# Questions tagged [monoidal-categories]

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### Tamari lattice and bicategory coherence

Reference links: Tamari lattice (Wikipedia): https://en.wikipedia.org/wiki/Tamari_lattice Associahedra: https://en.wikipedia.org/wiki/Tamari_lattice#/media/File:Tamari_lattice.svg The Tamari lattice ...
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### Is there a synthetic approach to (symmetric) monoidal infinity-categories?

Recent work of Riehl and Verity (e.g. the book "Elements of $\infty$-category theory") has established a "synthetic" / model-independent approach to the study of $\infty$-...
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### Equivalence between bialgebras and finite ring categories with fibre functor

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Rec{Rec}\DeclareMathOperator\End{End}\DeclareMathOperator\Mod{Mod}\newcommand\fdMod[1]{#1\text-{\Mod}^\text{fd}}$In the celebrated [EGNO, Thm 5.2.3], ...
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### Enriched tensor product of chain complexes

Question (idea): Is there a notion of tensor product of chain complexes in a $\mathcal{V}$-enriched monoidal category $\mathcal{C}$, for $\mathcal{V}$ a linear symmetric monoidal category? Let me ...
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### If a strong monoidal functor $F$ has an ambidextrous adjoint, then how close is the adjoint to being strong monoidal?

Let $F : C \to D$ be a strong (symmetric, say) monoidal functor. Suppose that $G : D \to C$ is both left and right adjoint to $F$ (an ambidextrous adjunction). Then by doctrinal adjunction $G$ is both ...
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### Permutative Gray monoid that can not be strictified

Strict monoidal 2-categories do not suffice to model all connected 3-types: the non-trivial coherence cells $\Sigma: (f \otimes 1)\circ (1 \otimes g) \cong (1 \otimes g)\circ (f \otimes 1)$ of a Gray ...
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### Equivalences induced from invertible objects in transported bifunctors along an adjoint pair

I'm interested in the following problem, similar in vein to this other question. To put it simply, I have an adjoint pair $F\dashv G$ between categories $\mathrm{C}$ and $\mathrm{D}$ and I suppose ...
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### How general is $TX \otimes X \simeq X \otimes TX$?

Let $C$ be a monoidal category, $X \in C$ an object, and $TX$ the free monoid object on $X$, assuming it exists. How often do we have an isomorphism $TX \otimes X = X \otimes TX$? is there a canonical ...
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### Group completion of a monoid (braid groups)

Let $B_n$ be the braid group on $n$ strands, $B_{\infty}$ the direct limit of braid groups. For a discrete group $G$, we let $BG$ to be the classifying space of $G$. After reading this question, I was ...
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### The evaluation and coevaluation maps for an object isomorphic to a dualisable object

Let $X$ be an object in a monoidal category with dual $X^*$ and evaluation and coevaluation maps $e$ and $c$. Now if we have an isomorphism $\sigma:X \to Y$, for some other object $Y$, then $Y$ must ...
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### Monoidal structure on presheaves

I am confused about the following monoidal structure, which gives a symmetric monoidal structure on R-modules (that I think is not Cartesian), even if R is not commutative. Let $C$ be a small category....
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### Coherence of the graphical language for pivotal categories

Throughout I follow A survey of graphical languages for monoidal categories, Peter Selinger, arXiv. A pivotal category is a monoidal category where each object $A$ has a dual $A^*$, together with a ...
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