TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 9, Pages 3611–3623 S 0002-9947(03)03322-1 Article electronically published on May 15, 2003 LEFT-DETERMINED MODEL CATEGORIES AND UNIVERSAL HOMOTOPY THEORIES J. ROSICK ´ Y AND W. THOLEN Abstract. We say that a model category is left-determined if the weak equiv- alences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not left-determined, we show that its non-oriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural left-determined model category struc- ture. This is used to give another and, as we believe simpler, proof of a recent result of D. Dugger about universal homotopy theories. 1. Introduction Recall that a model category K is a complete and cocomplete category K equipped with three classes of morphisms C , W and F , called cofibrations, weak equivalences and fibrations, such that (1) (C , F∩W) and (C∩W, F ) are weak factorization systems and (2) W is closed under retracts (in the category K → of morphisms of K) and has the 2-out-of-3 property (see [Q], [H], [Ho] or [AHRT2]). Model categories were introduced by D. Quillen to provide a foundation of homotopy theory. Here a weak factorization system is a pair (L, R) of morphisms such that every morphism has a factorization as an L-morphism followed by an R-morphism, and R = L  , L =  R where L  (  R) consists of all morphisms having the right (left) lifting property w.r.t. L (R, respectively). The morphism l has a left lifting property with respect to a morphism r (or r has a right lifting property w.r.t. l) if in every commutative square A u  l  C r  B v  D there exists a diagonal d : B → C. Received by the editors June 1, 2002. 2000 Mathematics Subject Classification. Primary 55U35. The first author was supported by the Grant Agency of the Czech Republic under Grant 201/99/0310. The hospitality of the York University is gratefully acknowledged. The second author was supported by the Natural Sciences and Engineering Council of Canada. c 2003 American Mathematical Society 3611 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use