離散幾何構造セミナー：An Introduction to First-Order Categorical Logic: Toward Topos-Theoretic Model Theory, 荒武永史氏(京都大学数理解析研究所)

開催日時

2018年7月12日 10時30分 ～ 2018年7月13日 12時00分

場所

3-413, 3-210

講演者

荒武 永史

離散幾何構造セミナー

• Lect 1: 2018/07/12 (Thu) 10:30-12:00 (Rm 3-413)
• Lect 2: 2018/07/12 (Thu) 13:30-15:00 (Rm 3-413)
• Lect 3: 2018/07/13 (Fri) 10:30-12:00 (Rm 3-210)

First-order categorical logic was introduced by Joyal, Makkai and Reyes as a categorical approach to first-order logic. Some theorems in logic can be rephrased and generalized in terms of categories. The principal goal of this talk is to illustrate connections between first-order logic and category theory.

Since this talk targets mathematicians, we begin with an overview of first-order logic. After introducing some requisites of category theory, we proceed to the central concepts of syntactic categories and classifying toposes. By using these concepts, we describe a classical example of interactions between logic and category theory; Gödel’s completeness theorem vs. Deligne’s theorem on coherent toposes.

Afterwards, we see recent developments on topos-theoretic aspects of model theory. Model theory is a discipline of mathematical logic and has many applications to “ordinary” mathematics, including algebra, (algebraic and analytic) geometry, number theory and (nonstandard) analysis. Classifying toposes are model-theoretic “cores” of theories and can be exploited to develop “language-free” model theory. To make the setting explicit, we introduce a bicategory of (classical) theories and show that the bicategory is biequivalent to the opposite of the 2-category of “classical” coherent toposes. As a consequence, we can show that some elementary notions in model theory have categorical counterparts. If time permits, we take a glance at more sophisticated results by Olivia Caramello. She has been developing the theory of classifying topos and her works give a new topos-theoretic viewpoint of model theory. At the end of this talk, we indicate future directions of topos-theoretic model theory and its applications to mathematics.

Notice:
We assume the audience familiar with elementary concepts of category theory (e.g. (co)limits, equivalence of categories and sheaf of sets on a topological space). The presentation will be given in Japanese with some materials written in English