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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 9th 2010
• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeSep 9th 2010

In 2-category this is standard, but if $n\gt 2$ I would rather talk about vertical composition of $n$-cells along $(n-1)$-cells, rather than of 2-cells along 1-cells anymore. What is standard ? In any case, the vertical composition of anything is always strictly associative.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeSep 9th 2010

Right, the discussion should be generalized.

On the other hand, “vertical” is unambiguous only for 2-categories. You seem to adopt the convention that “vertical” composition of $n$-morphisms is always that along $(n-1)$-morphisms. Is that universally accepted? I rarely seem to hear people talk about “horizontal” and “vertical” for $n$-categories above $n \gt 2$.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeSep 9th 2010

Is that universally accepted?

You may either not use the vertical terminology for any composition of $n$-cells or use for that one. As this is the only one which is just clean associative composition in general, and which does not restrict to any composition of lower level cells. Horizontal composition of $2$ cells, restricts to $1$ cells, and is not strictly associative; similarly any other composition of any cells in weak higher category, except the vertical composition which I mentioned. In the case of category of $(n-1)$-categories, you need Godement-like rules to define any composition except what I call above vertical. So it is logical to me to keep the terminology that way. But I do not know what is standard.