 Sarkar’s (1998) account of explanatory reduction draws attention to the requirement of representation: natural phenomena must be symbolized, embodied, pictured, or designated through media such as equations, scale miniatures, or abstract diagrams. Every reductive explanation in science involves representations of the systems or features to be related by reduction. (Note that this is not the question of how theory structure is represented; see, e.g., Schaffner 1996.) Almost all of the discussion surrounding one-many  (Section 4.1)  and many-one problems  (Section 4.2)  for theory and explanatory reduction assume or presume particular representations that directly impact the arguments offered (e.g., they involve approximations of different kinds). Sarkar distinguishes three representational criteria for a reductive explanation: (i) fundamentalism: the explanation of the phenomena relies entirely on features that pertain to a more fundamental realm; (ii) abstract hierarchy: the system is represented as a hierarchy, where the lower levels are regarded as more fundamental; and, (iii) spatial hierarchy: the abstract hierarchy is rendered physical by the requirement that the entities on the lower level are (spatial) parts of the entities at higher levels (see  Section 3.2).  Whether or not there are one-many or many-one relations between different hierarchical levels turns on the nature of the hierarchy invoked and how it is characterized. The decomposition of a system is not a univocal enterprise and, depending on the principles utilized, can generate both competing and complementary sets of part representations from the same system (Craver 2009, Kauffman 1971, Wimsatt 1974, Winther 2011). Certain governing principles are often visible, such as functional versus structural decompositions (Bechtel and Richardson 1993, Winther 2006). Therefore, prior to determinations of whether reductive explanations succeed or fail, questions of representational choice and adequacy need to be addressed explicitly.