X-Nico

Highly structured ring spectrum

The classical approach organizes this structure via operads, while the recent approach of Jacob Lurie deals with it using $\backslash infty$-operads in $\backslash infty$-categories.

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One can define an infinity-category of spectra (as done by Lurie).

Jacob Lurie

Another part of Lurie's work, is his article on topological field theories, where he sketches a classification of extended field theories using the language of infinity-categories.

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Infinity categories (in the form of Joyal's quasi-categories) are a convenient framework to do homotopy theory in abstract settings.

see also