RIGIDIFICATION OF ALGEBRAS OVER MULTI-SORTED THEORIES JULIA E. BERGNER Abstract.We define the notion of a multi-sorted algebraic theory, which * *is a generalization of an algebraic theory in which the objects are of diff* *erent "sorts." We prove a rigidification result for simplicial algebras over t* *hese the- ories, showing that there is a Quillen equivalence between a model categ* *ory structure on the category of strict algebras over a multi-sorted theory * *and an appropriate model category structure on the category of functors from a * *multi- sorted theory to the category of simplicial sets. In the latter model st* *ructure, the fibrant objects are homotopy algebras over that theory. Our two main examples of strict algebras are operads in the category of simplicial se* *ts and simplicial categories with a given set of objects. 1.Introduction Algebraic theories are useful in studying many standard algebraic objects, su* *ch as monoids, abelian groups, and commutative rings. An algebraic theory provides* * a functorial means of describing particular algebraic objects without specifying * *gen- erating sets for the operations to which the objects are subject, or for the re* *lations between these operations [12]. Given a category C of algebraic objects, the ass* *oci- ated algebraic theory TC (if it exists) is a small category with products satis* *fying the property that specifying an object of C is equivalent to giving a product-prese* *rving functor TC ! Sets. Consider a category C with an associated algebraic theory TC. If a functor fr* *om TC to the category of simplicial sets preserves products, then it is essentiall* *y a simplicial object in C and is thus a combinatorial model for a topological obje* *ct in C, such as a topological group when C is the category of groups. We call such a functor a strict T-algebra (Definition 2.3). If the functor preserves product* *s up to homotopy, we call it a homotopy T-algebra (Definition 2.4). A homotopy T- algebra can be viewed as a simplicial set with the appropriate algebraic struct* *ure "up to homotopy," in a higher-order sense. Using an appropriate notion of weak equivalence on homotopy T-algebras [2, 5.6], the following result due to Badzio* *ch relates strict and homotopy T-algebras: Theorem 1.1. [2, 1.4] Let T be an algebraic theory. Any homotopy T-algebra is weakly equivalent as a homotopy T-algebra to a strict T-algebra. As a motivation for the work in this paper, consider the category of monoids. There is an associated algebraic theory TM , and thus a simplicial monoid can be specified by a TM -algebra. However, the notion of simplicial monoid can be ____________ Date: August 8, 2005. 2000 Mathematics Subject Classification. Primary: 18C10; Secondary 18G30, 18* *E35, 55P48. Key words and phrases. algebraic theories, model categories, operads, simpli* *cial categories. 1 2 J.E. BERGNER generalized to that of a simplicial category, by which we mean a category enric* *hed over simplicial sets, since a simplicial monoid is a simplicial category with o* *ne object. We would like to have a generalization of Badzioch's theorem which appl* *ies to simplicial categories. From the point of view of algebraic structure, the m* *ain difference between a simplicial monoid and a simplicial category with more than one object is that in the latter case the description of the algebraic structur* *e is more complicated, in that two morphisms can be combined by the composition operation only if they satisfy certain compatibility conditions on the domain a* *nd range. Therefore, we would like to describe a more general notion of theory whi* *ch is capable of describing algebraic structures in which the elements have various s* *orts or types, and in which the operations which can be used to combine a collection* * of elements depend on these sorts. There is in fact such a "multi-sorted" theory, TOCat, such that a product- preserving functor TOCat! Sets is essentially a category with object set O (Exa* *m- ple 3.5). A simplicial category, analogously, can be viewed as a product-preser* *ving functor TOCat! SSets. A simpler example of an algebraic structure which requires the use of a multi- sorted theory, which we will describe in more detail in Example 3.2, is the cas* *e of a group acting on a set. There are two sorts of elements, namely, the elements * *of the group and the elements of the set. Two elements of the group can be combined via multiplication, or an element can be inverted. An element of the group and * *an element of the set can be combined via the group action. However, the elements of the set cannot be combined with one another in any nontrivial way, so the operations which we allow depend on the sort of element involved. The example of a module over a ring is constructed similarly in Example 3.3. Another application of the notion of a multi-sorted theory gives a convenient description of an operad. In Example 3.4, we characterize the theory Toperadof operads. An operad in the category of sets is then a product-preserving functor from Toperadto the category of sets. A multi-sorted theory T is a category with products, so we can define strict * *and homotopy T-algebras as before (see Definitions 3.6 and 3.7). Using a definition of weak equivalence for homotopy T-algebras (Proposition 4.12), the main result which we prove for multi-sorted theories is the following generalization of The* *orem 1.1: Theorem 1.2. Let T be a multi-sorted algebraic theory. Any homotopy T-algebra is weakly equivalent as a homotopy T-algebra to a strict T-algebra. As Badzioch does, we will actually prove a stronger statement in terms of a Quillen equivalence of model category structures (Theorem 5.1). Using our example of the theory Toperadof operads, an operad in the category of simplicial sets is a strict Toperad-algebra. A homotopy operad, or sequence* * of simplicial sets with the structure of an operad only up to homotopy, is then a homotopy Toperad-algebra and can be rigidified to a strict operad using this th* *eorem. Returning to the example of simplicial categories, let O be a set and SCO the category of simplicial categories with object set O in which the morphisms are * *the identity on the objects. In [3], we use Theorem 1.2 to prove a relationship bet* *ween SCO and the category of Segal categories with the same set O in dimension zero.