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Adjoint and Representable Functors What underlies dual-process cognition? Adjoint and representable functors Steven Phillips ([email protected]) Mathematical Neuroinformatics Group, Human Informatics Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8566 JAPAN Abstract & Stanovich, 2013, for the model and responses to the five criticisms). According to this model, cognition defaults to Despite a general recognition that there are two styles of think- ing: fast, reflexive and relatively effortless (Type 1) versus slow, Type 1 processes that can be intervened upon by Type 2 pro- reflective and effortful (Type 2), dual-process theories of cogni- cesses in response to task demands. However, this model is tion remain controversial, in particular, for their vagueness. To still far from the kinds of formal, computational explanations address this lack of formal precision, we take a mathematical category theory approach towards understanding what under- that prevail in cognitive science. lies the relationship between dual cognitive processes. From our category theory perspective, we show that distinguishing ID Criticism of dual-process theories features of Type 1 versus Type 2 processes are exhibited via adjoint and representable functors. These results suggest that C1 Theories are multitudinous and definitions vague category theory provides a useful formal framework for devel- C2 Type 1/2 distinctions do not reliably align oping dual-process theories of cognition. C3 Cognitive styles vary continuously, not discretely Keywords: dual-process; Type 1; Type 2; category theory; C4 Single-process theories provide better explanations category; functor; natural transformation; adjoint C5 Evidence for dual-processing is unconvincing Introduction Table 2: Five criticisms of dual-process theories (Evans & Intuitively, at least, human cognition involves two starkly dif- Stanovich, 2013). ferent styles of thinking: one style appears to be fast, reflexive and relatively effortless; the other slow, reflective and effortful We propose a category theory (Mac Lane, 1998) approach (Evans, 2003; Kahneman, 2011). The former is called Type 1 to understanding dual-process cognition. The attraction of and the latter Type 2 (Evans & Stanovich, 2013). Distinguish- this approach is the precise formalization of the relationships ing features are listed in Table 1. between cognitive processes that have some of the distinctive features of Type 1 versus Type 2 cognition. An intuitive basis ID Type 1 Type 2 for the category theory that supports this approach is given in D1 no working memory working memory the next section. Examples are sketched in the section that D2 autonomous decoupled, simulation follows, and this approach is discussed with regard to the five F1 fast slow criticisms in the final section. Technical support appears in F2 high capacity low capacity the appendix. F3 parallel serial F4 unconscious conscious Categories and functors F5 biased responses normative responses Here, we provide an intuitive overview of the supporting the- F6 contextualized abstract ory given in the appendix (see, e.g., Awodey, 2010; Mac Lane, F7 automatic controlled 1998, for introductions to category theory). To aid intuition, F8 associative rule-based formal concepts are interpreted in terms of the more familiar F9 experience-based consequence-based notion of an analogy between source and target domains. Our F10 ability-independent ability-dependent main interest is in adjoint and representable functors, which F11 evolved early evolved late depend on the concepts of category, functor and natural trans- F12 animal human formation. So, we proceed in that order. Some conceptual F13 implicit explicit correspondences are summarized in Table 3. F14 basic emotions complex emotions Category theory Cognitive science Table 1: Definitions/Features (D/F) of Type 1/2 processes category (sub)system, source/target (adapted from Evans & Stanovich, 2013, Table 1). functor construction, analogy natural transformation comparison (of analogies) Dual-process theories of such styles of thinking are con- adjoint (functor) inverse/obverse, Type 1 $ 2 troversial. Evans and Stanovich (2013) reviewed five general representable (functor) representation criticisms of dual-process theories, which are summarized in Table 2. Foremost is the acknowledgement that dual-process Table 3: Corresponding categorical and cognitive concepts. theories are multitudinous and vaguely defined. The Inter- vention model attempts to address this problem (see Evans A proportional analogy serves to bootstrap basic intuitions 2253 about categories, functors and natural transformations: Pre- of integers, thus avoiding a need for infinite precision. In decessor is to successor as one is to two; as Monday is