document: it is the purpose of this paper to extend the results of yetter @xcite , generalizing classical results of gerstenhaber @xcite and gerstenhaber and schack @xcite on the infinitesimal deformation theory of associative algebras and poset - indexed diagrams of associative algebras to a deformation theory for arbitrary pasting diagrams of @xmath0-linear categories , @xmath0-linear functors , and natural transformations . in particular , in @xcite the standard result that obstructions are cocycles was established only for the simplest parts of pasting diagrams : for pasting diagrams in which no compositions either 1- or 2-dimensional occur . in this paper 
   we will establish it for deformation complexes of pasting diagrams in general , by first giving a detailed and rigorous exposition of a method developed heuristically by shrestha @xcite , then applying the method to prove that obstructions are all cocycles in a family of pasting diagrams sufficient to imply the result in general .    along the way to proving that obstructions are closed in general , we will have occasion to consider deformations of pasting diagrams in which specified functors or natural transformations are required to be deformed trivially . 
   although in the present work such conditions will be used only to reduce the problem of showing obstructions are cocycles to simple instances , the ability to handle deformations subject to such restrictions could well be useful in other settings . 
   we will also make explicit a point overlooked in the statements and proofs of @xcite theorems 8.2 and 8.3 : the cochain maps constructed in those theorems depend on choices of association for 2-dimenensional compositions in the pasting digaram . 
   however , as we establish here , the cochain maps are independent of those choices , up to algebraic homotopy , and thus , the isomorphism type of the deformation complex for a pasting diagram is well - defined in either the homotopy category or derived category of cochain complexes over @xmath0 . 
   at the level of the hochschild cochain complexes @xmath1 , the interesting cochain maps are described fully in @xcite . 
   our purpose in this section is , rather , to describe chain maps induced by pasting compositions on the full deformation complexes @xmath2 , when @xmath3 is a composable pasting diagram . 
   proposition 4.5 of @xcite constructs two chain maps :    if @xmath4 , and @xmath5 are functors , then there is a cochain map @xmath6 given by    @xmath7    similarly if @xmath8 and @xmath9 are functors , there is a cochain map @xmath10 given by    @xmath11    proposition 4.6 of @xcite gives two more :    if @xmath12 is a natural transformation , then post- ( resp . 
   pre- ) composition by @xmath13 induces a cochain map @xmath14 ( resp . 
   @xmath15 for any functor @xmath16 .    here 
   @xmath17    and    @xmath18    and , @xcite proposition 4.7 give a cochain map , which ties together all of the deformations when a natural transformation , its source and target , and their common source and target are deformed simultaneously :    let @xmath19 be a natural transformation between @xmath0-linear functors @xmath20 . 
   let @xmath21 denote the cone on the cochain map    @xmath22 :                c^\bullet({\cal a})\oplus c^\bullet({\cal b } )                 \rightarrow c^\bullet(f ) \oplus c^\bullet(g).\ ] ]    the cochain groups are    @xmath23    with coboundary operators given by    @xmath24\ ] ]    proposition 4.7 was omitted in the statement of the proposition , though it is plainly present in the proof given in @xcite ] was then    let @xmath19 be a natural transformation , then    @xmath25 :   c^\bullet({\cal a } \stackrel{f}{g } { \cal b } ) \rightarrow c^\bullet(f , g)\ ] ]    is a cochain map . 
   in @xcite the chain maps of propositions 8.1 , 8.2 and 8.3 were used only to construct the deformation complex of a general pasting diagram . 
   in fact , they can be assembled into chain maps from the deformation complex of a composable pasting diagram to the simpler pasting diagram in which the compositions have been carried out . 
   propositions [ chainmapfrom2comp ] , [ chainmapfrom1precomp ] and [ chainmapfrom1postcomp ] give the map explicitly in the cases of a single 2 composition , precomposition of a natural transformation by a functor , and postcompostion of a natural transformation by a functor , respectively .    in each case 
   the proof begins by collecting the maps from the propositions of @xcite with values in the direct summands corresponding to cells of the pasting diagram in which the compositions have been performed , together with identity maps for those cells which remain from the original diagram , and arranging their summands in the correct places of a matrix of maps . 
   what is indicated in the sketches of proofs following each proposition are the main difficulties in the unedifying calculation which shows that the result is , in fact , a chain map between the deformation complexes .    [ chainmapfrom2comp ] 
   let @xmath26 be @xmath0-linear functors , and @xmath19 and @xmath27 be natural transformations . 
   let @xmath3 be the pasting diagram consisting of both @xmath28 and @xmath13 and their ( iterated ) sources and targets , and let @xmath29 be the pasting diagram consisting of the 2-dimensional composition @xmath30 and its ( iterated ) sources and targets . 
   then there is a chain map    @xmath31    induced by the 2-dimensional composition .    in particular if the summands of @xmath32 ( resp . 