* * In [4], we use the ideas of this proof to prove an analogous relationship between * *the category of all small simplicial categories and the category of all Segal categ* *ories. MULTI-SORTED THEORIES 3 Throughout this paper, we frequently work in the category of simplicial sets, SSets. Recall that a simplicial set is a functor op ! Sets, where denotes the cosimplicial category whose objects are the finite ordered sets [n] = (0, .* * .,.n) and whose morphisms are the order-preserving maps. The simplicial category op is then the opposite of this category. Some examples of simplicial sets are, f* *or each n 0, the n-simplex [n], its boundary `[n], and, for any 0 k n, the simplicial set V [n, k], which is [n] with the kth face removed. More informat* *ion about simplicial sets can be found in [8, I.1]. In this paper, we begin by recalling the definition of an algebraic theory and stating some of its basic properties. Using this definition as a model, we then* * define a multi-sorted theory. We should note here that this notion is not a new one; s* *imilar definitions are given by Ad'amek and Rosick'y [1, 3.14] and by Boardman and Vogt [5, 2.3]. (The still more general definition of a finite limit theory is used b* *y Johnson and Walters [11].) Because our perspective is slightly different, however, we * *will give a precise definition followed by some examples. Given a multi-sorted theory T, we define strict and homotopy T-algebras over a multi-sorted theory T and sh* *ow that the existence of a model category structure on the category of all T-algeb* *ras. We also show the existence of a model category structure on the category of all functors T ! SSets in which the fibrant objects are the homotopy T-algebras. We then show that there is a Quillen equivalence between these two model categorie* *s. Acknowledgments. I am grateful to Bill Dwyer for suggesting this approach to studying simplicial categories and operads. I would also like to thank Bernard Badzioch and Michael Johnson for helpful conversations about this work. 2. A Summary of Algebraic Theories We first recall the definition of an ordinary algebraic theory. More details * *about algebraic theories can be found in chapter 3 of [6]. Definition 2.1. An algebraic theory T is a small category with finite products and which has as objects Tn for n 0 together with, for each n, an isomorphism Tn ~=(T1)n. In particular, T0 is the terminal object in T. We can use theories to describe certain algebraic categories, namely those wh* *ich are determined by sets with n-ary operations for each n 2. Consider a category C such that there exists a forgetful functor : C ! Sets taking an object of C to its underlying set, and its left adjoint (a free funct* *or) L : Sets ! C. In other words, C is required to have free objects. If the category C and the a* *djoint pair ( , L) satisfy some additional technical conditions (see [6, 3.9.1] for de* *tails), we will call C an algebraic category. Given an object X of an algebraic category C, we have a natural map jX : L (X) ! X and given a set A, we have another map "A : A ! L(A). 4 J.E. BERGNER In order to discuss a theory over the algebraic category C, consider a set A together with a map mA : L(A) ! A satisfying two conditions: the composite map A _"X__// L(A)mA_//_A is the identity map on A, and the diagram L(mA) // mA ( L)2A __________//_ L(A)__//_A jLAL is a coequalizer. These maps define an algebraic structure on the set A, specif* *ically the structure possessed by the objects of C [12]. For example, if C = G, the category of groups, is the forgetful functor tak* *ing a group to its underlying set, and L is the free group functor taking a set to * *the free group on that set, then these two conditions are precisely the ones defini* *ng a group structure on the set A. We would like to discuss the algebraic theory T corresponding to C to simplify this way of talking about algebraic structure. Let X be an object of C. We cons* *ider natural transformations of functors C ! Sets _(-)_x_._.x.-(-)z_______"! (-). n Using the adjointness of and L, we have that (X) ~=Hom Sets({1}, (X)) ~=Hom C(L{1}, X) where {1} denotes the set with one object, and we can think of L{1} as the free object in C on one generator, since L is the free functor. Hence, we have (X)n = Hom Sets({1}, (X))n a = Hom Sets( {1}, (X)) n = Hom Sets({1, . .,.n}, (X)) = Hom C(L{1, . .,.n}, X). Now, by Yoneda's Lemma we have a bijection between the set of natural maps (X)n ! (X) and the set Hom C(L{1}, L{1, . .,.n}). The objects L{OE} = T0, L{1} = T1, . .,.L{1, . .,.n} = Tn, . . . are the objects of the algebraic theory T corresponding to C. The morphisms are the opposites of the ones in C between these objects. More precisely stated, T * *is the opposite of the full subcategory of representatives of isomorphism classes of f* *initely generated free objects of C. Given an object X of C, define a functor HX : T ! Sets such that HX (L{1, . .,.n}) = Hom C(L{1, . .,.n}, X) = (X)n. Now, the algebraic category C is equivalent to the category of the functors HX , namely, the full subcategory of the category of functors A : T ! Sets whose obj* *ects preserve products, or those for which the canonical map A(Tn) ! A(T1)n induced by the n projection maps is an isomorphism of sets for all n 0 [12]. MULTI-SORTED THEORIES 5 Example 2.2. Let G denote the category of groups. Consider the full subcategory of G whose objects Tn are the free groups on n generators for n 0 (where T0 is the trivial group). The opposite of this category is TG, the theory of groups. * *It can be shown that the category of product-preserving functors TG ! Sets is equivale* *nt to the category G. Product-preserving functors from the theory T to Sets are called algebras over T. We would also like to consider functors from an algebraic theory to the cate* *gory SSets of simplicial sets. To do so, we must first define a simplicial