nLab pseudonatural transformation

Idea

A pseudonatural transformation is a kind of homomorphism between parallel 2-functors: a lax natural transformation whose 2-morphism components are all invertible.

Definition

Definition

Given

• a pair $\mathcal{X}, \mathcal{Y}$ of 2-categories

• a pair $F,G \colon \mathcal{X} \longrightarrow \mathcal{Y}$ of 2-functors,

a pseudonatural transformation

$$\eta \colon F \Rightarrow G$$

consists of a function from objects of $\mathcal{X}$ to 1-morphisms of $\mathcal{Y}$, and a function from 1-morphisms of $\mathcal{X}$ to vertically invertible 2-morphisms of $\mathcal{Y}$, of the form

such that all the following equations hold among 2-morphisms in $\mathcal{Y}$:

(pseudo-naturality)

for all 2-morphisms in $\mathcal{X}$

we have

$$=$$

(unitality)

for all objects $x \in \mathcal{X}$ we have

$$=$$

(associativity)

for all pairs of composable 1-morphisms $x_1 \xrightarrow{f} x_2 \xrightarrow{g} x_3$ in $\mathcal{X}$, we have

$$=$$

Such a pseudonatural transformation is called a pseudonatural equivalence if each component $\eta(x)$ is an equivalence in the 2-category $\mathcal{Y}$. This is equivalent to $\eta$ itself being an equivalence in the 2-functor 2-category $[\mathcal{X},\mathcal{Y}]$ of 2-functors, pseudonatural transformations between these, and modifications between those.

Related concepts

• homotopy

• transfor

  • natural transformation

    • dinatural transformation, extranatural transformation

  • pseudonatural transformation

  • lax natural transformation

References

• John W. Gray, section I.4 of: Formal category theory: Adjointness for 2-categories, Lecture Notes in Mathematics 391, Springer (1974) [doi:10.1007/BFb0061280]

For review see most references at 2-category, such as

• Niles Johnson, Donald Yau, §4.2 in: 2-Dimensional Categories, Oxford University Press (2021) [arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001]

A generalization to extranatural transformations:

• Alexander S. Corner: A universal characterisation of codescent objects, Theory and Applications of Categories 34 24 (2019) 684-713 [tac:34-24]

See also:

• Camell Kachour: Définition algébrique des cellules non-strictes, Cahiers de Topologie et de Géométrie Différentielle Catégorique 1 (2008) 1-68 [numdam:CTGDC_2008__49_1_1_0, pdf]

• Camell Kachour: Steps toward the weak higher category of the weak higher categories in the globular setting, Categories and General Algebraic Structures with Applications 4 1 (2016) 9-42 [cgasa:11180, pdf]