convention. letters [tt-0010]

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The general idea is that we try to use visually distinct letters for different concepts:

1. uppercase calligraphic letters \({\cal C}, {\cal D}, {\cal E}, {\cal J}, {\cal S}\) denote categories
   • \({\cal C}, {\cal D}, {\cal E}\) are prefered in concepts about one to three categories, since \({\cal C}\) is the first letter of "category"
   • \({\cal J}\) is used in concepts like diagram, and assumed to be small
   • \({\cal S}\) is used in concepts like subcategory, or sometimes in concepts about a small category
2. boldface uppercase Roman letters denote specific categories, e.g. \(\mathbf {Cat}, \mathbf {Set}, \mathbf {Grp}, \mathbf {Top}, \mathbf {1}\)
3. uppercase Roman letters \(X, Y, Z, W, O, E, V\) denote objects in categories
   • \(X, Y\) usually mean objects in \({\cal C}\) and \({\cal D}\), respectively
   • \(O\) denotes any object
   • \(E\) denotes the equalizer object
   • in limit-related concepts, \(\mathrm {-}\) denotes any object, \(V\) denotes the vertex
4. lowercase Roman letters \(f, g, h, i, k, l, r\) and sometimes the lowercase Roman letter of the corresponding codomain or domain object denote arrows
   • occationally, when two arrows are closely related, they are denoted by the same letter with different subscripts, e.g. \(g_1, g_2\)
   • as special cases, \(\iota , p, i\) denote the inclusion, projection and injection arrows
5. uppercase script letters \(\mathscr {F}, \mathscr {G}, \mathscr {H}, \mathscr {I}, \mathscr {K}, \mathscr {L}, \mathscr {R}\) denote functors
   • \(\mathscr {D}\) denotes a diagram functor
   • \(\mathscr {L}\) and \(\mathscr {R}\) denote the left and right adjoint functors in an adjunction, respectively
   • \(\mathscr {H}\) denotes the Yoneda embedding functors
   • as a special case, functors with the terminal category (i.e. constant object functors) as the domain are identified with the objects in the codomain category, thus are denoted like an object: \(X : \mathbf {1} \to {\cal C}, \mathrm {*} \mapsto X\)
   • \(\mathscr {I}\) is only used in the inclusion functor (note that this is letter "I")
   • we do not use \(\mathscr {J}\) and \(\mathscr {S}\) because they are visually ambiguous
6. lowercase Greek letters \(\alpha , \beta , \eta , \epsilon , \sigma \) denote natural transformations, their components are denoted by them with subscripts.