**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\ngroups. 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\nsignificance of double categories. In this talk I will review 2-categories, recall Ehresmann's 1963 notion\nof double category, examine several examples, and describe connection pairs and folding structures on\ndouble categories. These double structures arise naturally when one categorifies the notion of category to\nthe 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\napplications, such as two dimensional Van Kampen theorems and extended field theory. Besides the\ndefinition of category, this talk will require no background in category theory and will be useful to\nanyone who is curious about double categories, their classical examples such as crossed modules, and\nmodern applications.

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