Topos
- For discussion of topoi in literary theory, see literary topos. For discussion of topoi in rhetorical invention, see Inventio.
Sheaves were introduced into mathematics in the 1940s, and a major theme since then has been to study a space by studying sheaves on that space. Grothendieck expounded on this idea by introducing the notion of a topos (plural: topoi or toposes - this is a contentious topic). A topos is a type of category that behaves like the category of sheaves of sets on a topological space. The main utility of this notion is in the abundance of situations in mathematics where topological intuition is very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the intuition. The greatest success of this yoga to date has been the introduction of the étale topos of a scheme.
References
- John Baez: Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html. A gentle introduction.
- F. William Lawvere and Stephen H. Schanuel: Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, Cambridge, 1997. An "introduction to categories for computer scientists, logicians, physicists, linguists, etc." (cited from cover text).
- Grothendieck and Verdier: "SGA4"