   @xmath33 ) are given in the order    @xmath34    ( resp . 
   @xmath35    @xmath36 is given by    @xmath37\ ] ]    ( sketch ) the only subtlety in the completely computational verification that this is a chain map involves one coordinate in which a relation of the sort in gerstenhaber and voronov @xcite or the proof of proposition 4.7 in @xcite is needed . 
   [ chainmapfrom1precomp ]    let @xmath38 and @xmath39 be @xmath0-linear functors , and @xmath40 be a natural transformation . 
   let @xmath3 be the pasting diagram consisting of @xmath28 and its ( iterated ) sources and targets together with @xmath41 and @xmath42 , and let @xmath29 be the pasting diagram consisting of the 1-dimensional composition @xmath43 and its ( iterated ) sources and targets . 
   then there is a chain map    @xmath44    induced by the 1-dimensional composition .    in particular if the summands of @xmath32 ( resp . 
   @xmath33 ) are given in the order    @xmath45    ( resp . 
   @xmath46    @xmath47 is given by    @xmath48\ ] ]    ( sketch ) here the `` hardest '' verification is a coordinate which vanishes by the naturality of @xmath43 . 
   [ chainmapfrom1postcomp ]    let @xmath49 and @xmath50 be @xmath0-linear functors , and @xmath19 be a natural transformation . 
   let @xmath3 be the pasting diagram consisting of @xmath28 and its ( iterated ) sources and targets , together with @xmath51 and @xmath52 , and let @xmath29 be the pasting diagram consisting of the 1-dimensional composition @xmath53 and its ( iterated ) sources and targets . 
   then there is a chain map    @xmath54    induced by the 2-dimensional composition . 
   in particular if the summands of @xmath32 ( resp . 
   @xmath33 ) are given in the order    @xmath55    ( resp . 
   @xmath56    @xmath57 is given by    @xmath58\ ] ]    ( sketch ) here , almost all of the coordinates in the bottom row present some minor `` difficulty '' , one depends on a gerstenhaber - voronov - style relation , while most of the others require unpacking the definitions of pullback or pushforward maps along functors and natural transformations to see some commutativity relation between them . 
   in this section we use the cochain maps of the previous section to give explicit examples of deformation complexes associated to various shapes commutative pasting diagrams . in one example 
   we see that cochain maps induced by two different associations of the 2-dimensional composition are not actually equal , but only chain homotopic ( or , to put it another way , are not equal in the abelian category of cochain complexes , but _ are _ in the homotopy category of cochain complexes and thus _ a fortiori _ in the derived category ) .    in the examples , we use commutative pasting diagrams because only in that context is the effect of the compositions of 2-arrows evident : if no commutativities are enforced , the iterative cone construction given explicitly in @xcite proposition 4.7 suffices to describe the entire structure of the deformation complex of the pasting diagram , as indeed it does in the case of commutative pasting diagrams in which no compositions occur ( for example , the commutative `` pillow '' consisting of two categories a pair of parallel functors between them , and two copies of the same natural transformation between the functors ) . 
   only when compositions are involved do the cochain maps of the previous section play a role .    in what follows 
   , we leave zero entries in matrices of maps giving coboundaries blank , as this seems to improve readability . 
   we begin with the simplest example , a `` commutative pillow with triangular cross - section '' :    [ triangularpillow ] consider the pasting diagram given by three functors @xmath59 and three natural transformations @xmath60 , @xmath61 and @xmath62 , with the obvious 0- , 1- and 2-cells and a single 3-cell enforcing the condition @xmath63 . 
   the deformation complex of the pasting diagram is then given by    @xmath64 @xmath65    with coboundary given by    @xmath66\ ] ]    note that here we have chosen to consider the composition @xmath67 to be the source of the 3-cell and the single natural transformation @xmath68 to be the target . 
   for the opposite choice the complex would be the same , except that the non - diagonal entries of the last row would be negated . 
   the complex @xmath69 of example [ triangularpillow ] is related to the map @xmath70 of proposition [ chainmapfrom2comp ] by    [ babyqi ] @xmath69 is quasi - isomorphic to the ( dual ) mapping cylinder on @xmath70 . 
   this follows immediately by applying the following general lemma about ( dual ) mapping cylinders of chain maps between mapping cones . 
   let @xmath71 and @xmath72 be chain maps , and suppose @xmath73 is a chain map between their mapping cones of the form    @xmath74 .\ ] ]    then the ( dual ) mapping cylinder on @xmath75 , @xmath76 \oplus cone(g)[1 ] \oplus cone(g)$ ] with differential given by    @xmath77 , \ ] ]    is quasi - isomorphic to @xmath78 with differential given by    @xmath79 .\ ] ]    include the last complex into the ( dual ) mapping cylinder by @xmath80 , the cokernel is plainly isomorphic to @xmath81})$ ] , which is acylic , so the subcomplex is quasi - isomorphic to the ( dual ) mapping cylinder as claimed . 
