Note on the construction of free monoids

Note on the construction of free monoids Stephen Lac k ∗ Sc ho ol of Computing and Mathematics Univ ersit y of W estern Sydney Lo c k ed Bag 1797 P enrith South DC NSW 1797 Australia email: s.lack@uws.edu. au Abstract W e construct free monoids in a monoidal category ( C , ⊗ , I ) with fin ite limits and coun table colimits, in whic h t ensoring on either side preserves reflexive coequ alizers and colimits of countable chains. In p articular this will b e the case if tensoring preserves sifted colimits. 1 Bac kground F or any mono idal ca tegory ( C , ⊗ , I ), one ca n form the categor y of monoids in C , and for suitable choice of C , this contains many imp ortant notions, such as monoids, rings, categ ories, differ en tial graded alg e br as, and mona ds : se e [8, Chapter VII]. F o r e a ch such C , the ca tegory Mo n C of monoids in C has a forgetful functor U : Mon C → C , and this forgetful functor often has a left adjoint, sending an ob ject of C to the free monoid on that ob ject. In particula r, if C has coun table co pr o ducts, and these are pr e served by tensoring on either side, then the fre e monoid on X is given by the well-known “ geometric series” I + X + X 2 + X 3 + . . . where X n stands for the n th“tensor p ow er” X ⊗ . . . ⊗ X of X . This case includes the free mo noid, the free ring (on an a dditiv e a belia n group), the free category ( on a graph) and the free differen tial graded algebra (on a chain complex), but it do es no t help with the case of free monads . Co nditio ns for the existence of free monads were g iv en by Bar r in [2]. F urther ana lysis of the fre e monoid constructio n was given by Dubuc in [3], o f which more will b e said b elow. The epic pap er [5] of Kelly analyzes many constructions of fre e monoids, free a lgebras, and colimits, g enerally re quiring transfinite pro cesses . It provides very general conditions for the existence of free monoids, in the case where the ∗ The supp ort of the A ustralian Research Council and DETY A is gratefully ac kno wledged. 1 category C is co co mplete a nd the functors − ⊗ C : C → C are co co n tinuous, for all ob jects C (the conditions on C ⊗ − are then quite mild). This allows the construction o f fre e monads in many cases, and also the co nstruction o f free op erads [9, 6]. F or example if eac h C ⊗ − preserves filtered colimits then the free monoid is given by a “facto rized” version of the geo metric series: I + X ( I + X ( I + X ( I + . . . A recent pape r of V allette [1 0] gave a constructio n of free monoids under m uch str onger a s sumptions on C ⊗ − but w eaker a ssumptions on − ⊗ C , and moreov er under the as sumption that C is abelian. These assumptions allowed the construction of free pr o per ads and other clos ely re la ted free structures, in the ab elian context. In this pap er we ge neralize and simplify the construction of V allette, remov- ing the assumption that C is ab elian. Sp ecifically , we supp ose that C has finite limits and co un table colimits, a nd that the functors − ⊗ C and C ⊗ − pr e- serve reflex iv e co equaliz ers and colimits o f coun table chains, and we cons truct free monoids under these assumptions . (This would b e the c ase for exa mple if tensoring preserved sifted colimits [1].) Some examples of such monoida l ca te- gories are given in Section 5. The questio n of whether thes e free monoids a re algebraic ally free [5] is br iefly discussed in Sectio n 6. Notation The tens or pro duct o f ob jects