## Abstract

There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms (object-to-object morphisms between objects of different categories) that parses an adjunction into two separate parts (left and right representations of heteromorphisms). Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory and is focused on the interpretation and application of the mathematical concepts .

## Keywords

Category theory Adjoint functors Heteromorphism Brain functors## Mathematics Subject Classification

18 92## References

- Awodey S (2006) Category theory. Clarendon Press, OxfordCrossRefGoogle Scholar
- Benabou J (1973) Les distributeurs, vol 33, Institut de Mathmatique Pure et AppliqueGoogle Scholar
- Boolos G (1971) The iterative conception of set. J Philos 68(April 22):215–231CrossRefGoogle Scholar
- Ehresmann AC, Vanbremeersch JP (2007) Memory evolutive systems: hierarchy, emergence, cognition. Elsevier, AmsterdamGoogle Scholar
- Eilenberg S, MacLane S (1945) General theory of natural equivalences. Trans Am Math Soc 58(2):231–294CrossRefGoogle Scholar
- Ellerman D (1988) Category theory and concrete universals. Erkenntnis 28:409–429Google Scholar
- Ellerman D (2006) A theory of adjoint functors with some thoughts on their philosophical significance. In: Sica G (ed) What is category theory?. Polimetrica, Milan, pp 127–183Google Scholar
- Ellerman D (2007) Adjoints and emergence: applications of a new theory of adjoint functors. Axiomathes 17(1 March):19–39CrossRefGoogle Scholar
- Goldblatt R (2006) Topoi: the categorical analysis of logic (revised ed.). Dover, MineolaGoogle Scholar
- Halford GS, Wilson WH (1980) A category theory approach to cognitive development. Cogn Psychol 12(3):356–411CrossRefGoogle Scholar
- Hungerford TW (1974) Algebra. Springer, New YorkGoogle Scholar
- Jacobson N (1985) Basic algebra I, 2nd edn. W.H. Freeman, New YorkGoogle Scholar
- Kainen PC (2009) On the Ehresmann–Vanbremeersch theory and mathematical biology. Axiomathes 19:225–244CrossRefGoogle Scholar
- Kan D (1958) Adjoint functors. Trans Am Math Soc 87(2):294–329CrossRefGoogle Scholar
- Kelly M (1982) Basic concepts of enriched category theory. Cambridge University Press, CambridgeGoogle Scholar
- Lambek J (1981) The influence of Heraclitus on modern mathematics. In: Agassi J, Cohen RS (eds) Scientific philosophy today: essays in honor of Mario Bunge. D. Reidel, Boston, pp 111–121CrossRefGoogle Scholar
- Lawvere FW (1969) Adjointness in foundations. Dialectica 23:281–295CrossRefGoogle Scholar
- Louie AH (1985) Categorical system theory. In: Rosen R (ed) Theoretical biology and complexity: three essays on the natural philosophy of complex systems. Academic Press, Orlando, pp 68–163Google Scholar
- Louie AH, Poli R (2011) The spread of hierarchical cycles. Int J Gen Syst 40(3 April):237–261CrossRefGoogle Scholar
- Lurie J (2009) Higher topos theory. Princeton University Press, PrincetonGoogle Scholar
- MacLane S (1948) Groups, categories, and duality. Proc Nat Acad Sci USA 34(6):263–267CrossRefGoogle Scholar
- MacLane S (1971) Categories for the working mathematician. Springer, New YorkGoogle Scholar
- MacLane S, Birkhoff G (1988) Algebra, 3rd edn. Chelsea, New YorkGoogle Scholar
- Magnan F, Reyes GE (1994) Category theory as a conceptual tool in the study of cognition. In: Macnamara J, Reyes GE (eds) The logical foundations of cognition. Oxford University Press, New York, pp 57–90Google Scholar
- Makkai M (1999) Structuralism in mathematics. In: Jackendoff R, Bloom P, Wynn K (eds) Language, logic, and concepts: essays in memory of John Macnamara. MIT Press (A Bradford Book), Cambridge, pp 43–66Google Scholar
- Pareigis B (1970) Categories and functors. Academic Press, New YorkGoogle Scholar
- Philips S (2014) Analogy, cognitive architecture and universal construction: a tale of two systematicities. PLOS One 9(2):1–9CrossRefGoogle Scholar
- Philips S, Wilson WH (2014) Chapter 9: a category theory explanation for systematicity: universal constructions. In: Calvo P, Symons J (eds) Systematicity and cognitive architecture. MIT Press, Cambridge, pp 227–249Google Scholar
- Rosen R (1958) The representation of biological systems from the standpoint of the theory of categories. Bull Math Biophys 20(4):317–342CrossRefGoogle Scholar
- Rosen R (2012) Anticipatory systems: philosophical, mathematical, and methodological foundations, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
- Russell B (2010) Principles of mathematics. Routledge Classics, LondonGoogle Scholar
- Samuel P (1948) On universal mappings and free topological groups. Bull Am Math Soc 54(6):591–598CrossRefGoogle Scholar
- Shulman M (2011) Cograph of a profunctorGoogle Scholar
- Taylor P (1999) Practical foundations of mathematics. Cambridge University Press, CambridgeGoogle Scholar
- von Humboldt W (1997) The nature and conformation of language. In: Mueller-Vollmer K (ed) The hermeneutics reader. Continuum, New York, pp 99–105Google Scholar
- Wood RJ (2004) Ordered sets via adjunctions. In: Pedicchio MC, Tholen W (eds) Categorical foundations. Encyclopedia of mathematics and its applications, vol 97. Cambridge University Press, Cambridge, pp 5–47Google Scholar
- Zafiris E (2012) Rosen’s modelling relations via categorical adjunctions. Int J Gen Syst 41(5):439–474CrossRefGoogle Scholar

## Copyright information

© Springer Science+Business Media Dordrecht 2015