nLab lax natural transformation

Idea

The notion of lax natural transformation is a generalization of the notion of natural transformation from category theory to higher category theory.

As a natural transformation is a morphism between two functors between categories, a lax natural transformation is a morphism between 2-functors between 2-categories:

Where a natural transformation has a commuting naturality square, a lax natural transformation has a 2-morphism filling that square. If that 2-morphism is required to be invertible, one speaks of a pseudonatural transformation, and if it is required to be an identity (which implies that the square commutes), then one speaks of a strict 2-natural transformation (this latter notion only really makes sense for strict 2-functors between strict 2-categories).

In the general terminology of higher category theory, a lax natural transformation may equivalently be called a (lax) 1-transfor.

Definitions

Given (possibly weak) 2-categories $C,D$ and (possibly lax or oplax) 2-functors $F, G : C\to D$, a lax natural transformation $\alpha:F\Rightarrow G$ is given by

• for each $A\in C$ a 1-morphism $\alpha_A:F(A)\to G(A)$ in $D$, as usual

• for each $f:A\to B$ in $C$ a 2-morphism $\alpha_f: G(f) \circ \alpha_A \Rightarrow \alpha_B \circ F(f)$:

such that

• for each $A,B$, the $\alpha_f$ are the components of a (strict) natural transformation $\alpha_{A,B}: (\alpha_A)^* \circ G_{A,B} \dot{\to} (\alpha_B)_* \circ F_{A,B}$

• the assignment $f\mapsto \alpha_f$ behaves sensibly with respect to identities and composition (see the references for details).

An oplax natural transformation is as above, only with the $2$-cells $\alpha_f$ reversed. This distinction is not entirely consistent in the literature; see the discussion of terminology below.

An (op)lax natural transformation $\alpha$ is a pseudonatural transformation if each $\alpha_f$ is invertible, and a strict natural transformation or strict 2-natural transformation if each is an identity.

In all of these cases, the word ‘natural’ is often dropped for brevity.

Categories and $n$-categories of lax transformations

Lax transformations from $F$ to $G$ and modifications between them form a category $Lax(F,G)$; likewise we have $Ps(F,G)$ and $Oplax(F,G)$ consisting of pseudo-natural and oplax transformations, respectively.

Pushing it up a notch, for 2-categories $C$ and $D$ we have hom-2-categories $2Cat_{lax}(C,D)$, $2Cat_{ps}(C,D)$, and $2Cat_{oplax}(C,D)$ whose objects are 2-functors (generally taken to be strong or strict), whose morphisms are lax, pseudo, or oplax transformations respectively, and whose 2-cells are modifications. These hom-2-categories are the internal homs for the various version of the Gray tensor product on $2Cat$.

Finally, there is a 3-category consisting of 2-categories, (strong or strict) 2-functors, pseudo-natural transformations, and modifications. No laxness is possible at this level (without “laxifying” the notion of 3-category).

“Lax” versus “oplax”

The choice of which direction to call “lax” and which to call “oplax” is not made consistently in the literature. The convention used above is Benabou’s original choice, as well as that of Leinster, Borceaux, and the majority of Australian writing on category theory. However, some references, such as the Elephant, make the opposite choice. One or two references use “left lax” and “right lax” instead.

It is arguably the case that the direction we call “oplax” occurs more commonly in practice. For instance, icons are a special sort of oplax transformations (although if “lax” and “oplax” were switched, then the acronymic derivation of the word “icon” would no longer work). Likewise, when monoidal categories are viewed as one-object 2-categories, monoidal natural transformations are special oplax transformations (in fact, they are precisely the icons).

On the other hand, the convention used above (besides having a little more weight of tradition) has the advantage that there is a 2-monad whose algebras are 2-functors, and for which lax and oplax algebra morphisms are precisely lax and oplax transformations, respectively, under this convention. Thus, for instance, theorems such as doctrinal adjunction can be applied to lax and oplax transformations without needing to switch back and forth between two different meanings of “lax.”

The Yoneda Lemma

Given a bicategory $C$, a lax functor $F:C^{op}\to Cat$ and a $0$-cell $A\in C$, there are adjoint functors

$$I : Lax(y A,F) \stackrel{\to}{\leftarrow} F(A) : J$$

such that $J \dashv I$. Here $y A:C \to Cat^{C^{op}}$ is the image of $A$ under the bicategorical Yoneda embedding.

Proof (sketch)

For $a\in F(A)$, let $J(a):y A\Rightarrow F$ be the lax transformation defined by $J(a)_B(f) = (F f)(a)$. The components

$$J(a)_g : F(g) \circ J(a)_X \dot{\to} J(a)_Y \circ g_*$$

for $g:X\to Y$ in $C$ are given by $F$‘s comparison map $(F_{g,-})_a$. The coherence conditions follow from those on $F$ wrt the associator and left unitor of $C$. For $x:a\to b$ in $F(A)$, $J(x)$ is a modification because $F_{g,f}$ is a natural transformation (i.e. 2-cell in $Cat$).

Let $I(\alpha) = \alpha_A(1_A)$ as usual. For $m:\alpha\ddot{\to}\beta$ a modification, $I(m)=(m_A)(1_A)$.

The unit $\eta: F A \dot{\to} I J$ at $a\in F A$ is the component $(F_A)_a$ of $F$‘s comparison map, which is natural in $a$ by definition. The counit $\epsilon_\alpha$ is obtained via the usual 1-chasing, and is given in components by

$$(\epsilon_\alpha)_B(f) = \alpha_B(r_f) \circ (\alpha_f)(1_A)$$

where $r$ is the right unitor of $C$. Some diagram-chasing confirms that this is indeed natural in everything, and so $\epsilon: J I \dot{\to} Lax(y A,F)$.

The triangle identities may be proved by expanding the definitions above and using the coherence conditions on lax functors and transformations and coherence of $C$.

See Gray for the case strict 2-categories and strict 2-functors.

Related concepts

• homotopy

• transfor

  • natural transformation

    • extranatural transformation, dinatural transformation

  • pseudonatural transformation

    • pseudo-extranatural transformation

  • lax natural transformation

  • 2-dinatural transformation

  • lax F-natural transformation

References

See the references at 2-category. For instance (note slightly outdated terminology)

• Gray, Adjointness For 2-Categories

The definition is spelled out explicitly in the following, where they are called lax transformations:

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

In the following oplax natural transformations are defined, but called, simply, “transformations”:

• Tom Leinster, Basic bicategories,

  arXiv:math/9810017.

• Ross Street, Two constructions on lax functors, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 13 (1972) no. 3 , p. 217-264 numdam