nLab
icon

Definition

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

• the assertion that $F$ and $G$ agree on objects.
• for each morphism $u:x\to y$ in $C$, a 2-cell ${\alpha }_u:F(u)\to G(u)$ in $D$ (note that this only makes sense because $F$ and $G$ agree on objects, so that $F(u)$ and $G(u)$ are parallel.
• for each 2-cell $\mu :u\to v$ in $C$, we have ${\alpha }_v.F(\mu )=G(\mu ).{\alpha }_u$.
• for each object $x$ of $C$, ${\alpha }_{1_x}$ is an identity (modulo the unit constraints of $F$ and $G$, if they are not strict functors).
• for each composable pair $x\stackrel{u}{\to }y\stackrel{v}{\to }z$ in $C$, we have ${\alpha }_v*{\alpha }_u={\alpha }_{vu}$ (modulo the composition constraints of $F$ and $G$, if they are not strict functors).

If $D$ 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 $D$. 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

• Stephen Lack, Icons. arxiv:0711.4657