Questions tagged [monoidal-categories]

A monoidal category, also called a tensor category, is a category $\mathcal{C}$ equipped with a bifunctor $\otimes\colon \mathcal{C}\times\mathcal{C}\to \mathcal{C}$ which is associative up to a natural isomorphism, and an object $\mathbb{1}$ which is both a left and right identity for $\otimes$ up to a natural isomorphism.

A monoidal category, also called a tensor category, is a category $\mathcal{C}$ equipped with a bifunctor $\otimes\colon \mathcal{C}\times\mathcal{C}\to \mathcal{C}$ which is associative up to a natural isomorphism, and an object $\mathbb{1}$ which is both a left and right identity for $\otimes$ up to a natural isomorphism. In symbols, we have natural isomorphisms:

$$A\otimes (B \otimes C) \cong (A \otimes B) \otimes C$$ $$\mathbb{1}\otimes A \cong A \cong A \otimes \mathbb{1}$$

The associated natural isomorphisms are subject to certain coherence conditions which ensure that all the relevant diagrams commute.

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Grothendieck's yoga of six operations - in relatively basic terms?

I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper inverse/direct images functors $f_!\dashv f^!$,…
Arrow
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Semirings induced by symmetric monoidal categories with finite coproducts

A symmetric monoidal category with finite coproducts is by definition a symmetric monoidal category $(\mathcal{C},\otimes,1,\dotsc)$ such that the underlying category $\mathcal{C}$ has finite coproducts (this includes an initial object $0$) with the…
Martin Brandenburg
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How to name these "ideals"?

Background. Let $\mathcal{C}$ be a symmetric monoidal category with unit $\mathbf{1}$. A subobject of $\mathbf{1}$ is just a monomorphism $I \to \mathbf{1}$. We may also call this an ideal of $\mathbf{1}$, because if $\mathcal{C}=\mathsf{Mod}(R)$,…
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Theory of promonads

I'm led to define a promonad in $\bf D$ as a monoid in the category of endo-profunctors of a category $\bf D$, where the product of two profunctors is their composition as profunctors: $$ F\odot G := \int^D F(-,D)\times G(D,-) $$ Is this theory…
fosco
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What's a double category with one object?

Categories with one object are equivalent to monoids. $2$-categories with one object are equivalent to monoidal categories. Therefore, I am wondering whether double categories with one object are equivalent to some known or interesting algebraic…
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What is the decategorification of a triangulated category?

The decategorification of an essentially small category $\mathcal C$ is the set $\lvert\mathcal C\rvert$ of isomorphism classes of $\mathcal C$. If $\mathcal C$ carries additional structure, then so does $\lvert\mathcal C\rvert$. For example, the…
Rasmus
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Motivation/Intuition for the Pentagon Axiom

I have just started reading a bit on monoidal categories, and there is I just can't make much sense of: the Pentagon Axiom. To provide some context, we have a category $\mathcal{C}$ together with a tensor product $\otimes \colon \mathcal{C} \times…
user313212
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Day convolution intuition

In the nLab, Day convolution is introduced as a generalisation of convolution of complex-valued functions, but I'm wondering how exactly to understand this. I can (just about) parse the definitions, but have absolutely no intuition or geometric…
Tim
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Is dualizablility of an object equivalent to tensoring with that object having a left adjoint?

Let $C$ be a closed symmetric monidal category. There is hence an adjunction $$ -\otimes X\colon C\leftrightarrows C\colon Map(X,-) $$ involving the internal Hom $Map(-,-)$ for every object $X$ of $C$ An object $X$ of $C$ is called dualizable if the…
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What is a good category for probability theory?

I am searching for a good category to think about probability theory in, with arrows as something like stochastic maps. There are certain nice structural features I would like the category to have (listed below) and I was hoping somebody could tell…
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Why can I choose to work in a strict monoidal category without loss of generality?

Let $\mathcal A$ be a monoidal category. We know that $\mathcal A$ is monoidally equivalent to a strict monoidal category $\mathcal A^{\mathrm{str}}$. In many books/papers it is assumed without loss of generality that $\mathcal A$ is strict to make…
Paul Slevin
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Tensor products from internal hom?

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…
user153312
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Coherence for symmetric monoidal categories

Let $\mathcal{C}$ be a monoidal category. By Mac Lane's coherence theorem for monoidal categories, there are strong monoidal functors $F : \mathcal{C} \to \mathcal{C}_s$ and $G : \mathcal{C}_s \to \mathcal{C}$ such that $\mathcal{C}_s$ is a strict…
Zhen Lin
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Pentagon diagram of Monoidal Categories

In Wikipedia we can find the following "Pentagon Diagram" for Monoidal Categories. What I don't understand is why that diagram is constructed in that way. Why do we start from $((A \otimes B) \otimes C) \otimes D$ and not from $(A \otimes B)…
Boris
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Details in applying the Barr-Beck monadicity theorem to Tannakian reconstruction

The Barr-Beck monadicity theorem gives necessary and sufficient conditions for a category $\mathcal{C}$ to be equivalent to a category of (co)algebras over a (co)monad. A functor $F:\mathcal{C}\to\mathcal{D}$ is said to be comonadic if it has a…
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