Multicategories Model All Connective Spectra

Abstract

There is a free construction from multicategories to permutative categories, left adjoint to the endomorphism multicategory construction. The main result shows that these functors induce an equivalence of homotopy theories. This result extends a similar result of Thomason, that permutative categories model all connective spectra.

Homotopy Equivalent Algebraic Structures in Multicategories and Permutative Categories

“…Because a natural transformation between functors induces a simplicial homotopy equivalence on nerves, we have the following application of Proposition 2.10. Its proof is similar to that of [JY∞b,2.12]. Proposition 2.12 ([GJO17, 2.9]).…”

“…◇ Proposition 6.12. [JY∞b,Proposition 5.13] The free permutative category construction given in Definitions 6.5 and 6.8 provides a 2-functor…”

“…Regarded as a 2-functor of underlying 2-categories, and restricting to the sub-2category PermCat st of strict symmetric monoidal functors, End has a left 2-adjoint F described in [EM09, Theorem 4.2] and [JY∞b,Section 5]. With respect to the stable Date: 28 February 2022.…”

“…Primary: 18M65; Secondary: 55P42, 18M05. 1 equivalences created by Segal's K-theory functor [Seg74], the adjoint pair (F, End) is shown to be an equivalence of homotopy theories in [JY∞b,Theorem 7.3].…”

“…The third of these sections recalls the Cat-multifunctor End from [JY∞a]. In Section 6 we review the definition of the left adjoint F from [EM06,JY∞b].…”