Birkhoff's Theorem from a geometric perspective: A simple example | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 2، دوره 4، شماره 1، اردیبهشت 2016، صفحه 1-8 اصل مقاله (336.86 K) | ||
| نوع مقاله: Research Paper | ||
| نویسنده | ||
| F. William Lawvere* | ||
| Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900, United States of America. | ||
| چکیده | ||
| From Hilbert's theorem of zeroes, and from Noether's ideal theory, Birkhoff derived certain algebraic concepts (as explained by Tholen) that have a dual significance in general toposes, similar to their role in the original examples of algebraic geometry. I will describe a simple example that illustrates some of the aspects of this relationship. The dualization from algebra to geometry in the basic Grothendieck spirit can be accomplished (without intervention of topological spaces) by the following method, known as Isbell conjugacy. | ||
| کلیدواژهها | ||
| Grothendieck spectrum؛ Cantor؛ Boole؛ Hilbert؛ Birkhoff: Existence and Sufficiency of generalized points؛ Reflexive Graphs | ||
| مراجع | ||
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[1] G. Birkhoff, Subdirect unions in universal algebra, Bull. Amer. Math. Soc. 50 (1944), 764-768. [2] G. Cantor, Beiträge zur Begründung der transfiniten Mengenlehre, Math. Ann. 46 (1895) 481-512. [3] J.R. Isbell, Small subcategories and completeness, Math. Syst. Theory 2 (1968), 27-50. [4] F.W. Lawvere, Unity and identity of opposites in calculus and Physics, Appl. Categ. Structures, 4 (1996), 167-174. [5] F.W. Lawvere, Taking categories seriously, Reprints in Theory Appl. Categ. 8 (2005), 1-24. [6] F.W. Lawvere, Axiomatic cohesion, Theory Appl. Categ. 19 (2007), 41-49. [7] F.W. Lawvere, Core varieties, extensivity, and rig geometry, Theory Appl. Categ. 20 (2008), 497-503. [8] F.W. Lawvere and M. Menni, Internal choice holds in the discrete part of any co- hesive topos satisfying stable connected codiscreteness, Theory Appl. Categ. 30(26) (2015), 909-932. [9] F.W. Lawvere and S.H. Schanuel, “Conceptual Mathematics”, Cambridge Univer- sity Press, 2nd edition, 2009. [10] W. Tholen, Nullstellen and subdirect representation, Appl. Categ. Structures 22 (2014), 907-929. | ||
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