to a cognitive context, the infinite real-world is represented by Tuesday. Analogy is generally recognized as a map of entities finite resources. This example also shows how representable in a source domain to entities in a target domain that pre- functors convey a second-order isomorphism even though a serves their relations (Gentner, 1983): e.g., next(predecessor, first-order isomorphism does not exist. successor) in the order domain maps to next(one, two) in the number domain; likewise, next(Monday, Tuesday) in the Categorical perspective on dual-processes days-of-the-week domain. In this context, we can interpret: Type 1 and Type 2 processes are distinguished by counter- posed features, suggesting adjoint and representable functors • a category (definition 1) as a source/target domain, which as the formal connection. We sketch cases centred on pairs of consists of a collection of objects (e.g., predecessor, suc- adjoint functors. In each case, we lead with an aspect of cog- cessor), a collection of morphisms (e.g., next) between nition that motivates a particular adjunction, which suggests objects, and a composition operation for combining mor- related distinctions that follow, phisms (e.g., next of next); Whole/part: diagonal-product functors • a functor (definition 3) as an analogy (e.g., days-of-the- One commonly held distinction is whether cognitive processes week) from one category (source) to another category (tar- operate on cognitive representations wholistically or compo- get) that “preserves structure” (e.g., next of next in the order nentially (whole/part). For example, kicked the bucket can domain maps to next of next in the number domain); and be interpreted wholistically (idiomatically) as died, or com- • a natural transformation (definition 4) as a comparison of ponentially as an act of kicking. The first interpretation is analogies (e.g., next number and next day) so that the result characteristically associative and the second rule-based (F8). of applying a relation to a transformation is the same as the This whole/part distinction is expressed by functors that transformation of the result of applying a relation (e.g. the pertain to a componential object called a product. Products day after the first day is the same as the second day). are constructed by the product functor, which is right adjoint to the diagonal functor (example 6(c)). This case is a conceptual Natural transformations also appear to be like analogies, but inverse in the sense of combining parts to form wholes in the appearance belies an important distinction: natural trans- one direction (product functor) and contextualizing wholes as formations are maps between functors, whereas functors are parts in the other direction (diagonal functor). maps between categories: 2nd-order contra 1st-order analogy. This adjoint situation was tested in a paired-associates task: With these concepts we can proceed to adjunction, re- subjects learned to associate pairs of letters to coloured shapes, garded as the centrepiece of ordinary category theory (Mac where say first letters were associated with colours and second Lane, 1998), and the related notion of representable func- letters to shapes (Phillips, Takeda, & Sugimoto, 2016). In this tor. Our motivation is the observation that Type 1/2 processes situation, the dual processes are a map of letter pairs versus are distinguished by counterposing features and immediacy of pair of letter maps. Performance on generalization (novel) (e.g., autonomous versus simulated). To illustrate, consider trials indicated associative versus rule-based processes, which approximation and precision as conceptual inverses, though correlated with conscious awareness (F4) of the underlying not actual inverses since approximation discards information: rule. Unaware subjects did not show correct responses to cues e.g., reals are approximated by integers, but their sets are not beyond the context of those seen in training trials: did not isomorphic. Yet, there is a “second-order isomorphism” be- infer an abstract rule that extended to novel letter pairs (F6). tween their orders affording comparisons with reals in terms Thus, the diagonal-product functor expresses three distinctive of integers without loss of precision. We can interpret: features of Type 1/2 processes: F4, F6 and F8. • an adjunction (definition 5) as an “inverse” (obverse) rela- Parallel/serial: product-exponential functors tion between two “anti-parallel” functors: e.g., one functor A second common distinction is between parallel versus serial 7! sends each real to its ceiling (approximation: e.g., 2:3 3) cognitive process (F3). Visual search, for example, is clas- and the other functor sends each integer to its corresponding sically regarded as
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