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It seems that every author has rather personal and unique conventions for designating "well-known" categories. This raises the question: Is there a reference available, on-line or otherwise, that lists known categories together with commonly accepted notation(s)? Adamek, et. al. in "The Joy of Cats" has a nice table of categories in the appendix that comes pretty close to what I'm looking for here. For anyone familiar with this text, could you comment on whether the notation they use to designate categories is reasonably standard?

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up vote 2 down vote accepted

I am familiar with "The Joy of Cats" and the categories mentioned there are quite standard, in most cases. You may also check Wikipedia definitions, Mac Lane's "Categories for the working mathematician" or Awodey's "Category theory". You will see that most definitions conincide, but I agree with @SL2 that it is always best to quickly, but clearly, describe what is meant with a specific category.

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The notation in your reference seems to be fairly understandable, but I would always make a point to explicitly state what your abbreviation is referring to, at least the first time you use it (The category $\mathbf{Top}$ of topological spaces is...). This is in part because often times a particular paper will call for a certain nice subcategory and then use the same abbreviation as for the full category. For example, depending on the paper, $\mathbf{Top}$ might refer to the full subcategory of compactly generated Hausdorff spaces, or of connected CW complexes.

I guess the moral is that even though the abbreviations you gave in your reference are easily understandable for the most part (I think $\mathbf{Top_*}$ is more common for pointed spaces than $\mathbf{pTop}$ for example), it is always best to make explicit what your notation is referring to so that there is no ambiguity.

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