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Icons

Icons

Definition

For 2-categories

Let C,DC,D be 2-categories and F,G:CDF,G:C\to D be 2-functors. An icon [Lack 2010, cf. Rem. ] α:FG\alpha \colon F\to G consists of:

  1. The assertion that FF and GG agree on objects.

  2. For each 1-morphism u:xyu \colon x\to y in CC, a 2-morphism 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.)

  3. For each 2-morphism μ:uv\mu \colon u\to v in CC, we have α v.F(μ)=G(μ).α u\alpha_v . F(\mu) = G(\mu).\alpha_u.

  4. 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).

  5. 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 2-functors).

Remark

If DD is a strict 2-category (or at least strictly unital), then an icon is identical to an oplax natural transformation whose 1-morphism 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.

Icons are also used to construct distributors in the context of enriched bicategories.

Relation to pseudo double categories

A bicategory can be viewed as a pseudo double category whose tight-cells are trivial. An icon is then precisely a transformation of oplax functors of pseudo double categories. See Paré for details.

References

An early observation that restricting to icons allows one to form a bicategory of bicategories and lax functors appears in:

The terminology “ICON” was introduced in:

Last revised on November 17, 2024 at 08:07:54. See the history of this page for a list of all contributions to it.