Abstract
The article is a discussion of the nature of structure in Bourbaki's “Elements de Mathematique”, contrasting it with the account of structure arising in category theory. It is also explained how Bourbaki’s concept of “mother structure” is given category-theoretic form.
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Notes
- 1.
Bourbaki (1939).
- 2.
Mashaal (2006).
- 3.
Bourbaki (1994)
- 4.
Bourbaki (1950).
- 5.
E.g. in Corry (1992).
- 6.
Eilenberg and Mac Lane (1945).
- 7.
Roughly speaking, Bourbaki’s species of structures correspond to “mapless” categories.
- 8.
It should be noted, however, in Chap. 4 of the Théorie des Ensembles Bourbaki does formulate versions of certain concepts—such as universal arrows and the solution set condition for their existence—which were later to become central to category theory. Mac Lane (1971) remarks that Bourbaki’s formulation “was cumbersome because [their] notion of ‘structure’ did not make use of categorical ideas”.
- 9.
A pioneering first step in this regard at an elementary level was undertaken by Lawvere and Schanuel in their work Conceptual Mathemtics (Lawvere and Schanuel 1997).
- 10.
See Johnstone for a full account of the representation theory for commutative rings.
- 11.
See, e.g. Bell (2018).
- 12.
See, e.g. Mac Lane and Moerdijk (1992) for an account of topos theory.
References
Bell, J. L. (2018). Categorical logic and model theory. In E. Landry (Ed.), Categories for the Working Philosopher. Oxford University Press.
Bourbaki, N. (1939) Éléments de Mathématique (Vol. 10). Paris: Hermann.
Bourbaki, N. (1950). The architecture of mathematics. American Mathematical Monthly, 67, 221–232.
Bourbaki, N. (1994). Éléments d’histoire des mathématiques. Berlin: Springer.
Corry, L. (1992). Bourbaki and the concept of mathematical structure. Synthese, 92(3), 311–348.
Eilenberg, S., & Mac Lane, S. (1945). General theory of natural equivalences. Transactions of the American Mathematical Society, 58, 231–294. Reprinted in Eilenberg, S., & Mac Lane, S. (1986). Eilenberg-Mac Lane: Collected Works. New York: Academic Press.
Lawvere, F. W., & Schanuel, S. (1997). Conceptual mathematics. Cambridge University Press.
Mac Lane, S. (1971). Categories for the working mathematician. Berlin: Springer.
Mac Lane, S., & Moerdijk, I. (1992). Sheaves in geometry and logic; a first introduction to topos theory. Berlin: Springer.
Mashaal. M. (2006). Bourbaki: a secret society of mathematicians. American Mathematical Society.
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Bell, J.L. (2020). Reflections on Bourbaki’s Notion of “Structure” and Categories. In: Peruzzi, A., Zipoli Caiani, S. (eds) Structures Mères: Semantics, Mathematics, and Cognitive Science. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 57. Springer, Cham. https://doi.org/10.1007/978-3-030-51821-9_1
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