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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 9th 2010
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   Author: Urs
   Format: MarkdownItexcreated [[vertical composition]]

   created vertical composition

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeSep 9th 2010
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   Author: zskoda
   Format: MarkdownItexIn 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 9th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexRight, 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 $n \gt 2$.

   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 $n \gt 2$.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeSep 9th 2010
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   Author: zskoda
   Format: MarkdownItex> 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.

   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.