straightforward to see that there are morphisms ϕ:F([X,Y])→[F(X),F(Y)] in D, which are natural in X and Y. What are sufficient conditions on F (or possibly the categories in question) to ensure that ϕ is … Read more

Monads which are monoidal and opmonoidal

Do monads which are monoidal and opmonoidal have a name? (Bimonoidal?) In case they have already been studied, who can point me to a reference? More in detail. Let $(C,\otimes)$ be a symmetric (or braided) monoidal category. Let $(T,\eta,\mu)$ be a monad on $C$ such that: $T:C\to C$ is a bilax monoidal functor (compatible lax … Read more

Category of (co)commutative Hopf monoids in an exact category

I’m transferring this question over from SE, since it didn’t get much attention over there. Let (C,⊗) be an exact monoidal category, and let H(C) be the category of cocommutative and commutative Hopf monoids in C with respect to ⊗. What can be said about this category? When is it abelian? If H(C) is abelian, … Read more

Tannaka-Krein reconstruction and rigidity

Let C be a rigid monoidal category together with a quasi-monoidal functor ω:C→veck to finite-dimensional vector spaces over a field k, i.e. we have isomorphisms φ0:k→ω(I) and φ=(φX,X:ω(X)⊗ω(Y)→ω(X⊗Y))X,Y∈C but these are not necessarily compatible with the constraints. In light of Majid’s Tannaka-Krein theorem for quasi-Hopf algebras and other results we know that there exists a … Read more

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 $\mathcal{V}-Cat$, which can be seen as the dual notion of a small monoidal $\mathcal{V}$-enriched category. In [Day, Ch.5] it is shown that from this we can build a promonoidal structure on $\mathcal{C}$, … Read more

Scaling Yetter–Drinfeld Modules

A braided vector space is a pair (V,σ) consisting of a vector space V, and a linear map σ:V⊗V→V⊗V, satisfying the Yang–Baxter equation. Ee can scale the braiding by λ∈C to produce a new braiding λσ. Given a Yetter–Drinfeld module (V,∙,δ), a braiding is given by σ:V⊗V→V⊗V,       v⊗w↦v(−1)∙w⊗v(0). As above, scaling this braiding again gives a … Read more

Categorical construction of comodule category of FRT algebra

Let B denote the braid groupoid, with objects being non-negative integers n∈Z≥0 and morphisms B(n,n)=Bn given by the braid group. Let C be a cocomplete rigid symmetric monoidal abelian category and F:B→C a monoidal functor. Moreover, let C:=∫b∈BF(b)∨⊗F(b) be the coend of F and V:=F(1). Is there a “purely categorical construction” of a category CV … Read more