will gene r ally b e denoted by juxtapo sition: X Y stands for X ⊗ Y (j ust as X 2 stands for X ⊗ X ). W e sometimes write as if the monoida l structure on C were strict. This is merely fo r conv enience; by the coherence theorem for monoidal catego ries (see [8, C ha pter VI I]) it co uld be av o ided. W e write π m,n : X m X n ∼ = X m + n for the canonica l is omorphism built up o ut of the asso ciativity is omorphisms. (If C really were strict this would b e the identit y; otherwise , in order to make sense of tensor p ow ers such as X n some particular br ack eting must be chosen.) W e sometimes write X for the identit y 1 X on an ob ject X , a nd r ow vector notation ( f g ) : A + B → C for morphisms out of a c o pro duct. The comp osite o f f : X → Y and g : Y → Z is written g .f . 2 The approac h of Dub u c The constr uctio n of a free monoid can be broken down into tw o parts. An ob ject Y is said to b e p ointe d if it is equipp ed with a map y : I → Y ; we write Pt C for the categ ory of p ointe d obje ct s in C . Then the fo r getful functor U : Mon C → C is the c o mpos ite of V : Mon C → Pt C which forgets the multiplication o f a monoid but remembers the unit, and W : Pt C → C , whic h forgets the point. Since a djunctions comp ose, to find a left adjoint to U = W V , it will suffice to find adjoints to V and to W . But W has a left a djoin t se nding C ∈ C to the copro duct injection I → I + C , provided that copro ducts with I exists, so in this case we are reduced to finding a left adjoin t to V . This reduction play ed 2 a key role in [3], which contained a construction that will b e impo rtant b elow (as well as v a rious transfinite v a riants which will not). W e describ e b elow one po in t of view (not contained in [3 ]) on this co nstruction. Thu s we seek a left adjoint to V : Mon C → Pt C . In order to motiv ate the construction, we r ecall here the connection between mono ids and the simplicial category [8, Chapter VI I]. W e follow Mac Lane in writing ∆ for the category of finite or dinals and order- preserving ma ps : this is the “alg ebraist’s simpli- cial catego ry”, as opp osed to the “top ologis t’s simplicial ca tegory” which o mits the empty ordinal, a nd reindexes the re maining ob jects. No w ∆ is mono idal with resp ect to ordinal sum, and “ classifies monoids in monoidal categor ies”, in the sense that for any monoida l ca teg ory ( C , ⊗ , I ), the catego ry Mon C of monoids in C is equiv alent to the c ategory M ( ∆ , C ) o f strong monoida l (=tensor-pr eserving) functors from ∆ to C , and mo noidal natural tra nsforma- tions. The strong monoidal functor c o rresp onding to a monoid M in C with m ultiplication µ : M 2 → M and unit η : I → M has imag e I η / / M ηM / / M η / / M 2 m o o ηM 2 / / M η M / / M 2 η / / M 3 . . . µM o o M µ o o There is an a nalogous descr iption of Pt C : let ∆ mon be the (non-full) sub- category of ∆ c ont aining all the ob jects but only the injectiv e order -preserving maps. This is still monoidal under ordinal s um, and now Pt C is equiv a len t to the ca tegory M ( ∆ mon , C ) of strong monoidal functors from ∆ mon to C and mono idal natural transfo r mations. Corresp onding to the p ointed ob ject ( Y , y : I → Y ) we hav e I y / / Y y Y / / Y y / / Y 2 y Y 2 / / Y y Y / / Y 2 y / / Y 3 . . . ( ∗ ) If we identify Mon C with M ( ∆ , C ) and Pt C with M ( ∆ mon , C ), then the for - getful V : Mon C → Pt C is identified with the functor M ( H , C ) : M ( ∆ , C ) → M ( ∆ mon , C ) g iven by compo