Let $\mathcal C$ be a symmetric monoidal closed category. This means that every functor $- \otimes B$ has a right adjoint $[B, -]$. Let $I$ be the unit and let $\rho \colon – \otimes I \to 1_{\mathcal C}$ be the right unitor. There are isomorphisms $$ \mathcal C (C \otimes I, C ) \cong \mathcal […]

I’m trying to follow Day’s argument to prove that $[\mathbf C,\mathbf{Sets}]$, where $\bf C$ is symmetric monoidal, is itself symmetric monoidal, but I’m stuck at the very beginning. Is there a way to prove that the convolution of two functors defines an associative “tensor” on $Psh(\mathbf C)$? I apologize if the question seems too boring, […]

Consider a coquasitriangular Hopf-algebra $(H,\mu,\eta,\Delta,\epsilon, S)$ over a field $\mathbb F$ with characteristic zero and the braided monoidal category $\mathcal C$ of $H$-right-comodules. We explicitly denote the coquasitriangular form of $H$ by $r$ and its convolution inverse by $r’$. A lengthy calculation yields the result that $H$ can be “transmutated” into an Hopf-algebra object in […]

Here by coherence conditions I mean those axioms imposed on associators and unities that grant that the groupoid generated by such morphisms is a poset, i.e. any two parallels morphisms in this groupoid are equal. These condition implies that such weak structures are equivalent (in a suitable sense) to strict ones. That said, where does […]

Monoidal categories come with tensor products, and sometimes, these categories are biclosed, i.e each restriction of the tensor bifunctor has a right adjoint. If the category happens to be symmetric, then restrictions of the $\otimes$ bifunctor are naturally isomorphic and we can talk about symmetric monoidal closed categories. If I understand correctly, the internal hom […]

Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both categories are complete, cocomplete, and have a closed symmetric monoidal structure, given by the projective tensor product (see here). The forgetful functor $\mathsf{Ban}_1 \to \mathsf{NormVect}_1$ is continuous (but not cocontinuous), […]

I’m having trouble understanding the following statement: “In a rigid category, duals are unique up to unique isomorphism.” It seems to me that this isomorphism is not unique. Let me try to give a counterexample: Let $(X,Y,\epsilon: X \otimes Y \to I,\eta: I: Y \otimes X)$ be a dual pair (satisfying the snake identities). Now, […]

Given a monoidal category ${\cal C}$, and $X \in {\cal C}$, we define a left dual of $X$ to be an object $X^*$ such that there exist morphisms $\epsilon:X^* \otimes X \to I$, and $\eta:I \to X \otimes X^*$, for $I$ the identity of the category, satisfying certain axioms, see here for details. When is […]

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