# 拓撲斯

## 格羅滕迪克拓撲斯（幾何中的拓撲斯）

### 等價構造

C為一範疇。Giraud的一個定理斷言，以下命題等價：

• 有小範疇D和包含關係C $\hookrightarrow$ Presh(D)使得其存在保持有限極限的左伴隨
• C是格羅滕迪克site上的層範疇。
• C滿足以下的Giraud公理

#### Giraud公理

• C生成元構成小集合，且允許所有小的余極限。進一步，余極限與纖維積可交換。
• C中的和是不交的。換句話說，XY在它們和上的纖維積C的初對象。
• C中所有等價關係皆為有效的。

$R \to X \times_{X/R} X \,\!$

#### 例子

Giraud定理已經給出了“sites上的層”作為例子的完全列表。注意不等價的sites常常給出等價的拓撲斯。如介紹所示，普通拓撲空間上的層激發了很多拓撲斯理論的基本定義和結果。

### 幾何態射

XY是拓撲空間，u是其間的連續映射，層上的前推和拉回給出相關拓撲斯間的幾何態射。

#### 拓撲斯的點

X是普通拓撲空間，xX的點，那麼把層F帶到它的莖Fx的函子有右伴隨（“摩天大樓層”函子），因此X的普通點同時決定了一個拓撲斯理論中的點。這些可以用沿連續映射x1X的拉回前推來構造。

## 參考資料

• F. William Lawvere and Stephen H. Schanuel (1997) Conceptual Mathematics: A First Introduction to Categories. Cambridge University Press. An "introduction to categories for computer scientists, logicians, physicists, linguists, etc." (cited from cover text).
• F. William Lawvere and Robert Rosebrugh (2003) Sets for Mathematics. Cambridge University Press. Introduces the foundations of mathematics from a categorical perspective.

• Grothendieck and Verdier: Théorie des topos et cohomologie étale des schémas (known as SGA4)". New York/Berlin: Springer, ??. (Lecture notes in mathematics, 269–270)

• Colin McLarty (1992) Elementary Categories, Elementary Toposes. Oxford Univ. Press. A nice introduction to the basics of category theory, topos theory, and topos logic. Assumes very few prerequisites.
• Robert Goldblatt (1984) Topoi, the Categorial Analysis of Logic (Studies in logic and the foundations of mathematics, 98). North-Holland. A good start. Reprinted 2006 by Dover Publications, and available online at Robert Goldblatt's homepage.
• John Lane Bell (2005) The Development of Categorical Logic. Handbook of Philosophical Logic, Volume 12. Springer. Version available online at John Bell's homepage.
• Saunders Mac Lane and Ieke Moerdijk (1992) Sheaves in Geometry and Logic: a First Introduction to Topos Theory. Springer Verlag. More complete, and more difficult to read.
• Michael Barr and Charles Wells (1985) Toposes, Triples and Theories. Springer Verlag. Corrected online version at http://www.cwru.edu/artsci/math/wells/pub/ttt.html. More concise than Sheaves in Geometry and Logic, but hard on beginners.

• Francis Borceux (1994) Handbook of Categorical Algebra 3: Categories of Sheaves, Volume 52 of the Encyclopedia of Mathematics and its Applications. Cambridge University Press. The third part of "Borceux' remarkable magnum opus", as Johnstone has labelled it. Still suitable as an introduction, though beginners may find it hard to recognize the most relevant results among the huge amount of material given.
• Peter T. Johnstone (1977) Topos Theory, L. M. S. Monographs no. 10. Academic Press. ISBN 0123878500. For a long time the standard compendium on topos theory. However, even Johnstone describes this work as "far too hard to read, and not for the faint-hearted."
• Peter T. Johnstone (2002) Sketches of an Elephant: A Topos Theory Compendium. Oxford Science Publications. As of early 2010, two of the scheduled three volumes of this overwhelming compendium were available.

• Maria Cristina Pedicchio and Walter Tholen, eds. (2004) Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory. Volume 97 of the Encyclopedia of Mathematics and its Applications. Cambridge University Press. Includes many interesting special applications.

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