sition with the inclusion H : ∆ mon → ∆ . If w e were dealing with o rdinary functors ra ther than strong monoidal o nes, in o ther words if we sought a left adjoint to Cat ( H, C ) : Cat ( ∆ , C ) → Cat ( ∆ mon , C ), then we could simply take the left Ka n extension a long H . In gener al this left Kan extension will not s e nd s tr ong mo noidal functors to strong monoidal functors, but in sp ecial ca ses it do es, and in fact provides the left adjoint to M ( H, C ). In such a cas e, if we form the strong mono idal functor ∆ mon → C corres p onding to a p ointed ob ject ( Y , y ), and take its left Kan extension along H , the re s ulting s trong mo no idal functor fro m ∆ to C will corres po nd, via the equiv alence M ( ∆ , C ) ≃ Mon C , to a monoid in C ; and the underlying ob ject o f this monoid is precisely the co limit o f the diag r am ( ∗ ) a bove, as a simple calcu- lation involving the co end formula for left K an extensions shows. This, then, is 3 the constructio n o f Dubuc (in its simplest form where transfinite cons tructions are not required): if the colimit of ( ∗ ) exists a nd is preserved by tensor ing on either side, then it has a monoid str ucture w hich is free o n the pointed ob ject ( Y , y : I → Y ). 3 The construction Colimits indexed by ∆ mon can b e co nstructed iteratively us ing co equalizer s and colimits of chains, a s we shall do below. Now many imp or ta n t functors do not preserve all co eq ua lizers, but do preser v e coequa lizers o f reflexive pair s (pairs which hav e a c ommon section). Ther e is also a g eneral w ay to repla ce a pa ir f , g : A ⇒ B by a reflexive pair with the same co equalizer: repla ce A by A + B , and then use the identit y map on B , as in A + B ( f B ) / / ( g B ) / / B . The constr uction given here a mount s to an a nalogous a daptation of the co n- struction of Dubuc desc r ibed in the previous section. O f course when b oth constructions work, they agr e e ; in par ticular this will b e the case if copr o ducts are preserved by tensoring on either side. Suppo se then that ( Y , y : I → Y ) is a p ointed ob ject, and form the cor re- sp onding diag r am ( ∗ ). The co limit of ( ∗ ) can be constructed as follows. F or each n and each k = 0 , . . . n − 2, for m the co equalize r Y n − 1 Y k y Y n − k − 1 / / Y k +1 y Y n − k − 2 / / Y n r n,k / / Y n k . Now for m the cointersection r n : Y n → Y n of the r n,k : Y n → Y n k as k runs from 0 to n − 2. A straightforw ard calc ula tion shows that for all j and k , the comp osites Y n − 2 Y k y Y n − k − 2 / / Y k +1 y Y n − k − 3 / / Y n − 1 Y j y Y n − j − 1 / / Y n r n / / Y n agree, while the comp osite r n .Y j y Y n − j − 1 is indep endent of j , and so by the universal pro per t y of r n − 1 : Y n − 1 → Y n − 1 there is a unique ma p h n : Y n − 1 → Y n such that the s quare Y n − 1 Y j y Y n − j − 1 / / r n − 1   Y n r n   Y n − 1 h n / / Y n 4 commutes for all j . The colimit of ( ∗ ), corres ponding to the constructio n of Dubuc, is the colimit of the chain consisting of the Y n with connecting maps h n . W e mo dify this a t the first step only , replacing co equalize rs by r eflexive co equalizers, as follows. The orig inal co equaliz e rs can b e written as Y n − 1 = Y k Y Y n − k − 2 Y k y Y Y n − k − 2 / / Y k Y y Y n − k − 2 / / Y n r n,k / / Y n k and our new reflexive co equalizers are Y k ( Y + Y 2 ) Y n − k − 2 f n,k / / g n,k / / Y n d n,k o o q n,k / / Z n k where f n,k = Y k ( y Y Y 2 ) Y n − k − 2 , g n,k = Y k ( Y y Y 2 ) Y n − k − 2 , a nd