A jump operator on the Weihrauch degrees

A partial order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⩽</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> admits a jump operator if there is a map <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>j</mml:mi> <mml:mo>:</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>P</mml:mi> </mml:math> that is strictly increasing and weakly monotone. Despite its name, the jump in the Weihrauch lattice fails to satisfy both of these properties: it is not degree-theoretic, and there are functions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>f</mml:mi> </mml:math> such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>f</mml:mi> <mml:msub> <mml:mo>≡</mml:mo> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">W</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:msup> <mml:mi>f</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> . This raises the question: Is there a jump operator in the Weihrauch lattice? We answer this question positively and provide an explicit definition for an operator on partial multi-valued functions that, when lifted to the Weihrauch degrees, induces a jump operator. This new operator, called the totalizing jump , can be characterized in terms of the total continuation, a well-known operator on computational problems. The totalizing jump induces an injective endomorphism of the Weihrauch degrees. We study some algebraic properties of the totalizing jump and characterize its behavior on some pivotal problems in the Weihrauch lattice.