Icons

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.

💡 Research Summary

The paper introduces a novel 2‑cell notion called an “icon” and shows how bicategories can be regarded as objects of a genuine 2‑category. Traditionally, mathematicians view ordinary algebraic structures (groups, rings, topological spaces) as forming a 1‑category, 1‑dimensional categorical structures (categories, monoidal categories, categories with finite limits) as forming a 2‑category, and 2‑dimensional categorical structures (2‑categories, bicategories) as forming a 3‑category. This hierarchy suggests that bicategories belong one level higher than a 2‑category, making it difficult to speak of a 2‑category whose objects are bicategories because the appropriate notion of 2‑cell is missing.

An icon is defined as follows. Given two bicategories 𝔅 and ℂ and a lax functor F : 𝔅 → ℂ, an icon η : F ⇒ G is a lax natural transformation whose components η_X are identity 2‑cells for every object X of 𝔅. In other words, it is an “oplax natural transformation with identity components”. By forcing the component 2‑cells to be identities, the usual coherence problems of oplax transformations disappear, and both horizontal and vertical composition of icons become strictly associative and unital. Moreover, icons satisfy the interchange law, so the collection of bicategories, lax functors, and icons forms a bona‑fide 2‑category, which the authors denote by Bicat.

The paper proves several basic properties of icons. First, identity icons exist for every lax functor, providing the required 2‑cell identities. Second, the vertical composition of icons coincides with the usual composition of lax natural transformations, while horizontal composition is given by whiskering on either side; the interchange law holds strictly because the component identities prevent any “interference”. Third, icons are stable under whiskering with ordinary 2‑cells, which allows one to incorporate additional structure such as monoidal constraints or finite‑limit preservation without leaving the 2‑category. Fourth, any equivalence of bicategories can be expressed as an invertible icon, showing that icons capture the essential notion of bicategorical equivalence.

Four substantial applications are presented.

1. Monoidal categories – A monoidal category can be seen as a one‑object bicategory. Using icons, strong monoidal functors and monoidal natural transformations become ordinary 1‑ and 2‑cells in Bicat, while lax monoidal functors correspond to lax functors. This unifies the various flavors of monoidal morphisms within a single 2‑categorical framework.

2. 2‑nerve of a bicategory – The classical nerve of a category yields a simplicial set. For bicategories, the authors construct a “2‑nerve” that lands in the 2‑category of icons rather than in simplicial sets. The resulting object retains the full bicategorical coherence data, because the 2‑cells of the nerve are precisely icons. This provides a more faithful embedding of bicategories into higher‑dimensional combinatorial models.

3. 2‑dimensional Lawvere theories – Lawvere theories describe algebraic structures via finite‑product categories. By replacing the underlying category with a bicategory and the morphisms between theories with icons, the authors obtain a 2‑categorical notion of Lawvere theory that can express operations with coherent associativity constraints (e.g., monoidal or braided structures). Models of such theories are then functors into a target bicategory, again respecting icons.

4. Bundles of bicategories – A bundle whose fibers are bicategories and whose transition data are lax functors can be organized as a pseudofunctor into Bicat. When the transition natural transformations are icons, the total space inherits a strict 2‑categorical structure, simplifying the theory of bicategorical fibrations and making it possible to apply standard 2‑categorical tools (e.g., limits, colimits) fiberwise.

Conceptually, icons reveal that the “missing” 2‑cells needed to embed bicategories into a 2‑category are not exotic but simply oplax transformations constrained to be identity on objects. This insight collapses the apparent gap between 2‑ and 3‑dimensional categorical worlds, allowing practitioners to work with bicategories using the familiar language of 2‑categories. The authors suggest that further developments—such as higher‑dimensional analogues of icons, applications to higher‑dimensional algebra, and connections with homotopy‑theoretic models—are promising directions for future research.