d n,k = Y k bY n − k − 2 , with b the copro duct injection Y 2 → Y + Y 2 . Then, as b efore, we shall form the cointersection q n : Y n → Z n of the q n,k , the induced maps j n : Z n − 1 → Z n (see b elow) satisfying j n .q n − 1 = q n . ( Y k y Y n − k − 1 ), a nd the colimit Z of the chain consisting of the Z n and the j n ; we w r ite z n : Z n → Z for the legs of the colimit co cone. It is , how ever, worth taking a little more time to justify the existence of the j n . W e must show that fo r all j and k , the comp osites Y k ( Y + Y 2 ) Y n − k − 3 Y k ( y Y Y 2 ) Y n − k − 3 / / Y k ( Y y Y 2 ) Y n − k − 3 / / Y n − 1 Y j y Y n − j − 1 / / Y n q n / / Z n are equal. This is co mpletely s traightforw ard if either j ≤ k or j ≥ k + 2, but the case j = k + 1 is a bit more complica ted; it c a n b e broken down a s in the following diagra m: Y n q n   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Y n − 1 Y k +1 y Y n − k − 2 4 4 i i i i i i i i i i i i Y k +2 y Y n − k − 3 * * U U U U U U U U U U U U Y n q n % % K K K K K K Y k ( Y + Y 2 ) Y n + k − 3 Y k ( y Y Y 2 ) Y n + k − 3 j j j j j j j j j 5 5 j j j j j j j j j j j Y k ( Y + Y 2 ) y Y n − k − 3 / / Y k ( Y y Y 2 ) Y n − k − 3 T T T T T T T T T ) ) T T T T T T T T T T T Y k ( Y + Y 2 ) Y n − k − 2 Y k ( y Y Y 2 ) Y n − k − 2 j j j 4 4 j j j Y k ( Y y Y 2 ) Y n − k − 2 T T T * * T T T Y n Y n q n 9 9 s s s s s s Y n − 1 Y k +2 y Y n − k − 3 4 4 i i i i i i i i i i i i Y k +1 y Y n − k − 2 * * U U U U U U U U U U U U Y n q n F F                5 in which the individua l regions ar e ea sily seen to commute. In the follo wing sec tion we prove that Z ca n b e made into a monoid which is free o n ( Y , y ). W e record the gene r al re sult as : Theorem 1 L et C b e a monoidal c ate gory with finite limits and c ountable c ol- imits, and the functors − ⊗ C and C ⊗ − pr eserve r eflex ive c o e qualizers and c olimits of c ount able chains. This includes in p articular the c ase wher e C has the state d limits and c olimits and C ⊗ − and − ⊗ C pr eserve sifte d c olimits. Then the fr e e monoid on a p ointe d obje ct ( Y , y ) exists, and its u nderlying obje ct Z c an b e c alculate d as ab ove. The fr e e monoid on an obje ct X is found by taking Y to b e I + X and y t o b e the c opr o duct inje ction. 4 The pro of Suppo se that C s atisfies the conditions of the theor em. Our construction in- volv ed three types o f co limit: r eflexive c oe q ualizers, finite co intersections o f regular epimor phisms, and c olimits of chains. By assumption, the firs t a nd third of these are preserved by tensoring; w e shall see that the se c ond is also preserved. W e defer for the moment the proo f, merely no ting that the case of binary cointersections suffices, and reco r ding: Lemma 2 If q : B → C and q ′ : B → C ′ ar e r e gular epimorphisms, then their c ointerse ct ion ( pu s hout) B q / / q ′   C r   C ′ r ′ / / D is pr eserve d by ten soring on either side. W e also need the following form o f the “ 3-by-3 lemma” [4] for refle x iv e co equalizers, whose pro o f is o nc e again deferred. Lemma 3 (3-by-3 lem ma) If A 1 h 1 / / h 2 / / A 2 o o h / / A 3 B 1 k 1 / / k 2 / / B 2 o o k / / B 3 ar e r eflexive c o e qualizers, pr eserve d by tensoring on either side, then A 1 ⊗ B 1 h 1 ⊗ k 1 / / h 2 ⊗ k 2 / / A 2 ⊗ B 2 o o h ⊗ k / / A 3 ⊗ B 3 6 is also a r eflexive c o e qualizer, and A 3 ⊗ B 3 is the c ointerse ction of A 3 ⊗ B 2 and A 2 ⊗ B 3 (as quotients of A 2 ⊗ B 2 ). This shows in p articular that r e gular epimorphi sms ar e close d under tensoring. W e need to construct a multiplication µ : Z Z → Z . The idea will b e first to construct µ m,n : Z m Z n → Z m + n , then show that they pass to the colimit to give the desired µ . 