Swallowtail butterfly Swallowtail butterfly Swallowtail butterflies are large, colorful butterflies in the family Papilionidae, and include over 550 species. Though the majority are tropical, members of the family inhabit every continent except Antarctica. The family includes the largest butterflies in the world, the birdwing butterflies of the genus "Ornithoptera". Swallowtails have a number of distinctive features; for example, the papilionid caterpillar bears a repugnatorial organ called the osmeterium on its prothorax. The osmeterium normally remains hidden, but when threatened, the larva turns it outward through a transverse dorsal groove by inflating it with fluid. The forked appearance of the swallowtails' hindwings, which
Comma category by Lawvere, which involved the comma punctuation mark. Although standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category" (Lawvere, 1963 p. 13), the name persists. The most general comma category construction involves two functors with the same codomain. Often one of these will have domain 1 (the one-object one-morphism category). Some accounts of category theory consider only these special cases, but the term comma category is actually much more general. Suppose that formula_1, formula_2, and formula_3 are categories, and formula_4 and formula_5 (for source