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We rework and generalize equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. There is a classical version which gives classical $\mathrm{\Omega <\#comment/>}$- $G$-spectra for any topological group $G$, but our focus is on the construction of genuine $\mathrm{\Omega <\#comment/>}$- $G$-spectra when $G$is finite. We also show what is and is not true when $G$is a compact Lie group. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for classical $G$-spectra for general $G$but fails for genuine $G$-spectra. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving an illuminating direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving a number of corrections to results and proofs in the literature. 

Abstract We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics . In the case of a finite total order [ n ], we enumerate all model structures, exhibiting a rich combinatorial structure encoded by Shapiro’s Catalan triangle. This is an application of previous work of the authors on the theory of $$N_\infty $$ N ∞ -operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of [ n ].