4.1 Construction of µ m,n Consider first the diagram Y m Z n l ∼ = / / Z m + n m + l Y m Y n Y m q n l O O q m k Y n   π m,n / / Y m + n q m + n m + l O O q m + n k   Z m k Y n ∼ = / / Z m + n k which we shall build o ut of the canonical isomo rphism π m,n : Y m Y n ∼ = Y m + n . Since q m k : Y m → Z m k was co nstructed as the co equalizer o f maps f m,k and g m,k , th us q m k Y n : Y m + n = Y m Y n → Z m k Y n can b e constructed as the co equalizer of f m k Y n and g m k Y n ; tha t is, of f m + n k and g m + n k ; thus we get the induced isomorphism Z m k Y n ∼ = Z m + n k at the b ottom of the diagram. Similarly the co equalizer defining Z n l is pres erved by tenso ring o n the left by Y m and so we get the induced isomo rphism Y m Z n l ∼ = Z m + n m + l at the top. Thus Z m Y n is the cointersection of all the Z m + n p with 0 ≤ p ≤ m − 2, and Y m Z n is the cointersection of all the Z m + n p with m ≤ p ≤ m + n − 2. By the 3-by-3 lemma, Y m Y n Y m q n / / q m Y n   Y m Z n q m Z n   Z m Y n Z m q n / / Z m Z n is a cointersection, a nd s o Z m Z n is the cointersection of all the Z m + n p with 0 ≤ p ≤ m − 2 or m ≤ p ≤ m + n − 2. On the other hand Z m + n is the cointersection of all the Z m + n p with 0 ≤ p ≤ m + n − 2, and so ther e is a canonical quotient map µ m,n : Z m Z n → Z m + n fitting into the commutativ e diagram Y m Y n π m,n / / q m q n   Y m + n q m + n   Z m Z n µ m,n / / Z m + n 7 4.2 Construction of µ Since tensor ing preser ves c olimits o f chains, we have Z Z = Z ⊗ Z = (colim m Z m ) ⊗ (colim n Z n ) ∼ = colim m colim n ( Z m ⊗ Z n ) so there will be a unique map µ : Z Z → Z ma king Z m Z n µ m,n / /   Z m + n   Z Z µ / / Z commute pr ovided that the µ m,n are co mpatible with the maps j n : Z n → Z n +1 (and j m ); in other words that the maps µ m,n are na tural in m a nd n . W e expla in the naturality in n ; the cas e of m is similar . In the first dia gram b elow, the left square commutes by definition of µ m,n , and the r ight square commutes by definition of j m + n +1 . In the second dia gram, commutativit y o f the left square follows from the definition of j n +1 , while the right square co mm utes by definition of µ m,n +1 . Y m Y n π m,n / / q m q n   Y m + n q m + n   Y m + n y / / Y m + n +1 q m + n +1   Z m Z n µ m,n / / Z m + n j m + n +1 / / Z m + n +1 Y m Y n Y m Y n y / / q m q n   Y m Y n +1 q m q n +1   π m,n +1 / / Y m + n +1 q m + n +1   Z m Z n Z m j n +1 / / Z m Z n +1 µ m,n +1 / / Z m + n +1 Now the comp osites across the top of the tw o diagra ms a gree, by naturality of asso ciativity , and the (common) left vertical q m q n is a regula r epimorphism, so that the comp osites a cross the b ottom agree. T his gives the desir e d naturality in n , and so we obtain the r equired map µ : Z Z → Z . 4.3 V erification of asso ciativ e and unit la ws The asso ciative law µ.µZ = µ.Z µ will ho ld provided that Z m Z n Z p µ m,n Z p / / Z m µ n,p   Z m + n Z p µ m + n,p   Z m Z n + p µ m,n + p / / Z m + n + p commutes for all m , n , and p . Now the tw o paths ar ound this square will agr ee provided that they a gree when co mpos e d with the regular epimor phis m q m q n q p : Y m Y n Y p → Z m Z n Z p , and this in turn follows fr om the ev iden t commutativit y 8 of Y m Y n Y p π m,n Y p / / Y m π n,p   Y m + n Y p π m + n,p   Y m Y n + p π m,n + p / / Y m + n + p . The unit is given by the comp o site I y / / Y = Z 1 z 1 / / Z where z 1 is the r elev ant leg of the colimit co cone; the verification of the unit law is similar to but eas ie r than the verification o f a s so ciativity . 