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For the sake of having an answer that's not in the form of a comment... What you describe are graded morphisms of degree $h$. Presumably you mean to allow all $h$ (otherwise it's not a category). I …

This is more of a comment than an answer. I think a nicer way to construct the quotient $\def\cC{\mathcal{C}} \cC/G$ is as follows. The objects of $\cC/G$ are the same as the objects of $\cC$. The …

For what it's worth, the construction you describe features prominently in https://arxiv.org/abs/0807.4146, though that paper does not use 2-categorical language. More generally, given a 2-category $C …

I'm looking for the name of a certain n-category definition. (Someone explained it to me a couple of years ago. I remember the definition, but not the name. Without the name it's difficult to searc …

In the original WW paper, we distinguish between "crude" and "topological" boundary conditions. (I think "algebraic" would be a better name than "topological" here, but perhaps it is too late to chan …

Recall the notion of an $n$-cateogry $C$ enriched in a symmetric monoidal category. Instead of a set of $n$-morphisms $mor(a, b)$ (where $a$ and $b$ are compatible $(n{-}1)$-morphisms), we have an ob …

I think it's more natural to take advantage of the monoidal structure and regard the vector spaces as functors rather than objects. For simplicity, consider only finite dimensional vector spaces. Gi …

This doesn't directly answer your question, but I think that it is relevant. Also, it's ridiculously long, but hopefully that will make for easier reading than a shorter, vaguer version. Let's think …

I describe below a categorified version of the coend construction, "2-coend" for short. It takes as input a collection of 1-categories $\{W_{xy}\}$ which afford left and right representations of a (w …

Since no one else has posted an answer I'll take the opportunity to plug a recent paper with Scott Morrison, arxiv.org/abs/1009.5025. Section 6 of that paper gives a definition of "$A_\infty$ $n$-cat …

This is a reference request. I'm looking for a definition of Morita equivalence of *-algebras, as described below. If anyone thinks that this is not the right way to define Morita equivalence of *-a …

Yes, the Walker-Wang model is related to the Crane-Yetter-Kauffman TQFT in the same way the Levin-Wen model is related to the Turaev-Viro TQFT. See, for example, the table on page 14 of the notes fro …

If the dimension of $Z(S^{n-1})$ is greater than 1, then the TQFT is not even approximately multiplicative under connect sum. If $Z(S^{n-1})$ is 1-dimensional, then a simple cut and paste argument sh …

Consider the case $a=b=c$. Then the fusion space $V_{aaa}$ affords a representation of $\mathbb Z/3$ via a $2\pi/3$ rotation. The F-move in your question is essentially this $2\pi/3$ rotation, and t …

Khovanov homology can be thought of as a braided monoidal 2-category with duals, i.e. a 4-category with duals where the 0- and 1-morphisms are trivial. 0-morphisms: an unmarked point 1-morphisms: an …