algebra * *over a theory T. For simplicity, we will also use the term "algebra" to refer to th* *ese simplicial algebras. Definition 2.3. [2, 1.1] Given an algebraic theory T, a (strict simplicial) T-a* *lgebra A is a product-preserving functor A : T ! SSets. Namely, the canonical map A(Tn) ! A(T1)n, induced by the n projection maps Tn ! T1, is an isomorphism of simplicial sets. In particular, A(T0) is the one-point space [0]. The category of all T-algebras will be denoted AlgT. Similarly, we have the notion of a homotopy algebra, for which we only require products to be preserved up to homotopy: Definition 2.4. [2, 1.2] Given an algebraic theory T, a homotopy T-algebra is a functor X : T ! SSets which preserves products up to homotopy. The functor X preserves products up to homotopy if for each n the canonical map X(Tn) ! X(T1)n is a weak equivalence of simplicial sets. In particular, we assume that X(T0) * *is weakly equivalent to [0]. There exists a forgetful functor, or evaluation map, UT : AlgT ! SSets such that UT(A) = A(T1). This functor has a left adjoint, the free T-algebra fu* *nctor FT : SSets ! AlgT where, if Y is any simplicial set, a FT(Y )(T1) = Hom T(Tn, T1) x Y n= ~ n 0 where the identifications come from the structure of the algebraic theory [2, 2* *.1]. 3.Multi-Sorted Algebraic Theories We now generalize the definition of an algebraic theory to that of a multi-so* *rted theory. Definition 3.1. Given a set S, an S-sorted algebraic theory (or multi-sorted th* *eory) T is a small category with objects Tff_nwhere ff_n=< ff1, . .,.ffn > for ffi 2 * *S and n 0 varying, and such that each Tff_nis equipped with an isomorphism Yn Tff_n~= Tffi. i=1 6 J.E. BERGNER For a particular ff_n, the entries ffi can repeat, but they are not ordered. In* * other words, ff_nis a an n-element subset with multiplicities. There exists a termin* *al object T0 (corresponding to the empty subset of S). Notation. Lower-case Greek letters (with or without subscripts), say ff or ffi,* * will be used to denote objects of S, whereas underlined ones, say ff_nor simply ff_,* * will denote an n-element subset of objects of S (with multiplicities) for n 1. Notice that a theory with a single sort is a theory in the sense of the previ* *ous section. We would like to speak of multi-sorted theories corresponding to categories w* *hich are analogous to the algebraic categories which we had in the ordinary case. Ho* *w- ever, because we have several objects (or "sorts") Tffwhere we only had the obj* *ect T1 in an ordinary theory, we have many pairs of adjoint functors, one for each * *sort. Let C be a category with coproducts such that given any element fi 2 S, we have a forgetful functor fi: C ! Sets and its left adjoint, the free functor Lfi: Sets ! C. For each object X in C and element fi 2 S, we have a map jX,fi: Lfi fi(X) ! X and, for each set A a map "A,fi: A ! fiLfi(A). As before, in order to make sense of the notion of theory, we consider a set A together with, for each fi 2 S, a map mA,fi: fiLfi(A) ! A satisfying two conditions: the composite map "A,fi mA,fi A ____//_ fiLfi(A)_//_A is the identity map on A, and the diagram _fiLfi(mA,fi)//_ mA,fi ( fiLfi)2A_________// fiLfi(A)_//_A fijLfiA,fiLfi is a coequalizer. These maps define a "multi-sorted algebraic structure" on C. * *In particular, we have a notion of composition for certain elements of C depending on their sorts. Given this structure, we can now construct the S-sorted theory corresponding to the category C. Given ffi, fi 2 S, we consider the natural transformations of functors C ! Se* *ts ff1(-) x . .x. ffn(-) ! fi(-). As before, we can apply these functors to an object X of C and rewrite to obtain a map Hom Sets({1}, ff1(X)) x . .x.Hom Sets({1}, ffn(X)) ! Hom Sets({1}, fi(X)) which, by adjointness, is equivalent to Hom C(Lff1{1}, X) x . .x.Hom C(Lffn{1}, X) ! Hom C(Lfi{1}, X). MULTI-SORTED THEORIES 7 Since C has coproducts, we can rewrite this map as Hom C(Lff1{1} q . .q.Lffn{1}, X) ! Hom C(Lfi{1}, X). Then, by Yoneda's Lemma, there is a bijection between the set of natural transf* *or- mations ff1(-) x . .x. ffn(-) ! fi(-) and the set a Hom C (Lfi{1}, Lffk{1}). k The objects of the theory T corresponding to C are given by finite coproducts of "free" objects Lffk{1} of C for all choices of ffk, and the morphisms are the o* *pposites of those of C. Let X be an object of C and (ff1, . .,.ffn, fi) 2 Sn+1 an (n + 1* *)-tuple of elements in S. We define the map HX,ff1,...,ffn,fi: Top ! Sets such that an an HX,ff1,...,ffn,fi( Lffk{1}) = Hom C( Lffk{1}, X) = ff1(X) x . .x. ffn(X). k=1 k=1 If the category C satisfies analogous conditions to those of [6, 3.9.1], then C* * is equivalent to the category of all such functors. We now consider with some examples. Example 3.2. Consider pairs (G, X), where G is a group and X is a set. We can obtain two different 2-sorted theories from these pairs, one corresponding * *to the category of unstructured pairs, and the other corresponding to the category* * of pairs (G, X) with a given action of the group G on the set X. In each case, we have two forgetful functors and their respective left adjoin* *ts. We begin with the category of unstructured pairs, which we denote P. The objects are the pairs (G, X) and the morphisms (G, X) ! (H, Y ) consist of pairs (', f) where ' : G ! H is a group homomorphism and f : X ! Y is a map of sets. For each sort i = 1, 2 we have a forgetful map i: P ! S and its left adjoint Li: S! P. When i = 1, we have, for any group G and set X, 1(G, X) = G (where on the right hand side G denotes the underlying set of the group G) and for any set S L1(S) = (FS, OE) where FS denotes the free group on the set S. Similarly, when i = 2, we define 2(G, X) = X and L2(S) = (e, S) where e denotes the trivial group. In order to determine the objects of our theory, consider functors Fi,j: P ! Sets 8 J.E. BERGNER such that Fi,j(G, X) = Gix Xj. In other words, Fi,j(G, X) = Hom P(L1(i) q L2(j), (G, X)) where i denotes the set with i elements and similarly for j. The objects of the theory will be representatives of the isomorphism classes of the L1(i) q L2(j) * *for all choices of i