4.4 Univ ersal prop ert y The unit of the adjunction will be the map Y = Z 1 z 1 / / Z of p ointed o b jects; we must show that this has the appro priate universal pro p- erty . In other words, for every mono id M = ( M , µ, η ) and every morphism f : ( Y , y ) → ( M , η ) of po in ted ob jects, we must show that there is a uniq ue monoid morphism g : ( Z , µ, η ) → ( M , µ, η ) with g z 1 = f . F or each n , we have the comp osite f n as in Y n f n / / M n µ ( n ) / / M where µ ( n ) is the n -a ry multiplication o per ation for the monoid M . W e m ust show that these maps f n = µ ( n ) .f n pass to the quotient to give g n : Z n → M . W e chec k o nly that the co mposites in Y + Y 2 ( y Y Y 2 ) / / ( Y y Y 2 ) / / Y 2 f 2 / / M 2 µ / / M are e q ual; the other cases all follow by functoria lit y of ⊗ . Now the tw o displayed comp osites ar e maps out o f a co pro duct, so will ag r ee if their components do; for the comp onents o n Y 2 this is trivial, and for the comp onents on Y we hav e µ.f 2 .y Y = µ.η M .f = f = µ.M η .f = µ.f 2 .Y y . Thu s the maps µ ( n ) .f n induce maps g n : Z n → M , which clearly pass to the colimit to give g : Z → M . W e must show that this is a monoid ma p, a nd is the unique s uch which extends f . 9 Now g pre s erves the unit by co ns truction, and will preserve the multiplication provided that Z m Z n g m g n / / µ m,n   M M µ   Z m + n g m + n / / M commutes. But Z m Z n is a quotient of Y m Y n , s o this in turn r estricts to com- m utativity of Y m Y n f m f n / / π m,n   M M µ   Y m + n f m + n / / M which holds by co nstruction of the f n and asso ciativity of µ . This prov es that g is a monoid ma p; it remains to show the uniqueness . Suppo se then that h : Z → M is a monoid map, with h.z 1 = f . In order to show that h = g , it will s uffice to s how that h.z n = g n for all n . This in turn will ho ld if h.z n .q n = f n for a ll n . Th us we must show that the exterior of the diagram Y n z n 1 / / q n   Z n h n / / µ ( n )   M n µ ( n )   Z n z n / / Z h / / M commutes. The r ight square co mm utes b ecause h is a monoid ho momorphism, so it suffices to show that the le ft square co mmutes, and this follo ws from the definition of µ : Z 2 → Z by a straightforward induction. 4.5 Pro of of lemmas Consider a dia g ram A 11 f 1 / / f 2 / / f ′ 2   f ′ 1   A 12 g ′ 2   g ′ 1   A 21 g 1 / / g 2 / / A 22 in which g i .f ′ j = g ′ j .f i for i, j ∈ { 1 , 2 } , and supp ose als o that there exist s : A 12 → A 11 and s ′ : A 21 → A 11 with f 1 .s = f 2 .s = 1 a nd f ′ 1 .s ′ = f ′ 2 .s ′ = 1. Then a map x : A 22 → B s atisfies x.g ′ 1 .f 1 = x.g ′ 2 .f 2 if and only if it satisfies x.g ′ 1 = x.g ′ 2 and x.g 1 = x.g 2 . F o r if the for mer equatio n holds then we have x.g ′ 1 = x.g ′ 1 .f 1 .s = x.g ′ 2 .f 2 .s = x.g ′ 2 10 x.g 1 = x.g 1 .f ′ 1 .s ′ = x.g ′ 1 .f 1 .s ′ = x.g ′ 2 .f 2 .s ′ = x.g 2 .f ′ 2 .s ′ = x.g 2 while if the latter tw o e quations ho ld then x.g ′ 1 .f 1 = x.g ′ 2 .f 1 = x.g 1 .f ′ 2 = x.g 2 .f ′ 2 = x.g ′ 2 .f 2 . As a res ult we have: Prop osition 4 In the situ ation ab ove, t he c o e qualizer of g ′ 1 .f 1 and g ′ 2 .f 2 is the c ointerse ct ion of the c o e qualizer of g 1 and g 2 and the c o e qualizer of g ′ 1 and g ′ 2 . T o prove