All Questions
Tagged with 2-categories monoidal-categories
9 questions
5
votes
0
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66
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Bicategories in which the composition functors $\circ_{A, B, C}$ admit right adjoints
In Bénabou's 1967 Introduction to Bicategories, he mentions that, in a forthcoming part II, he would study bicategories $\mathcal K$ in which each composition functor $$\circ_{A, B, C} \colon \mathcal ...
- 10.7k
5
votes
1
answer
300
views
3-functoriality of the lax Gray tensor product
In Formal category theory: adjointness for 2-categories, Gray defines a tensor product of 2-categories, now more commonly known as the lax Gray tensor product, which I will denote by $\otimes_l$. For ...
- 10.7k
2
votes
0
answers
71
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2-morphism between circuits in a monoidal category
We are used to seeing equations between circuits in monoidal categories like this
I am wondering about morphisms between string diagrams. I think they are 2-cells. I found an example of a 2-cell ...
- 331
7
votes
0
answers
184
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Strictifying monoidal 2-functors
Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be a (weak) monoidal 2-functor between two strict monoidal 2-categories. Up to replacing $\mathcal{C}$ by an equivalent strict monoidal 2-category, can I ...
- 1,001
4
votes
0
answers
95
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Coherence for closed bicategories
A right-closed bicategory [1] is a bicategory that has all right extensions (i.e. right adjoints to precomposition with a fixed 1-cell). A one-object right-closed bicategory is therefore a right-...
- 10.7k
7
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428
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Left Kan extensions of "strong" monoidal functors
Consider the 2-category $\mathsf{MonCat}$ where objects are monoidal categories,
1-cells are strong monoidal functors, and 2-cells are monoidal natural transformations.
Given arrows $f: \mathsf{C} \to ...
3
votes
1
answer
187
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On the Group Structure of Morphism Set of a Strict 2-Group
The standard definition of a strict 2-group says that it is a strict monoidal category in which every morphism is invertible and each object has a strict inverse.
Also it is a well known fact that a ...
- 1,215
8
votes
2
answers
472
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How to construct a free 2-group on a groupoid?
Let G
be a groupoid. I'm wondering how to construct the free 2-group on G.
By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$
equipped with a functor $i:G\longrightarrow\mathcal{F}\...
- 83
8
votes
1
answer
530
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Is a weak monoidal category a monoid object in some category?
A monoid in the Category Cat is a strict monoidal category according to Wikipedia. Is it possible to weaken the monoid so that its realisation in Cat is a weak monoidal category? Do we shift up a ...
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