and j. This coproduct in P is defined to be the coproduct of e* *ach element in the pairs. Thus we have (G, X) q (G0, X0) = (G * G0, X x X0) where G * G0denotes the free product of groups. So, our corresponding theory is the opposite of the full subcategory of P whose objects are of the form L1(i)qL* *2(j). When we equip each pair (G, X) with an action of G on X to obtain another category which we denote PA , the process is identical until we have to specify* * the coproduct, since in this case we need to take the group actions into account. We then have the coproduct in PA (G, X) q (G0, X0) = (H, (H xG X) q (H xG0X0)) where H = G * G0and we have defined H xG X = {(h, x)|h 2 H, x 2 X}= ~ when (hg, x) ~ (h, gx) for any g 2 G. We can now take the opposite of a full subcategory of PA as above to obtain the corresponding theory. Example 3.3. A very similar example is the case of a commutative ring R and an R-module A. Again, we have two different 2-sorted theories: one where we simply have a ring R and regard A merely as an abelian group, and the other where we consider the R-module structure on A. As before, we begin with PR , the category of pairs with no additional struct* *ure. We have the forgetful map 1 : PR ! Sets where 1(R, A) = R for any ring R and abelian group A, where on the right side R is the underlying set of the ring R. Its left adjoint is the functor L1 : Sets ! PR where for any set S, L1(S) = (Z[S], e), where Z[S] is the free commutative ring* * on the set S and e denotes the trivial (abelian) group. Then we have the map 2 : PR ! Sets such that 2(R, A) = A, where again on the right hand side A is the underlying set of the abelian group A. Its left adjoint is the map L2 : Sets ! PR where L2(S) = (Z, F AS) where F AS denotes the free abelian group on the set S. To know what the objects of this 2-sorted theory are, we need to know what the coproduct is. We have that (R, A) q (R0, A0) = (R Z R0, A A0), and from there we can obtain a theory as in the previous example. MULTI-SORTED THEORIES 9 Now consider the category PM whose objects are pairs (R, A) where R is a ring and A is a module over A. If A and A0 are modules over R and R0, respectively, we have a coproduct similar to that in the group action example. So, we say that (R, A) q (R0, A0) = (R Z R0, (R0 Z A) (R Z A0)). Example 3.4. Another example of a multi-sorted theory is the N-sorted theory of operads. Recall that an operad in the category of sets is a sequence of sets {P (k)}k 0, a unit map 1 2 P (1), and operations P (k) x P (j1) x . .x.P (jk) ! P (j1 + . .+.jk) satisfying associativity, unit, and equivariance conditions [14, II.1.4]. There is a notion of a free operad on n generators at levels m1, . .,.mn [14, xII.1.9]. Specifically, such a free operad has, for each 1 i n, a generato* *r in P (mi). Note that the values of mi can repeat. For example, one can think of the free operad on n generators, each at level 1, as the free monoid on n generator* *s. In the category of operads, consider the full subcategory of free operads. Ea* *ch object in this category, then, can be described as the free operad on n generat* *ors at levels m1, . .,.mn for some n 0 and m1, . .,.mn. The opposite of this categor* *y is the theory of operads. Using the notation we have set up for multi-sorted theor* *ies, we have that Tfffor ff 2 N is just the free operad on one generator at level ff* * and for ff_n=< ff1, . .,.ffn >, we have that Tff_nis the free operad on n generator* *s at levels ff1, . .f.fn. There is also a notion of non- operads, where we no longer have an action of the symmetric group or an equivariance condition [14, II.1.14]. We can define t* *he theory of non- operads analogously, taking the opposite of the full subcategor* *y of free non- operads in the category of all non- operads. Example 3.5. Consider the category OCat whose objects are the categories with a fixed object set O and whose morphisms are the functors which are the identity map on the objects. There is a theory TOCat associated to this category. The objects of the theory are categories which are freely generated by directed gra* *phs with vertices corresponding to the elements of the set O. This theory will be s* *orted by pairs of elements in O, corresponding to the morphisms with source the first element and target the second. In other words, this theory is (O x O)-sorted. In particular, consider ff = (x, y) 2 O x O. Then, if x 6= y, Tffis the categ* *ory with object set O and one nonidentity morphism with source x and target y. If x = y, then Tffis the category freely generated by one morphism from x to itself and no other nonidentity morphisms. In general, if ff_=< ff1, . .,.ffn >, then Tff_is the category with object se* *t O and morphisms freely generated by the morphisms given for each ffk as in the previo* *us case. Consider the forgetful functor ff: OCat ! Sets where, for any object X in C, ff(X) = Hom X(x, y). Its left adjoint then is the free functor Lffdefined by, for a set A, ( Lff(A) = C withHom C (x, y) = A ifx 6= y C withHom C (x, y) = FA ifx = y where FA is the free monoid generated by the set A and where in each case there are no other nonidentity morphisms in the category C. 10 J.E. BERGNER As with ordinary algebraic theories, we can define strict and homotopy T-alge* *bras for a multi-sorted theory T. Definition 3.6. Given an S-sorted theory T, a (strict simplicial) T-algebra is a product-preserving functor A : T ! SSets. Here, product-preserving means that the canonical map Yn A(Tff_n) ! A(Tffi) i=1 induced by the projections Tff_n! Tffifor all 1 i n is an isomorphism of simplicial sets. As before, we will denote the category of strict T-algebras by AlgT. Definition 3.7. Given an S-sorted theory T, a homotopy T-algebra is a functor X : T ! SSets which preserves products up to homotopy. The functor X preserves products up to homotopy if the canonical map Yn X(Tff_n) ! X(Tffi) i=1 induced by the projection maps Tff_n! Tffifor all 1 i n is a weak equivalen* *ce of simplicial sets. We would like to prove a rigidification result similar to Theorem 1.1 above. * *We begin by finding model category structures for