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Fixpoint and loop constructions as colimits

Part I

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1488)

Abstract

The constructions of fixpoints and while-loops in a category of domains can be derived from the colimit, loop(f) of a diagram which consists of a single endomorphism f : D → D. If f is increasing then the colimiting map is the least-fixpoint function Y and

$$loop\left( f \right) = fix\left( f \right)$$

the subobject of fixpoints. If f=cond(b, g, 1) is the conditional of a while-program then

$$loop\left( f \right) = \left( {D_{\neg b} + loop_\infty \left( g \right)} \right)_ \bot$$

the lifted sum of the terminating values (where b is false) and the infinite loops.

Keywords

  • Full Subcategory
  • Enrich Category
  • Infinite Loop
  • Dose Category
  • Countable Chain

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Supported by grants GR/E 78487 and GR/F 07866 from SERC, and OGPIN 016 from NSERC.

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References

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© 1991 Springer-Verlag

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Jay, C.B. (1991). Fixpoint and loop constructions as colimits. In: Carboni, A., Pedicchio, M.C., Rosolini, G. (eds) Category Theory. Lecture Notes in Mathematics, vol 1488. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084220

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  • DOI: https://doi.org/10.1007/BFb0084220

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54706-8

  • Online ISBN: 978-3-540-46435-8

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