nLab 2-morphism

• object

• morphism

• 2-morphism

• 3-morphism

• k-morphism

---

Definition

A 2-morphism in an n-category is a k-morphism for $k = 2$: it is a higher morphism between ordinary 1-morphisms.

So in the hierarchy of $n$-categories, the first step where 2-morphisms appear is in a 2-category. This includes cases such as bicategory, 2-groupoid or double category.

Shapes

There are different geometric shapes for higher structures: globes, simplices, cubes, etc. Accordingly, 2-morphisms may appear in different guises:

A globular $2$-morphism looks like this:

$$a\mathrlap{\begin{matrix}\begin{svg} <svg width="76" height="37" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="79929"> <g> <title>Layer 1</title> <path marker-end="url(#se_marker_end_svg_79929_2)" id="svg_79929_2" d="m2,18.511721c31.272522,-14.782231 42.439789,-16.425501 71.625,-1.25" stroke="#000000" fill="none"/> <path id="svg_79929_13" marker-end="url(#se_marker_end_svg_79929_2)" d="m2,24.511721c33.286949,14.464769 40.259941,16.4624 71.500008,1.75" stroke="#000000" fill="none"/> </g> <defs> <marker refY="50" refX="50" markerHeight="5" markerWidth="5" viewBox="0 0 100 100" orient="auto" markerUnits="strokeWidth" id="se_marker_end_svg_79929_2"> <path stroke-width="10" stroke="#000000" fill="#000000" d="m100,50l-100,40l30,-40l-30,-40l100,40z" id="svg_79929_3"/> </marker> </defs> </svg> \end{svg}\includegraphics[width=56]{curvearrows}\end{matrix}}{\phantom{a}\space{0}{0}{13}\Downarrow\space{0}{0}{13}\phantom{a}} b$$

A simplicial $2$-morphism looks like this:

$$\begin{matrix} && b \\ & \nearrow &\Downarrow& \searrow \\ a &&\to&& c \end{matrix}$$

A cubical $2$-morphism looks like this:

$$\begin{matrix} & & b \\ & \nearrow & & \searrow \\ a & & \Downarrow & & d \\ & \searrow & & \nearrow \\ & & c \end{matrix}$$

Of course, using identity morphisms and composition, we can turn one into the other; which is more fundamental depends on which shapes you prefer.

Eric: Are there any consistency requirements for a 2-morphism? For example, in the bigon above, if $f:a\to b$, $g:a\to b$, and $\alpha:f\to g$, are there requirements on $\alpha:f\to g$ regarding $f$ and $g$? For example, should $\alpha$ come with component 1-morphisms $\alpha_a:a\to a$ and $\alpha_b:b\to b$ such that

$$\alpha_a\circ g = f\circ\alpha_b$$

or maybe

$$\alpha_a\circ g \simeq f\circ\alpha_b$$

? Could there be a 2-morphism without the corresponding 1-morphism components?

Urs Schreiber: in any given 2-category you have to specify which 2-morphisms exactly there are supposed to be, what $\alpha$ exactly you allow between $f$ and $g$. When you ask about components, it seems you are thinking of 2-morphisms specifically in the 2-category Cat. Here, yes, the allowed 2-morphisms are those that are natural transformations between their source and target 1-morphisms, which are functors.

Eric: I think the exchange law might be what I had in mind.

Jon Slaughter: Yes, clearly any higher order structure superimposed cannot corrupt the lower structure. Given any $g = \alpha \circ f$ must be a structurally valid/preserving map between $f$ and $g$. As Urs mentions, for Cat the 2-morphisms are natural transforms and must preserve the structure of the functors below them. In general what must be preserved depends on the sub-structure. Since composable morphisms must be reducible, this requires 2-morphisms to preserve the composability rule. For 2-morphisms this does work out to be the exchange laws. For higher order morphisms one can expect compositions of higher order composiblity rules. E.g., if natural transformations must preserve functorality in Cat then 3-morphisms will have to preserve naturality(but in terms of the functor category these 3-morphisms reduce to 2-morphisms and their structural preserving rules reduce to the exchange laws). It generally is not valid to talk about components with n-morphisms since one does not know the object structure. In Cat, with functors, we know by definition they have components(the image of an arrow under the functor) and the natural transformation that must preserve the composibility of components. The exchange laws are more general in that they are not expressed in terms of components.

Examples

• In the 2-category Cat, 2-morphisms are natural transformations between functors.

• In a path 2-groupoid 2-morphisms are certain surfaces or images of surfaces in a space, going between paths in that space.