T-algebras and homotopy T-algebr* *as. We then find a Quillen equivalence between these model category structures T- algebras for any multi-sorted theory T. 4. Model Category Structures In this section, we define, given a multi-sorted theory T, model category str* *uc- tures on the category of diagrams T ! SSets and on the category of T-algebras. We begin with a review of model category structures. Recall that a model category structure on a category C is a choice of three d* *is- tinguished classes of morphisms: fibrations, cofibrations, and weak equivalence* *s. A (co)fibration which is also a weak equivalence will be called an acyclic (co)fi* *bration. With this choice of three classes of morphisms, C is required to satisfy axioms MC1-MC5 [7, 3.3]. An object X in C is fibrant if the unique map X ! * from X to the terminal object is a fibration. Dually, X is cofibrant if the unique map OE ! X from the* * initial object to X is a cofibration. The factorization axiom MC5 guarantees that each object X has a weakly equivalent fibrant replacement bXand a weakly equivalent cofibrant replacement eX. These replacements are not necessarily unique, but th* *ey can be chosen to be functorial in the cases we will use [10, 1.1.3]. The model category structures which we will discuss are all cofibrantly gener* *ated. A cofibrantly generated model category C is a model category for which there are two sets of morphisms, one of generating cofibrations and one of generating acy* *clic cofibrations, such that a map is a fibration if and only if it has the right li* *fting property with respect to the generating acyclic cofibrations, and a map is an a* *cyclic fibration if and only if it has the right lifting property with respect to the * *generating cofibrations [9, 11.1.2]. To describe such model categories, we make the follow* *ing definition. MULTI-SORTED THEORIES 11 Definition 4.1. [9, 10.5.2] Let M be a category and I a set of maps in C. Then an I-injective is a map which was the right lifting property with respect to ev* *ery map in I. An I-cofibration is a map with the left lifting property with respect* * to every I-injective. We are now able to state the theorem that we will use to prove our model category structures in this paper. Theorem 4.2. [9, 11.3.1] Let M be a category which has all finite limits and co* *l- imits. Suppose that M has a class of weak equivalences which satisfies the "two* * out of three property" (model category axiom MC2) and which is closed under retract* *s. Let I and J be sets of maps in M which satisfy the following conditions: (1) Both I and J permit the small object argument [9, 10.5.15]. (2) Every J-cofibration is an I-cofibration and a weak equivalence. (3) Every I-injective is a J-injective and a weak equivalence. (4) One of the following conditions holds: (i)A map that is an I-cofibration and a weak equivalence is a J-cofibra* *tion, or (ii)A map that is both a J-injective and a weak equivalence is an I- injective. Then there is a cofibrantly generated model category structure on M in which I * *is a set of generating cofibrations and J is a set of generating acyclic cofibrati* *ons. We will refer to the standard model category structure on the category SSets * *of simplicial sets. In this case, a weak equivalence is a map of simplicial sets f* * : X ! Y such that the induced map |f| : |X| ! |Y | is a weak homotopy equivalence of topological spaces. The cofibrations are inclusions, and the fibrations are the* * maps with the right lifting property with respect to the acyclic cofibrations [8, I.* *11.3]. This model category structure is cofibrantly generated; the generating cofibrat* *ions are the maps `[n] ! [n] for n 0, and the generating acyclic cofibrations are the maps V [n, k] ! [n] for n 1 and 0 k n. We will also need the notion of a simplicial model category M, or one for whi* *ch an object X K is defined for any object X of M and simplicial set K. In particu* *lar, M is a simplicial category with a model structure which is required to satisfy * *several axioms [9, 9.1.6]. Definition 4.3. For any objects X and Y in a simplicial model category M, the function complex is the simplicial set Map (X, Y ). It is important to note that a function complex is only homotopy invariant in the case that X is cofibrant and Y is fibrant. For the general case, we have t* *he following definition: Definition 4.4. [9, 17.3.1] A homotopy function complex Map h(X, Y ) in a sim- plicial model category M is the simplicial set Map (Xe, bY) where Xe is a cofib* *rant replacement of X in M and bYis a fibrant replacement for Y . Several of the model category structures that we will use will be obtained by localizing a given model category structure with respect to a map or a set of m* *aps. Suppose that P = {f : A ! B} is a set of maps with respect to which we would like to localize a model category M. 12 J.E. BERGNER Definition 4.5. A P -local object X is a fibrant object of M such that for any f : A ! B in P , the induced map on homotopy function complexes f* : Map h(B, W ) ! Map h(A, W ) is a weak equivalence of simplicial sets. A map g : X ! Y in M is then a P -loc* *al equivalence if for every local object W , the induced map on homotopy function complexes g* : Map h(Y, W ) ! Map h(X, W ) is a weak equivalence of simplicial sets. Given a multi-sorted theory T, let SSetsT denote the category of functors T ! SSets. Note that the category AlgT of strict T-algebras is a full subcategory * *of SSetsT. The category SSetsT is an example of a category of diagrams. In general, given any small category D, there is a category SSetsD of D-diagrams in SSets, or fun* *ctors D ! SSets. We can obtain two model category structures on SSetsD by the following results. Theorem 4.6. [8, IX 1.4] Given the category SSetsD of C-diagrams of simplicial sets, there is a simplicial model category structure SSetsDfin which the weak e* *quiva- lences and fibrations are objectwise and in which the cofibrations are the maps* * which have the left lifting property with respect to the maps which are both fibratio* *ns and weak equivalences. Theorem 4.7. [8, VIII 2.4] There is a