CATEGORICAL COMPLEXITY

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 commo...

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Bibliographic Details
Published inForum of Mathematics, Sigma Vol. 8
Main Authors BASU, SAUGATA, ISIK, UMUT
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 2020
Subjects
14Q20
18A10
68Q15
Boolean algebra
Boolean functions
Circuits
Complexity theory
Discrete Mathematics
Mathematical analysis
Polynomials
Rings (mathematics)
Vector spaces
68Q15
18A10
14Q20
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ISSN2050-5094
2050-5094
DOI10.1017/fms.2020.26

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Summary: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.
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ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2020.26