the 3-by-3 lemma (Lemma 3), apply this in the ca se of A 1 ⊗ B 1 A 1 ⊗ k 1 / / A 1 ⊗ k 2 / / h 2 ⊗ B 1   h 1 ⊗ B 1   A 1 ⊗ B 2 h 2 ⊗ B 2   h 1 ⊗ B 2   A 2 ⊗ B 1 A 2 ⊗ k 1 / / A 2 ⊗ k 2 / / A 2 ⊗ B 2 noting that the A 1 ⊗ k i hav e a common section A 1 ⊗ t , and the h 1 ⊗ B 1 hav e a common section s ⊗ B 1 . T o prov e L e mma 2, let q : B → C and q ′ : B → C ′ be the coequa lizers of the reflexive pair s A h 1 / / h 2 / / B s o o A ′ h ′ 1 / / h ′ 2 / / B s ′ o o and form the universal ob ject P with morphisms P k 1 / / k 2 / / k ′ 2   k ′ 1   A ′ h ′ 2   h ′ 1   A h 1 / / h 2 / / B satisfying equatio ns as ab ov e . In terms of elements, this would b e formed as { x 1 , x 2 ∈ A, x ′ 1 , x ′ 2 ∈ A ′ | h i ( x ′ j ) = h ′ j ( x i ) } . It is straig h tforward to show that the relev ant pair s a r e reflexive; for ex ample x ′ 7→  sh ′ 1 ( x ′ ) , sh ′ 2 ( x ′ ) , x ′ , x ′  provides a common sec tion to k 1 and k 2 . The prop ositio n then r educes the cointersection of q and q ′ to the reflexive co equalizer of h ′ 1 .k 1 and h ′ 2 .k 2 , which by a ssumption is preserved by tensor ing. 5 Examples If C is any v ariety , equipped with the car tesian pro duct × as tensor pr oduct, then pr o ducts, refle x iv e co eq ua lizers, a nd colimits o f chains are a ll computed 11 as in Set , and s ince the pro duct in Set with a fix e d ob ject is co contin uous, it follows that tensor ing in C with a fixed ob ject pr eserves the relev ant co limits. F or a fixed set A , the c a tegory Span ( A, A ) of s pans fro m A to A is the category of all se ts ov er A × A . This is mo noidal via pullback X × A Y w w o o o o o o o o ' ' O O O O O O O O X w w o o o o o o o o o ' ' O O O O O O O O O Y w w o o o o o o o o o ' ' O O O O O O O O O A A A and a mono id in the res ulting monoidal catego ry is precisely a category with ob ject-set A . T ensoring on either side is co contin uous (a nd in fac t has a n adjoint) b ecaus e pullbacks in Set are co co n tinuous. But now we can move from Set to a categor y E with finite limits in which pullback may not b e co contin uous, but do es pr e s erve reflexive co equaliz ers and colimits o f chains: this is the case, for example, in any v ar iet y . If we consider a n ob ject A ∈ E , and the catego ry Span ( E )( A, A ) of internal s pa ns in E from A to A this is once again mono idal, and once again a monoid is a c a tegory in E with A as its ob ject of ob jects. But this time tensoring on either side pr e serves r eflexive co equalizers and colimits of c hains, but not ar bitrary colimits. Thus our cons truction gives free internal categorie s in E . F or a mo r e s tructured example, o ne co uld consider not Span ( E ) but Prof ( E ), the bicategor y o f internal ca tegories in E and pr ofunctors b etw een them. Fixing an in ternal catego ry A , we g et a monoidal category Prof ( E )( A, A ), and once again the c o nditions fo r our co nstruction will b e sa tisfied. T aking E = Mon , the category of monoids, a nd A to b e (a suitable str ict version of ) the mo noidal cate- gory P o f finite sets and bijections, we get a mono idal categ o ry Prof ( Mon )( P , P ) in which monoids ar e precisely PROPs (see [7]), and so a different notio n of free PROP to that g iven in [10]. Finally , for a slig h tly childish exa mple, tak e the categor y C to be the cat- egory Grp of g roups and gr oup ho momorphisms, with the cartesian monoidal structure (with the pro duct as tens or pro duct). F or a gro up G the functors G × − : Grp → Grp and − × G : Grp → Grp