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In [[type theory]], '''containers''' are abstractions which permit various "collection types", such as [[list (computing)|lists]] and [[tree (data structure)|trees]], to be represented in a uniform way. A ([[unary operation|unary]]) container is defined by a type of ''shapes'' S and a type family of ''positions'' P, indexed by S. The ''extension'' of a container is a family of dependent pairs consisting of a shape (of type S) and a function from positions of that shape to the element type. Containers can be seen as [[canonical form]]s for collection types.<ref name='containers'>{{cite journal |author=Michael Abbott, Thorsten Altenkirch and Neil Ghani|year=2005 |title=Containers: Constructing strictly positive types  |journal=Theoretical Computer Science |volume=342 |issue=1 |pages=3–27 |doi=10.1016/j.tcs.2005.06.002}}</ref>
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For lists, the shape type is the [[natural number]]s (including zero). The corresponding position types are the types of natural numbers less than the shape, for each shape.
 
For trees, the shape type is the type of trees of units (that is, trees with no information in them, just structure). The corresponding position types are isomorphic to the types of valid paths from the root to particular nodes on the shape, for each shape.
 
Note that the natural numbers are isomorphic to lists of units. In general the shape type will always be isomorphic to the original non-generic container type family (list, tree etc.) applied to unit.
 
One of the main motivations for introducing the notion of containers is to support [[generic programming]] in a [[dependent type|dependently-typed]] setting.<ref name='containers'/>
 
== Categorical aspects ==
{{Expand section|date=October 2008}}
The extension of a container is an [[endofunctor]]. It takes a function <code>g</code> to
 
<math>\lambda\left(s,f\right).\left(s,g\circ f\right)</math>
 
This is equivalent to the familiar <code>map g</code> in the case of lists, and does something similar for other containers.
 
== Indexed containers ==
{{Expand section|date=October 2008}}
'''Indexed containers''' (also known as '''dependent polynomial functors''') are a generalisation of containers, which can represent a wider class of types, such as vectors (sized lists).<ref name='indexed-containers'>{{cite journal
| author        = Thorsten Altenkirch, Neil Ghani, Peter Hancock, [[Conor McBride]], and Peter Morris
| title          = Indexed Containers
| version        = Unpublished manuscript
| url            = http://strictlypositive.org/indexed-containers.pdf
| format        = PDF
| accessdate    = 2008-10-30
}}
</ref>
 
The element type (called the ''input type'') is indexed by shape and position, so it can vary by shape and position, and the extension (called the ''output type'') is also indexed by shape.
 
== See also ==
* [[Container (abstract data type)]]
* [[Functor (type theory)]]
 
== References ==
{{reflist}}
 
[[Category:Type theory]]
 
 
{{type-theory-stub}}

Latest revision as of 21:00, 3 September 2014

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