1
vote
0answers
113 views

The bar construction or the quotient for monoidal category action on a category

Given a monoid $C$ acting (from the right) on a set $M$, there is a bar construction giving a simplicial set or equivalently a translation/action category, $$ N_\bullet (M\rtimes C)= \cdots M\times ...
7
votes
1answer
306 views

String diagrams for (weak) monoidal categories

Hi, In a strict monoidal category, where the associator, left and right unitor are identity morphisms we have the following relations between (string) diagrams: where $i_x$ and $e_x$ are the unit ...
5
votes
1answer
222 views

What is the right way to define the nerve of an unbiased monoidal category?

I've been toying around with unbiased composition in higher categorical structures on and off for a while now. In particular, I've been playing around with unbiased monoidal 2-categories. One ...
7
votes
2answers
342 views

Effects of “weak” vs. “strict” categories in Eckmann-Hilton arguments

A standard example for demonstrating the need for genuinely weak n-categories is that a weak 3-category with unique 0- and 1-cells amounts to the same thing as a braided monoidal category (by an ...
9
votes
2answers
2k views

Is there a meaningful difference between biased and unbiased composition?

In higher category theory, there are notions of biased and unbiased definitions of composition of $n$-morphisms (or, as a special case, tensor products of objects). In the biased framework, we define ...
7
votes
4answers
505 views

What's the right object to categorify a braided tensor category?

The yoga of categorification has gained a lot of popularity in recent years, and some techniques for it have made a lot of progress. It's well-understood that, for example, a ring is probably ...