| Title | Topology, Algebra, Diagrams |
| Publication Type | Journal Article |
| Authors | Rotman, Brian |
| Journal | Theory, Culture & Society |
| Volume | 29 |
| Issue | 4-5 |
| Pagination | 247-260 |
| ISSN | 0263-2764 |
| Abstract | Starting from Poincaré?s assignment of an algebraic object to a topological manifold, namely the fundamental group, this article introduces the concept of categories and their language of arrows that has, since their mid-20th-century inception, altered how large areas of mathematics, from algebra to abstract logic and computer programming, are conceptualized. The assignment of the fundamental group is an example of a functor, an arrow construction central to the notion of a category. The exposition of category theory?s arrows, which operate at three distinct but deeply interconnected levels, is framed by a comparison with the language and outlook of set theory founded on the concept of membership; sets and their theorization having provided, famously through the Bourbaki initiative, the basic ontological and epistemological vocabulary for defining and handling all mathematical entities. The comparison with sets emphasizes how categories offer a form of diagrammatic argument and thought against set theory?s fidelity to syntax-based proofs; how categories invert set theory?s priority of objects and their attributes over relations by making the relations of an object to others of its kin primary; and how categories replace identity, that is, equality, between objects, by the weaker notion of isomorphism, restricting equality to identity between arrows. The article concludes with a return to topology and some remarks about the question of its possible use in articulating/characterizing cultural dynamics. |
| URL | https://doi.org/10.1177/0263276412444472 |
| DOI | 10.1177/0263276412444472 |
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