Correction to "Simplicial monoids and Segal categories"

Correction to “Simplicial monoids and Segal categories” Julia E. Bergner In the pap er [ 4 ], there is an error in the statement of Prop osition 3.1 2. The given pa ir of maps is not in fact an a djoint pa ir, since the left adjoint do es not preserve copro ducts. Therefore , we give the following revised statemen t. Prop ositio n 1. Ther e is a Quil len e quivalenc e of mo del c ate gories L : LSS ets T M ∗ / / A l g T M : N . o o The r ight adjoint functor N is just the for g etful functor ; if we regar d a T M - algebra as strictly lo cal T M -diagra m of simplicia l s ets, then this functor forgets the strictly local structure. This propos ition is then just a v a riation on Badzio ch’s rig idification theorem [ 1 ]. W e give a pro o f of this pr op osition, not just for the theory T M of monoids, but for any (pos sibly m ulti-so rted) a lgebraic theor y , following the pro of of the generaliza tion of Badzio ch’s rigidification theorem found in [ 3 ]. Prop ositio n 2. L et T b e a (multi-sorte d) algebr aic the ory. Then ther e is a Quil len e quivalenc e of mo del c ate gories b etwe en A l g T and LSS ets T O . The pro of o f the result tha t we would like to us e is given by letting T = T M , the theory of monoids, and letting O = ∗ , the set with a single element. Here, if T is an O -so rted theor y , LSS ets T O denotes the categ o ry of functor s T → SS ets with the homotopy T -algebra model s tructure, but with the additional condition that the image of the terminal ob ject in T is actually isomo rphic to the constant simplicial set giv en by O , r ather than just weakly equiv ale n t to it. W e need to find an adjoin t pair of functors betw een A l g T and LSS ets T O and prov e that it is a Quillen equiv alenc e . Let J T : A l g T → SS ets T O be the inclusion functor. W e need to show we have a n adjoin t functor taking an arbitrar y diag ram in SS ets T to a T - a lgebra. Here, w e use the idea that a T -a lgebra is a strictly lo cal diagram, as given by the following definition. Definition 3. Let D b e a small ca tegory a nd SS ets D the c a tegory of functors D → SS ets . Let P be a s et o f morphisms in SS ets D . An ob ject Y in SS ets D is strictly P - lo c al if for ev ery morphism f : A → B in P , the induced map on function complexes f ∗ : Map( B , Y ) → Map( A, Y ) 1 2 JULIA E. BERGNER is an isomorphism o f s implicial sets. A map g : C → D in SS e ts D is a strict P - lo c al e quivalenc e if for ev er y strictly P -loca l ob ject Y in SS e ts D , the induced map g ∗ : Map( D , Y ) → Ma p( C, Y ) is an isomor phism of simplicial sets. Now, g iven a category of D -diagrams in SS ets and the full subcatego ry of strictly P -lo cal dia grams for some set P of maps, we hav e the following result which can be prov ed just as in [ 3 , 5.6]. Lemma 4. Consider two c ate gories, the c ate gory of strictly lo c al diagr ams with r esp e ct to the set of maps P = { f : A → B } , and t he t he c ate gory of diagr ams X : D → SS ets whi ch ar e strictly lo c al with r esp e ct to only one of the maps in P . Then the for getful functor fr om the first c ate gory t o the se c ond has a left adjoint. Now, we can a pply this lemma using the fact that a strictly lo cal T -diagra m is precisely a T -algebra, and noting that the o b jects of LSS ets T O are strictly lo ca l with r esp ect to the map sp ecifying the data on the ima ge o f the terminal ob ject o f T . Applying Lemma 4 to the fun ctor J T , w e obtain its left adjoint functor K T : SS ets T O → A l g T . Then t he follo wing propositio n follows just as in the gener al ca se [ 3 , 5.9]. Prop ositio n 5. The adjoint p air of funct ors K T : SS ets T O / / A l g T : J T . o o is a Qu il len p air. Then, we can ex tend to the loca lized model structure LSS ets T O just as in [ 3 , 5.11]. Prop ositio n 6. The adjoint p air K T : LSS e ts T / / A l g T : J T o o is a Qu il len p air. The only difference in the proof is that w e can remov e o ne map (t he one with resp ect to which the ob jects of SS ets T O are already strictly lo cal) as we lo caliz e to get LSS ets T O . With this minor c ha nge in the localiza tion, t he pro of of Prop osition 2 follo w s just as the pro of of the genera l case [ 3 , 5.13]. Then, the other co rrection that should be noted is that the mo del structure LSS ets T M should be r emov ed from the c hain of Quillen eq uiv alences just preceding Section 5 of [ 4 ]. Although this chain is no t explicitly given for the man y-ob ject case in Section 5, the s a me c hange s should b e made ther e. The author is gra teful to the referee of [ 2 ] who po int ed out th is error. References [1] Bernard Badzioch, Algebraic theories in homotop y theory , A nn. of Math. (2) 155 (2002), no. 3, 895–913. [2] J.E. Bergner, Adding i nv erses to diagrams encoding algebraic structures, preprin t av ail able at math.A T/0610291. [3] J.E. Bergner, Rigidification of algebras ov er multi-sorted theories, Algebr. Ge om. T op ol. 6 (2006) 1925-1955. CORRECTION TO “SIMPLICIAL MONOIDS AND SEGAL CA TEGORIES” 3 [4] J.E. Bergner, Simpli cial monoids and Segal categories, Contemp. Math. 431 (2007) 59-83. Kansas St a te Un iversity, 138 Cardwell Hall, Manha tt an, KS 6 6506 E-mail addr ess : bergnerj@member. ams.org