model category, model -category Definitions Morphisms Universal constructions Refinements Producing new model structures Presentation of -categories Model structures for -groupoids for equivariant -groupoids fine model structure on topological G-spaces coarse model structure on topological G-spaces (Borel model structure) for rational -groupoids for rational equivariant -groupoids for -groups for -algebras general -algebras specific -algebras for stable/spectrum objects model structure on spectra model structure on ring spectra model structure on parameterized spectra model structure on presheaves of spectra for -categories for stable -categories for -operads for -sheaves / -stacks A model structure on spectra for orthogonal spectra. The category of orthogonal spectra is a presentation of the symmetric monoidal (∞,1)-category of spectra, with the special property that it implements the smash product of spectra in such a way as to give itself a symmetric monoidal model category of spectra: the model structure on orthogonal spectra. This implies in particular that with respect to this symmetric smash product of spectra an E-∞ ring is presented simply as a plain commutative monoid in orthogonal spectra. When defining a commutative ring as an abelian group equipped with an associative, commutative and untial bilinear pairing one evidently makes crucial use of the tensor product of abelian groups . That tensor product itself gives the category Ab of all abelian groups a structure similar to that of a ring, namely it equips it with a pairing that is a functor out of the product category of Ab with itself, satisfying category-theoretic analogs of the properties of associativity, commutativity and unitality. One says that a ring is a commutative monoid in the category Ab of abelian groups, and that this concept makes sense since itself is a symmetric monoidal category. Now in stable homotopy theory, as we have seen above, the category Ab is improved to the stable homotopy...