nLab
2-morphism

• object

• morphism

• 2-morphism

• 3-morphism

• k-morphism

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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.

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.