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Taut functors and the difference operator
Theory and applications of categories, Lawvere Festschrift, Tome 43 (2025), pp. 281-362 Cet article a éte moissonné depuis la source Theory and Applications of Categories website

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We establish a calculus of differences for taut endofunctors of the category of sets, analogous to the classical calculus of finite differences for real valued functions. We study how the difference operator interacts with limits and colimits as categorical versions of the usual product and sum rules. The first main result is a lax chain rule which has no counterpart for mere functions. We also show that many important classes of functors (polynomials, analytic functors, reduced powers, ...) are taut, and calculate explicit formulas for their differences. Covariant Dirichlet series are introduced and studied. The second main result is a Newton summation formula expressed as an adjoint to the difference operator.
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Classification : Primary: 18A22, 12H10, secondary: 18C15, 18F50
Keywords: Taut functor, polynomial, reduced power, analytic functor, Dirichlet series, difference operator 
@article{TAC_2025_43_a9,
     author = {Robert Par\'e},
     title = {Taut functors and the difference operator},
     journal = {Theory and applications of categories},
     pages = {281--362},
     year = {2025},
     volume = {43},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2025_43_a9/}
}
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Robert Paré. Taut functors and the difference operator. Theory and applications of categories, Lawvere Festschrift, Tome 43 (2025), pp. 281-362. http://geodesic.mathdoc.fr/item/TAC_2025_43_a9/