do not of cours e pr e serves all colimits, but they do pr eserve r eflexive co equalizers and co limits o f chains, as would b e the case with a n y v ariety in place of Grp . N ow by the “ Eckmann- Hilton arg umen t”, a monoid in Grp is precisely an a b elian group. So our construction reduces to the ab elianization o f a g roup. 6 Algebraically free monoids The free monoid construction we have b een lo o king at inv olves a p ointed ob ject ( Y , y ) a mono id ( Z , µ, η ), and a mo rphism o f p ointed o b jects k : ( Y , y ) → ( Z , η ). But there is a nother pos sible univ ersal prop e r t y that such data might satisfy . W rite C ( Z,µ,η ) for the categor y of o b jects of C equipped with an action of the 12 monoid ( Z, η , µ ), and write C ( Y ,y ) for the categ ory o f ob jects of C equipp ed with an action of the p ointed ob ject ( Y , y ): in o ther words, a morphism α : Y A → A satisfying α.y A = 1. There is a functor k ∗ : C ( Z,η, µ ) → C ( Y ,y ) sending A equipp e d with Z -action β : Z A → A to A eq uipped with Y -ac tion Y A kA / / Z A β / / A. When this functor k ∗ is an isomo r phism of categor ies we say that k exhibits ( Z, µ, η ) as the algebr aic al ly fr e e monoid on ( Y , y ) [5]. The a lgebraically fr e e monoid, if it exists , is free, but it is p ossible for a free monoid to exist w itho ut being a lg ebraically free; nonetheles s under fairly gener al conditions the alge - braically free monoid exists (a nd is free); see [5 , Section 23]. This mea ns that the free monoid on ( Y , y ) can b e found by c a lculating free ( Y , y )-a ctions, an idea that implicitly go es ba c k to [2]. All cases where the free monoid is co mputed in [5] are done in this wa y . Under the h yp otheses of our theor em, a necessar y and sufficient condition for the free mono id o n ( Y , y ) to b e algebr aically free is that for any a ction α : Y A → A of ( Y , y ) on A , the compo sites ( Y + Y 2 ) A ( y Y Y 2 ) A / / ( Y y Y 2 ) A / / Y 2 A Y α / / Y A α / / A agree. In gener al there se ems to b e no reason why this should alw ays b e true, al- though we do not hav e a sp ecific example w her e it fails. W e therefor e conjecture that the hypotheses o f our theorem ar e not sufficient to guar ant ee that the free monoid is algebr aically free. References [1] J. Ad´ amek and J. Rosick´ y. On sifted colimits and gener alized v a rieties. The ory A ppl. Cate g. , 8:33 – 53, 200 1. [2] Michael B arr. Co equa lizers a nd free tr iples. Math. Z. , 116:30 7 –322, 1970. [3] Edua r do J. Dubuc. F r ee monoids. J. Algeb r a , 29:208 –228, 1974. [4] P . T. J o hnstone. T op os the ory . Academic Press [Ha rcourt B r ace Jov anovich Publishers], Londo n, 19 77. London Mathematica l So ciety Monogr aphs, V ol. 10. [5] G. M. Kelly . A unified treatment of transfinite constructio ns fo r free alge- bras, free monoids, co limits, asso ciated sheav es , a nd so on. Bul l. Austr al. Math. So c. , 22(1):1– 83, 1980 . [6] G. M. Kelly . On the op erads of J. P. May . R epr. The ory A ppl. Cate g. , 13:1–1 3 (electronic), 2005. 13 [7] Stephen La ck. Comp osing PROPS. The ory Appl. Cate g. , 13 :No. 9, 147 –163 (electronic), 2 004. [8] Saunder s Mac Lane. Cate gories for the working mathematician . Spr inger- V erlag, New Y ork, 1 971. [9] J. P . May . The ge ometry of iter ate d lo op sp ac es . Springer-V e r lag, Ber lin, 1972. Lecture s Notes in Mathematics, V ol. 271 . [10] Bruno V allette. F ree mono id in monoidal ab elian c a tegories. A ppl. Cate g. Structures , to app ear. 14