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CATEGORICAL COMPLEXITY

Part of: Theory of computing General theory of categories and functors Computational aspects in algebraic geometry

Published online by Cambridge University Press:  30 June 2020

SAUGATA BASU Open the ORCID record for SAUGATA BASU [Opens in a new window]  and
UMUT ISIK
SAUGATA BASU
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47906, USA; [email protected]
UMUT ISIK
Affiliation:
Department of Mathematics, University of California, Irvine, Irvine, CA 92697, USA; [email protected]

Abstract

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We introduce a notion of complexity of diagrams (and, in particular, of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest such as finite sets, Boolean functions, topological spaces, vector spaces, semilinear and semialgebraic sets, graded algebras, affine and projective varieties and schemes, and modules over polynomial rings. We show that on one hand categorical complexity recovers in several settings classical notions of nonuniform computational complexity (such as circuit complexity), while on the other hand it has features that make it mathematically more natural. We also postulate that studying functor complexity is the categorical analog of classical questions in complexity theory about separating different complexity classes.

MSC classification

Primary: 18A10: Graphs, diagram schemes, precategories 68Q15: Complexity classes (hierarchies, relations among complexity classes, etc.)
Secondary: 14Q20: Effectivity, complexity
Type
Discrete Mathematics
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e34
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
Copyright
© The Author(s) 2020

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