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seen Jan 25 at 17:13

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answered classifying $\infty$-toposes for topological/localic groups?
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comment Projective objects in HTT
What you describe works exactly the same way in the setting of quasi-categories: if C is a category which admits small colimits, then any cosimplicial object of C determines a pair of adjoint functors relating C to the quasicategory of simplicial spaces. If C is the quasi-category of spaces and your cosimplicial space is contractible in each degree, the resulting functor from simplicial spaces to spaces is given by taking the colimit.
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answered Projective objects in HTT
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comment Can motivic E_∞-ring spectra be strictified to commutative motivic symmetric ring spectra?
The example that you cite is a mistake: the category of symmetric spectra is not freely powered in the sense of DAG III. While it is possible to prove strictification results in the setting of symmetric spectra, one cannot do so simply by applying the results of DAG III.
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answered Is projective morphism with projective fiber flat?
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answered Lower Algebra: Modules over the monoidal category of abelian groups
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answered Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the groupoid objects?
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comment Stabilization of $\infty$-categories versus SW stabilization
That's right. To put it differently: stabilization is for "big" $\infty$-categories, and $SW_{\Sigma}$ is for "small" $\infty$-categories; passage to $\infty$-categories of inductive limits intertwines the two constructions. The construction $SW_{\Sigma}$ is perhaps more concrete than stabilization because you can perform it at the level of homotopy categories: this is because passage from an $\infty$-category $\mathcal{C}$ to its homotopy category commutes with sequential direct limits, but not with sequential inverse limits.