nLab
icon

Icons

Definition

Let C,DC,D be 2-categories and F,G:CDF,G:C\to D be functors. An icon α:FG\alpha:F\to G consists of the following:

  • the assertion that FF and GG agree on objects.
  • for each morphism u:xyu:x\to y in CC, a 2-cell α u:F(u)G(u)\alpha_u:F(u) \to G(u) in DD (note that this only makes sense because FF and GG agree on objects, so that F(u)F(u) and G(u)G(u) are parallel.
  • for each 2-cell μ:uv\mu:u\to v in CC, we have α v.F(μ)=G(μ).α u\alpha_v . F(\mu) = G(\mu).\alpha_u.
  • for each object xx of CC, α 1 x\alpha_{1_x} is an identity (modulo the unit constraints of FF and GG, if they are not strict functors).
  • for each composable pair xuyvzx\overset{u}{\to} y \overset{v}{\to} z in CC, we have α v*α u=α vu\alpha_v {*} \alpha_u = \alpha_{v u} (modulo the composition constraints of FF and GG, if they are not strict functors).

If DD is a strict 2-category (or at least strictly unital), then an icon is identical to an oplax natural transformation whose 1-cell components are identities. In general, there is a bijection between icons and such oplax natural transformations, obtained by pre- and post-composing with the unit constraints of DD. The name “icon” derives from this correspondence: it is an Identity Component Oplax Natural-transformation.

Applications

Icons have technical importance in the theory of 2-categories. For instance, there is no 2-category (or even 3-category) of 2-categories, functors, and lax or oplax transformations (even with modifications), but there is a 2-category of 2-categories, functors, and icons. (In fact, this 2-category is the 2-category of algebras for a certain 2-monad.)

Additionally, if monoidal categories are regarded as one-object 2-categories, then monoidal functors can be identified with 2-functors, and monoidal transformations can be identified with icons.

References

Revised on September 3, 2010 20:45:10 by Toby Bartels (173.190.153.41)