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The | The theory of '''accessible categories''' was introduced in 1989 by mathematicians [[Michael Makkai]] and Robert Paré in the setting of [[category theory]]. While the original motivation came from [[model theory]], a branch of [[mathematical logic]],<ref name=ref1>J. Rosicky [http://arxiv.org/abs/0708.2185 "On combinatorial model categories"], ''[[Arxiv]]'', 16 August 2007. Retrieved on 19 January 2008.</ref> it turned out that accessible categories have applications in [[homotopy theory]].<ref name=ref1/><ref name=ref3>J. Rosicky, Injectivity and accessible categories</ref> Some properties of accessible categories depend on the [[Axiomatic set theory|set universe]] in use, particularly on the [[cardinal number|cardinal]] properties.<ref name=ref2>J. Adamek and J. Rosicky, Locally Presentable and Accessible Categories, Cambridge University Press 1994</ref> | ||
==Definition== | |||
Let <math>K</math> be an infinite [[regular cardinal]] and let <math>C</math> be a [[category (mathematics)|category]]. | |||
An object <math>X</math> of <math>C</math> is called '''<math>K</math>-presentable''' if the [[Hom functor]] <math>Hom(X,-)</math> preserves <math>K</math>-directed [[colimit]]s. | |||
The category <math>C</math> is called '''<math>K</math>-accessible''' provided that : | |||
* <math>C</math> has <math>K</math>-directed colimits | |||
* <math>C</math> has a set <math>P</math> of <math>K</math>-presentable objects such that every object of <math>C</math> is a <math>K</math>-directed colimit of objects of <math>P</math> | |||
A category <math>C</math> is called '''accessible''' if <math>C</math> is <math>K</math>-accessible for some infinite regular cardinal <math>K</math>. | |||
A <math>\aleph_0</math>-presentable object is usually called '''finitely presentable''', and | |||
an <math>\aleph_0</math>-accessible category is often called '''finitely accessible'''. | |||
==Examples== | |||
*The category [[Category of modules|<math>R</math>-Mod]] of (left) <math>R</math>-modules is finitely accessible for any ring <math>R</math>. The objects that are finitely presentable in the above sense are precisely the [[finitely presented module]]s (which are not necessarily the same class of objects than [[finitely generated module]]s unless <math>R</math> is [[noetherian ring|noetherian]]). | |||
*The category of [[simplicial set]]s is finitely-accessible. | |||
*The category Mod(T) of models of some [[first-order theory]] T with countable signature is <math>\aleph_1</math> -accessible. <math>\aleph_1</math> -presentable objects are models with a countable number of elements. | |||
==Further notions== | |||
When the category <math>C</math> is [[cocomplete]], <math>C</math> is called a locally presentable category. | |||
Locally presentable categories are also [[Complete category|complete]]. | |||
==References== | |||
{{reflist}} | |||
==Further reading== | |||
* {{Citation | |||
| last = Makkai | first = Michael | |||
| last2 = Paré | first2 = Robert | |||
| title = Accessible categories: The foundation of Categorical Model Theory | |||
| publisher = AMS | |||
| series = Contemporary Mathematics | |||
| year = 1989 | |||
| isbn = 0-8218-5111-X }} | |||
* {{Citation | |||
| last = Adámek | first = Jiří | |||
| last2 = Rosicky | first2 = Jiří | |||
| title = Locally presentable and accessible categories | |||
| publisher = CUP | |||
| series = LNM Lecture Notes | |||
| year = 1994 | |||
| isbn = 0-521-42261-2 }} | |||
[[Category:Category theory]] | |||
{{categorytheory-stub}} | |||
Revision as of 21:44, 26 November 2012
The theory of accessible categories was introduced in 1989 by mathematicians Michael Makkai and Robert Paré in the setting of category theory. While the original motivation came from model theory, a branch of mathematical logic,[1] it turned out that accessible categories have applications in homotopy theory.[1][2] Some properties of accessible categories depend on the set universe in use, particularly on the cardinal properties.[3]
Definition
Let be an infinite regular cardinal and let be a category. An object of is called -presentable if the Hom functor preserves -directed colimits. The category is called -accessible provided that :
- has -directed colimits
- has a set of -presentable objects such that every object of is a -directed colimit of objects of
A category is called accessible if is -accessible for some infinite regular cardinal .
A -presentable object is usually called finitely presentable, and an -accessible category is often called finitely accessible.
Examples
- The category -Mod of (left) -modules is finitely accessible for any ring . The objects that are finitely presentable in the above sense are precisely the finitely presented modules (which are not necessarily the same class of objects than finitely generated modules unless is noetherian).
- The category of simplicial sets is finitely-accessible.
- The category Mod(T) of models of some first-order theory T with countable signature is -accessible. -presentable objects are models with a countable number of elements.
Further notions
When the category is cocomplete, is called a locally presentable category. Locally presentable categories are also complete.
References
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Further reading
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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
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