simplicial model category SSetsDcin which the weak equivalences and the cofibrations are objectwise and in which the fibr* *ations are the maps which have the right lifting property with respect to the maps whi* *ch are cofibrations and weak equivalences. We now return to the situation where our small category is a multi-sorted the* *ory T. We would like to have an evaluation map and its left adjoint as in the ordin* *ary case (see the end of section 2 above), but here we will have one for each fl 2 * *S. These evaluation maps look like Ufl: AlgT ! SSets such that Ufl(A) = A(Tfl) for any T-algebra A. Each functor Uflhas a left adjoint, the free functor Ffl: SSets ! AlgT such that, given a simplicial set Y and object Tfi_in T, a Ffl(Y )(Tfi_) = (Hom (Tfl,...,fl, Tfi_) x Y n)= ~ n 0 where the equivalence is as in the ordinary case (see the end of section 2 abov* *e). Given a theory T (regular or multi-sorted), define a weak equivalence in the category AlgT of T-algebras to be a map which induces a weak equivalence of simplicial sets after applying the evaluation functor Ufffor each sort ff. Simi* *larly, define a fibration of T-algebras to be a map f such that Uff(f) is a fibration * *of MULTI-SORTED THEORIES 13 simplicial sets. Then define a cofibration to be a map with the left lifting pr* *operty with respect to the maps which are fibrations and weak equivalences. The following theorem is a generalization of a result by Quillen [15, II.4]. Theorem 4.8. Let T be an S-sorted theory. There is a cofibrantly generated model category structure on AlgT with the weak equivalences, fibrations, and cofibrat* *ions as defined above. We first need to define sets I and J which will be our candidates for generat* *ing sets of cofibrations and acyclic cofibrations, respectively. We first define I * *to be the set of maps Uff`[n] ! Uff [n] for each n 0 and ff 2 S. Similarly, define J to* * be the set of all maps UffV [n, k] ! Uff [n] for each n 1, 1 k n, and ff 2 S* *. We now use these sets to prove our model category structure. Proof.We need to show that the conditions of 4.2 are satisfied for these sets I* * and J. The existence of limits and colimits and the conditions on the weak equivale* *nces follow just as they do in the case where T is an ordinary theory [15, II.4]. We now show that I and J satisfy the small object argument. Consider some T- algebra X, which can be written as a directed colimit colimm(Xm ) and can there* *fore be computed objectwise. Thus, we can show that `[n] is small: Hom AlgT(Fff`[n], colimm(Xm=))HomSSets( `[n], Uffcolimm(Xm )) = Hom SSets( `[n], colimm(UffXm )) = colimmHom SSets( `[n], UffXm ) = colimmHom AlgT(Fff`[n], Xm ). The object V [n, k] can be shown to be small analogously, so we have proved sta* *te- ment (1). We first prove statements (3) and (4)(ii), namely that an I-injective is prec* *isely a J-injective and a weak equivalence. An I-injective f : X ! Y by definition has the right lifting property with respect to the maps in I, but using the adjoint* *ness of Uffand Fff, this fact is equivalent to f's being an acyclic fibration. But t* *hen f is a weak equivalence and has the right lifting property with respect to the ma* *ps in J, again by adjointness. It remains to prove statement (2). Suppose i : A ! B is a J-cofibration. Then it has the right lifting property with respect to the fibrations. Another adjo* *int- ness argument shows that i therefore is an I-cofibration and a weak equivalence, completing the proof. We now need a model category structure on the category of homotopy T-algebras. However, the category of homotopy T-algebras does not have all finite limits and colimits (axiom MC1). Thus, we instead define a model category structure on all diagrams T ! SSets in such a way that the fibrant objects are homotopy T-algebr* *as. The following theorem holds for model categories M which are left proper and cellular. We will not define these conditions here, but refer the reader to [9,* * 13.1.1, 12.1.1] for more details. It can be shown that SSetsT satisfies both these cond* *itions [9, 13.1.14, 12.5.1]. Theorem 4.9. [9, 4.1.1] Let M be a left proper cellular model category and P a * *set of morphisms of M. There is a model category structure LP M on the underlying category of M such that: 14 J.E. BERGNER (1) The weak equivalences are the P -local equivalences. (2) The cofibrations are precisely the cofibrations of M. (3) The fibrations are the maps which have the right lifting property with r* *espect to the maps which are both cofibrations and P -local equivalences. (4) The fibrant objects are the P -local objects. To localize the model structure SSetsTf, we first need an appropriate map. To do so for ordinary algebraic theories, Badzioch [2, 2.9] uses free diagrams whi* *ch are corepresented by the objects Tn of the theory T. In particular the n projec* *tion maps Tn ! T1 induce maps a Hom (T1, -) ! Hom (Tn, -). n He defines his localization with respect to these maps. We would like to define similar free diagrams in a multi-sorted theory. For each ff_n=< ff1, . .,.ffn > and 1 i n, there exists a projection map Tff_n! Tffiinducing a map Hom T(Tffi, -) ! Hom T(Tff_n, -). Taking the coproduct of all such maps results in a map an ~ff_n: Hom T(Tffi, -) ! Hom T(Tff_n, -). i=1 These maps are the ones which we will use to localize SSetsT. We define P to be the set of all such maps ~ff_nfor each ff_nand n 0. Proposition 4.10. There is a model category structure LSSetsT on the category SSetsT with weak equivalences the P -local equivalences, cofibrations as in SSe* *tsTf, and fibrations the maps which have the right lifting property with respect to t* *he maps which are cofibrations and weak equivalences. Proof.This proposition is a special case of Theorem 4.9. The following propositions are proved by Badzioch for ordinary theories, and * *his proofs follow for multi-sorted theories. Proposition 4.11. [2, 5.5] An object Z of LSSetsT is fibrant if and only if it * *is a homotopy T-algebra which is fibrant as an object of SSetsTf. Proposition 4.12. [2, 5.6] If X and X0 are homotopy T-algebras in SSetsT and there is a P -local weak equivalence