nLab
horizontal composition

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2-Category theory

2-category theory

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Transfors between 2-categories

Morphisms in 2-categories

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Contents

Definition

In a 2-category, the composition of 2-morphisms along objects is called horizontal composition .

Layer 1 A ⇓ α B ⇓ β C ↦ A ⇓ α * β C A\qquad\Downarrow\mathrlap{\alpha}\qquad B\qquad\Downarrow\mathrlap{\beta}\qquad C\;\;\mapsto\;\; A\qquad\; \Downarrow\mathrlap{\alpha\ast\beta}\qquad\; C F 1 F_1 F 2 F_2 G 1 G_1 G 2 G_2 F 1 : F 2 F_1\colon F_2 G 1 : G 2 G_1\colon G_2

This is in contrast to the vertical composition of 2-morphisms, which is their composition along 1-morphisms.

Properties

Horizontal and vertical composition are subject to the compatibility condition called the interchange law.

Examples

Revised on September 30, 2011 16:21:34 by Tim Porter (193.48.130.2)
Edit | Back in time (8 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: natural transformation, Cat, bicategory, 2-category, simplex category, Godement product, Eckmann-Hilton argument, exchange law, 2-pullback, Theta category, whiskering, vertical composition, Lie 2-group, infinity-Chern-Weil theory introduction, coherence law, bigroupoid, premonoidal category, exchange structure, (infinity,1)-bimodule, geometry of physics, prequantum field theory, TwoVect, Hilbert bimodule