Arrow category

Arrow categories are a construction from the mathematical branch of category theory . The arrow category (or or ) exists for each category , its objects are the morphisms from , their morphisms are commutative squares. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}^{\mathbf {2} }}$ ${\displaystyle [\mathbf {2} ,{\mathcal {C}}]}$ ${\displaystyle {\mathcal {C}}^{\rightarrow }}$ ${\displaystyle {\mathcal {C}}}$

Concrete definition

Be a category. The objects in the category are made up of morphisms . A morphism between the objects and in the arrow category is then given by the commutative square ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}^{\mathbf {2} }}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle (\varphi ,\psi )}$ ${\displaystyle f}$ ${\displaystyle g}$

given, whereby the composition is done by vertical linking of these diagrams.

As a functor category

Let be a category consisting of two objects and a morphism between them (indicated by the diagram ). The arrow category is defined as the functor category of all functors from to with natural transformations as morphisms. The diagonal functor allows a fully faithful embedding of in this category. ${\displaystyle \mathbf {2} }$ ${\displaystyle 0\to 1}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mathbf {2} }$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$

As a comma category

The arrow category can be defined as a comma category , where the identical functor denotes. ${\displaystyle (id_{\mathcal {C}}\downarrow id_{\mathcal {C}})}$ ${\displaystyle id_{\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$

Full sub-categories

Often one also looks at full sub-categories of the arrow category for a category, i. H. you limit yourself to certain morphisms which you select as objects. In topology, for example, the category of pairs of spaces is defined as the full sub-category of which only has the embeddings (that is, the extreme monomorphisms ) as objects. In homology theories according to the Eilenberg-Steenrod axioms , the relative homology functors form functors on this category of space pairs. ${\displaystyle \mathbf {Top} ^{\mathbf {2} }}$

Web links

• arrow category , entry in nLab . (English)

Individual evidence

1. ^ Saunders Mac Lane and Ieke Moerdijk: Sheaves in Geometry and Logic . A First Introduction to Topos Theory. Springer , New York 1992, ISBN 0-387-97710-4 , pp. 25 .
2. ↑ Mac Lane, Moerdijk, p. 27