f : Z ! X0, then f is also a weak equivale* *nce in SSetsTf, i.e. an objectwise weak equivalence. Proposition 4.13. [2, 5.8] A map f : X ! X0 is a P -local equivalence if and only if for any T-algebra Y which is fibrant in SSetsTc, the induced map of fun* *ction complexes f* : Map(X0, Y ) ! Map(X, Y ) is a weak equivalence of simplicial sets. These results can actually be stated in more generality; they are really just statements about the fibrant objects in a localized model category structure (s* *ee chapter 3 of [9] for more details). Hence, we can consider the category LSS etsT to be our homotopy T-algebra model category structure. MULTI-SORTED THEORIES 15 5.Rigidification of Algebras over Multi-Sorted Theories We are now able to prove the following statement, which is a stronger version* * of Theorem 1.2: Theorem 5.1. There is a Quillen equivalence of model categories between AlgT and LSSetsT. We begin with the necessary definitions. Recall that an adjoint pair F : C____//Do:oR_ (where F is the left adjoint and R is the right adjoint) is defined by a map ' : Hom D(F X, Y ) ! Hom C(X, RY ) and is sometimes written as the triple (F, R, ') [13, IV.1]. Definition 5.2. [10, 1.3.1] If C and D are model categories, then the adjoint p* *air (F, R, ') is a Quillen pair if one of the following statements is true: (1) F preserves cofibrations and acyclic cofibrations. (2) R preserves fibrations and acyclic fibrations. Definition 5.3. [10, 1.3.12] A Quillen pair is a Quillen equivalence if for all* * cofi- brant X in C and fibrant Y in D, a map f : F X ! Y is a weak equivalence in D if and only if the map 'f : X ! RY is a weak equivalence in C. We need to find an adjoint pair of functors between AlgT and LSSetsT and prove that it is a Quillen equivalence. Let JT : AlgT ! SSetsT be the inclusion functor. We need to show we have an adjoint functor taking an arbitrary diagram in SSetsT to a T-algebra. We first make the following definit* *ion. Definition 5.4. Let C be a small category and SSetsD the category of functors C ! SSets. Let P be a set of morphisms in SSetsD . An object Y of SSetsD is strictly P -local if for every morphism f : A ! B in P , the induced map on fun* *ction complexes (Definition 4.3) f* : Map (B, Y ) ! Map (A, Y ) is an isomorphism of simplicial sets. A map g : C ! D in SSetsD is a strict P -* *local equivalence if for every strictly P -local object Y in SSetsD , the induced map g* : Map (D, Y ) ! Map (C, Y ) is an isomorphism of simplicial sets. Now, given a category of C-diagrams in SSets and the full subcategory of stri* *ctly P -local diagrams for some set P of maps, we have the following result. Lemma 5.5. Consider two categories, the category of all diagrams X : C ! SSets and the category of strictly local diagrams with respect to the set of maps P =* * {f : A ! B}. Then the forgetful functor from the category of strictly local diagrams* * to the category of all diagrams has a left adjoint. 16 J.E. BERGNER Proof.Without loss`of generality, assume that we have just one map f in P ; oth* *er- wise replace f by fffff. Given an arbitrary diagram X, we would like to const* *ruct a strictly local diagram from X. So, suppose that X is not strictly local, i.e.* * the map f* : Map (B, X) ! Map (A, X) is not an isomorphism. First suppose that f* fails to be surjective. Then we ob* *tain an object X0 as the pushout in the following diagram: ` // A!X A _____X | | | | ` |fflffl fflffl| A!X B _____//X0 where each coproduct is taken over all maps A ! X. If X fails to be injective, * *we obtain X0 by taking the pushout ` ` (B A B) _____//X | | | | `|fflffl |fflffl B ________//X0 ` again where the coproduct is over all maps B A B ! X, and where the map a B B ! B A is the fold map. (If f* is neither injective nor surjective, apply one of the a* *bove pushouts, then apply the other to the new object X0 rather than to the original X.) In the first case, i.e. where f* is not surjective, for any strictly local ob* *ject Y we obtain a commutative diagram Map (X0, Y )___//_Map(B, Y ) |~=| |~=| fflffl| fflffl| Map (X, Y )____//_Map(A, Y ) showing that the map X ! X0 is a strict local equivalence since f : A ! B is. In the second case, where f* is not injective, we obtain a similar diagram, b* *ut it takes more work to show that the map X ! X0 is a strict local equivalence. We first obtain the diagram Map (X0, Y )______//_Map(B, Y ) | | | | fflffl| fflffl|` Map (X, Y )____//Map(B A B, Y ) It then suffices to show that the right hand vertical arrow is an isomorphism. MULTI-SORTED THEORIES 17 ` Recall that the object B A B is defined as the pushout in the diagram A _______//_B | | | | fflffl| fflffl|` B ____//_B A B which enables us to look at the diagram ` Map(B A B, Y )____//_Map(B, Y ) | ~| | =| fflffl| fflffl| Map (B, Y )______//_Map(A, Y ). Hence the map a B ! B B A is a strict local equivalence. But, this map fits into a composite * * * * * * ` B_____//__________33__________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_______________________________B//A_BB id Since the identity map is a strict local equivalence, it follows that the map a B B ! B A is a strict local equivalence, since it can be shown that the strictly local eq* *uivalences satisfy the "two out of three property" (model category axiom MC2). Therefore, in either case, the map X ! X0is a strict local equivalence. Howev* *er, we still do not know that the map Map (B, X0) ! Map (A, X0) is an isomorphism. So, we repeat this process, taking a (possibly transfinite) * *colimit to obtain a local object eXsuch that there is a local equivalence X ! eX. It suffices to show that the functor which takes a diagram X to the local dia* *gram eXis left adjoint to the forgetful functor. So if J is the forgetful functor fr* *om the category of strictly local diagrams to the category of all diagrams and K is the functor we have just defined, we need to show that Map (X, JY ) ~=Map (KX, Y ) for any diagram X and strictly local diagram Y . But, proving this statement is equivalent to showing that Map (X, Y ) ~=Map (Xe, Y ) which was shown above for each step, and it still holds for the colimit. We consider the category SSetsT of T-diagrams and the strict localization with respect to the set of maps an P = { Hom T(Tffi, -) ! Hom T(Tff_, -)} i=1 defined in the last section to obtain the model category structure of homotopy T-algebras. 