Theta-categories are symmetric monoidal infinity-categories with an LSym monad, and every neutralized Tannakian Theta-category is equivalent to the ind-perfect complexes on its stack of LSym-fiber functors.
Lax limits of model categories
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abstract
For a diagram of simplicial combinatorial model categories, we show that the associated lax limit, endowed with the projective model structure, is a presentation of the lax limit of the underlying $\infty$-categories. Our approach can also allow for the indexing category to be simplicial, as long as the diagram factors through its homotopy category. Analogous results for the associated homotopy limit (and other intermediate limits) directly follow.
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Theta-Categories and Tannakian duality
Theta-categories are symmetric monoidal infinity-categories with an LSym monad, and every neutralized Tannakian Theta-category is equivalent to the ind-perfect complexes on its stack of LSym-fiber functors.