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Spaces, Categories, and the Cobar Construction

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Mahmoud Zeinalian
The Graduate Center, CUNY Mathematics/MPIM
Thu, 2019-08-15 15:00 - 16:00
MPIM Lecture Hall
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In topology, there are two ways to think about fiber bundles on connected spaces. Generally speaking, parallel transport along loops starting and ending at a basepoint gives rise to a map of groups from the loop space (i.e., the space of loops) to the group of automorphisms of the fiber. This map determines, and is determined by, the bundle. A bundle also determines, and is determined by, a map of spaces from its base to the classifying space of the automorphisms of its fibre. 

Thinking of fiber bundles in two ways establishes a duality between connected spaces and groups. Every space has a loop space, which is a group, and every group has a classifying space, which is a space. Moreover, the classifying space of the loop space of a space is the space itself, and the loop space of the classifying space of the group is the group itself.

In algebra, one half of this duality assigns to a coalgebra (an avatar of a space) an algebra (an avatar of a group). This assignment, known as the cobar construction, was introduced in a 1956 paper of Frank Adams under the assumption of simply-connectedness. A deeper look at the categorical meaning of the construction reveals that the simply-connectedness assumption can be removed. This is joint work with Manuel Rivera which started when he visited the Max-Planck-Institute for a week in 2016. Some newer results are with Manuel Rivera and Felix Wierstra whom I met at Max-Planck last year.
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