Abstract
Categorical orthodoxy has it that collections of ordinary mathematical structures such as groups, rings, or spaces, form categories (such as the category of groups); collections of 1-dimensional categorical structures, such as categories, monoidal categories, or categories with finite limits, form 2-categories; and collections of 2-dimensional categorical structures, such as 2-categories or bicategories, form 3-categories. We describe a useful way in which to regard bicategories as objects of a 2-category. This is a bit surprising both for technical and for conceptual reasons. The 2-cells of this 2-category are the crucial new ingredient; they are the icons of the title. These can be thought of as “the oplax natural transformations whose components are identities”, but we shall also give a more elementary description. We describe some properties of these icons, and give applications to monoidal categories, to 2-nerves of bicategories, to 2-dimensional Lawvere theories, and to bundles of bicategories.
Similar content being viewed by others
References
Bénabou, J.: Introduction to bicategories. In Reports of the Midwest Category Seminar, pp. 1–77. Springer, Berlin (1967)
Blackwell, R., Kelly, G.M., Power, A.J.: Two-dimensional monad theory. J. Pure Appl. Algebra 59(1), 1–41 (1989)
Cheng, E., Gurski, N.: The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories. In: Davydov et al. (ed.) Categories in Algebra, Geometry and Mathematical Physics (=Contemporary Math. 431), pp. 143–164. AMS, Washington, DC (2007)
Gordon, R., Power, A.J., Street, R.: Coherence for tricategories. Mem. Am. Math. Soc. 117(558), vi+81 (1995)
Grandis, M., Pare, R.: Adjoint for double categories. Addenda to: “Limits in double categories” [Cah. Topol. Géom. Différ. Catég. 40 (1999), no. 3, 162–220]. Cah. Topol. Geom. Differ. Categ. 45(3), 193–240 (2004)
Gray, J.W.: Formal Category Theory: Adjointness for 2-Categories. Springer, Berlin (1974)
Hess, K., Parent, P.-E., Scott, J.: Co-rings over operads characterize morphisms, arXiv:math/0505559
Kelly, G.M.: Structures defined by finite limits in the enriched context. I. Cah. Topol. Geom. Differ. 23(1), 3–42 (1982)
Kelly, G.M., Power, A.J.: Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads. J. Pure Appl. Algebra 89(1–2), 163–179 (1993)
Lack, S.: On the monadicity of finitary monads. J. Pure Appl. Algebra 140(1), 65–73 (1999)
Lack, S.: Limits for lax morphisms. Appl. Categ. Structures 13(3), 189–203 (2005)
Lack, S., Paoli, S.: 2-nerves for bicategories. K-Theory 38(2), 153–175 (2008)
Mac Lane, S., Paré, R.: Coherence for bicategories and indexed categories. J. Pure Appl. Algebra 37(1), 59–80 (1985)
Power, J.: Enriched Lawvere theories. Theory Appl. Categ. 6, 83–93 (1999). (electronic). The Lambek Festschrift
Power, J.: Three dimensional monad theory. In: Davydov et al. (ed.) Categories in Algebra, Geometry and Mathematical Physics (=Contemporary Math. 431), pp. 405–426. AMS, Washington, DC (2007)
Power, J., Robinson, E.: Premonoidal categories and notions of computation. Math. Struct. Comput. Sci. 7(5), 453–468 (1997). Logic, domains, and programming languages (Darmstadt, 1995)
Street, R.: Fibrations in bicategories. Cah. Topol. Geom. Differ. 21(2), 111–160 (1980)
Street, R.: Categorical structures. In: Handbook of algebra, vol. 1, pp. 529–577. North-Holland, Amsterdam (1996)
Wolff, H.: V-cat and V-graph. J. Pure Appl. Algebra 4, 123–135 (1974)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lack, S. Icons. Appl Categor Struct 18, 289–307 (2010). https://doi.org/10.1007/s10485-008-9136-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-008-9136-5