**Title:** Double Categories and Double Structures

**Speaker:** Professor Tom Fiore

**Speaker Info:** University of Chicago

**Abstract:**

In 1946, Whitehead introduced the notion of crossed module in his study of adding relations to homotopy groups. A few years later, Mac Lane and Whitehead proved that crossed modules model homotopy 2-types. Since crossed modules are equivalent to certain double groupoids, these classical results allude to the significance of double categories. In this talk I will review 2-categories, recall Ehresmann's 1963 notion of double category, examine several examples, and describe connection pairs and folding structures on double categories. These double structures arise naturally when one categorifies the notion of category to the notion of pseudo algebra over the 2-theory of categories as in the context of conformal field theory. After a long gestation period, double categories (and their weakened version) are finding more and more applications, such as two dimensional Van Kampen theorems and extended field theory. Besides the definition of category, this talk will require no background in category theory and will be useful to anyone who is curious about double categories, their classical examples such as crossed modules, and modern applications.

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