18 J.E. BERGNER Recall that we defined the inclusion map (or forgetful functor) JT : AlgT ! SSetsT. Applying the above lemma to this functor of diagrams, we obtain its left adjoint functor KT : SSetsT ! AlgT. The following proposition holds in the more general situation of an arbitrary diagram category. Proposition 5.6. The adjoint pair of functors KT : SSetsT____//AlgTo:oJT._ is a Quillen pair. Proof.As categories, AlgT is a subcategory of SSetsT, and the map JT is an incl* *u- sion. Since in both cases, the fibrations and weak equivalences are defined obj* *ect- wise, JT preserves fibrations and acyclic fibrations. Lemma 5.7. Each map KT(~ff_) is an isomorphism in AlgT. Proof.Let A be a T-algebra. Notice that by Yoneda's Lemma we have that Map SSetsT(Hom T(Tff_, -), A) ' A(Tff_). Then we have the following weak equivalences of simplicial sets: a a Map AlgT(KT( Hom T(Tffi, -)),'A)MapSSetsT( Hom T(Tffi, -), A) i Y i ' MapSSetsT(Hom T(Tffi, -), A) Yi ' A(Tffi) i ' A(Tff_n) ' Map SSetsT(Hom T(Tff_n, -), A) ' Map AlgT(KT(Hom T(Tff_n, -)), A) It then suffices to show that the map KT(~) actually induces this isomorphism. This fact follows from the commutativity of the diagram KT // T SSetsT_____AlgOO::u uu JTOFfl|| uuu |uuu Ffl SSets which follows since a a a KT( JT(Hom T(Tff_, -))) ' KTJT(Hom T(Tff_, -)) ' Hom T(Tff_, -). Now, we need to show that the same adjoint pair is still a Quillen pair when * *we replace the model structure SSetsT with the model structure LSSetsT. MULTI-SORTED THEORIES 19 Proposition 5.8. The adjoint pair KT : LSSetsT_____//AlgTo:oJT_ is a Quillen pair. Proof.Consider again the set of maps a P = {~ff_: Hom T(Tffi, -) ! Hom T(Tff_, -)}. i The model category structure LSS etsT is obtained by localizing with respect to these maps. Then using Lemma 5.7, we have that each map KT(~ff_n) is an isomor- phism in AlgT. Hence, it follows from [9, 3.3.20] that the pair of adjoints for* *ms a Quillen pair even after the localization on SSetsTf. Before stating the main theorem, that the above Quillen pair is actually a Qu* *illen equivalence, we first need a lemma. Badzioch's proof [2, 6.5] for ordinary theo* *ries is slightly different than the one here, but it would follow for multi-sorted t* *heories as well. Lemma 5.9. If X is cofibrant in LSS etsT, then the unit map j : X ! KTX = JTKTX is a weak equivalence in LSSetsT. Proof.Case 1: The cofibrant object X is a free diagram. Then X looks like a Hom T(Tff_n, -). ff_ The proof for such an object follows from the proof of Lemma 5.7. Case 2: Let X be any cofibrant diagram. Then X ' hocolim opXi where each Xi is a free diagram. It then suffices to show that Map (KTX, Y ) ' Map (X, Y )* * for any T-algebra Y which is fibrant in SSetsTcof. Using case 1, we have the follow* *ing: Map(X, Y )' Map (hocolim opXi, Y ) ' holim Map (Xi, Y ) ' holim Map (KTXi, Y ) ' Map (hocolim opKTXi, Y ) ' Map (KTX, Y ). The lemma follows. Now, the proof of the main theorem follows from this lemma exactly as it does for ordinary theories in [2, 6.4]. Theorem 5.10. The Quillen pair of functors KT : LSSetsT____//_AlgTo:oJT._ is a Quillen equivalence. Proof.Let X be a cofibrant object in LSS etsT, A a fibrant object in AlgT, and f : X ! A = JTA a map in LSS etsT. We need to show that f is a P -local 20 J.E. BERGNER equivalence if and only if its adjoint map g : KTX ! A is a weak equivalence in AlgT. There is a commutative diagram j X ____//_EEKTX EEE |g fEEE""Efflffl|| A First assume that f is a P -local equivalence. Then g must also be a P -local equivalence since j is, by the previous lemma. However, g is a map in AlgT, and so it is an objectwise weak equivalence, or a weak equivalence in AlgT. Conversely, suppose that g is a weak equivalence in AlgT. Then it is a P -loc* *al equivalence. Hence, f = g O j is also a P -local equivalence. Hence, we have a Quillen equivalence of model categories between strict T- algebras and homotopy T-algebras. References [1]Ji~r'i Ad'amek and Ji~r'i Rosick'y, Locally Presentable and Accessible Cate* *gories, London Math- ematical Society Lecture Note Series 189, Cambridge University Press, 1994. [2]Bernard Badzioch, Algebraic theories in homotopy theory, Ann. of Math. (2) * *155 (2002), no. 3, 895-913. [3]J. E. Bergner, Simplicial monoids and Segal categories, in preparation. [4]J.E. Bergner, Three models for the homotopy theory of homotopy theories, pr* *eprint available at math.AT/0504334. [5]J.M. Boardman and R.M. Vogt, Homotopy invariant algebraic structures on top* *ological spaces. Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, 1973. [6]Francis Borceux, Handbook of Categorical Algebra, Volume 2, Encyclopedia of* * Mathematics and its Applications 51, Cambridge University Press, Cambridge, 1994. [7]W.G. Dwyer and J. Spalinski, Homotopy theories and model categories, in Han* *dbook of Algebraic Topology, Elsevier, 1995. [8]P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Math,* * vol. 174, Birkhauser, 1999. [9]Philip S. Hirschhorn, Model Categories and Their Localizations, Mathematica* *l Surveys and Monographs 99, AMS, 2003. [10]Mark Hovey, Model Categories, Mathematical Surveys and Monographs 63, AMS, * *1999. [11]Michael Johnson and R.F.C. Walters, Algebra objects and algebra families fo* *r finite limit theories, J. Pure Appl. Algebra 83(1992), 283-293. [12]F. William Lawvere, Functorial semantics of algebraic theories, Proc. Nat. * *Acad. Sci U.S.A. 50(1963) 869-872. [13]Saunders MacLane, Categories for the Working Mathematician, Second Edition,* * Graduate Texts in Mathematics 5, Springer-Verlag, 1997. [14]Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology* * and physics. Mathematical Surveys and Monographs, 96. American Mathematical Society, 2002. [15]Daniel Quillen, Homotopical Algebra, Lecture Notes in Math 43, Springer-Ver* *lag, 1967. Kansas State University, 138 Cardwell Hall Manhattan, KS 66506 E-mail address: bergnerj@member.ams.org