   continuing with pasting diagrams which only involve 2-dimensional compositions , consider next the two `` pillows with square cross - sections '' :    consider the pasting diagram given by four functors @xmath82 and four natural transformations @xmath60 , @xmath61 , @xmath83 and @xmath84 , with the obvious 0- , 1- and 2-cells and a single 3-cell enforcing the condition @xmath85 . 
   the deformation complex of the pasting diagram is then given by    @xmath86 @xmath87    with coboundary given by    @xmath88\ ] ]    where @xmath89    consider the pasting diagram given by four functors @xmath90 and four natural transformations @xmath60 , @xmath61 , @xmath91 and @xmath92 , with the obvious 0- , 1- and 2-cells and a single 3-cell enforcing the condition @xmath93 . 
   the deformation complex of the pasting diagram is then given by    @xmath94 @xmath95    with coboundary given by    @xmath96\ ] ]    where @xmath97    now , observe that in this example , we made a choice : we computed the bottom row , and in paricular the map @xmath98 by left - associating @xmath28 , @xmath13 and @xmath68 . had we right - associated them , the bottom row , other than @xmath98 would have remained the same ( though it would have been more natural to write @xmath99 , @xmath100 and @xmath101 as the names for the other maps ) . 
   the second entry , rather than being @xmath98 would have been    @xmath102 . 
   now , it is easy to see that the chain maps from    @xmath103 @xmath104    to @xmath105 given by the entries of the bottom row other than @xmath106 are not equal , and their difference has a single non - zero entry @xmath107 . 
   there is however a contracting homotopy for this difference , given by the map with all by the second entry zero , and that entry given by @xmath108 , due to the relationship between the brace and coboundary discovered in the classical case by gerstenhaber and voronov @xcite and seen to apply in the categorical setting in @xcite . 
   thus in the homotopy category ( or the derived category , if one prefers ) our choice of left - association was a matter of indifference . 
   it is easy to compute more examples in which no 1-dimensional compositions occur , for instance a `` pillow '' with a pentagonal cross - section with a triple composition equal to a double composition , by combining the techniques of the last three examples .    rather than 
   write this out explicitly , we now turn to examples involving 1-dimensional composition :    [ equalspost1comp ] consider the pasting diagram consisting of three categories @xmath42 , @xmath109 and @xmath52 , three functors @xmath20 and @xmath50 , two natural transformations @xmath60 and @xmath110 and a single 3-cell enforcing the condition that @xmath111 
   .    the deformation complex of the pasting diagram is given by    @xmath112 @xmath113    with coboundary given by    @xmath114\ ] ]    [ equalspre1comp ] consider the pasting diagram consisting of three categories @xmath42 , @xmath109 and @xmath52 , three functors @xmath115 and @xmath39 , two natural transformations @xmath116 and @xmath117 and a single 3-cell enforcing the condition that @xmath118 
   .    the deformation complex of the pasting diagram is given by    @xmath112 @xmath119    with coboundary given by    @xmath120\ ] ]    consider the pasting diagram consisting of three categories @xmath42 , @xmath109 and @xmath52 , four functors @xmath20 , @xmath50 and @xmath121 , and three natural transformations @xmath60 , @xmath122 and @xmath123 , with the obvious 0- , 1- and 2- cells and a single 3-cell enforcing the condition that @xmath124 
   .    the deformation complex of the pasting diagram is given by    @xmath125 @xmath126    with coboundary given by    @xmath127^ * & -id & d_{h(f),k }    \end{array } \right]\ ] ]    where @xmath128    consider the pasting diagram consisting of three categories @xmath42 , @xmath109 and @xmath52 , four functors @xmath115 , @xmath39 and @xmath121 , and three natural transformations @xmath129 , @xmath130 and @xmath131 , with the obvious 0- , 1- and 2- cells and a single 3-cell enforcing the condition that @xmath132 . 
   the deformation complex of the pasting diagram is given by    @xmath125 @xmath133    with coboundary given by    @xmath134^ * & \sigma_*(f^ * ) & -id & d_{h(f),k }    \end{array } \right]\ ] ]    where @xmath135    consider the pasting diagram consisting of three categories @xmath42 , @xmath109 and @xmath52 , four functors @xmath136 , @xmath137 and @xmath138 , and three natural transformations @xmath139 , @xmath130 and @xmath140 , with the obvious 0- , 1- and 2- cells and a single 3-cell enforcing the condition that @xmath141 . 
   the deformation complex of the pasting diagram is given by    @xmath125 @xmath142    with coboundary given by    @xmath143 _ * & \sigma^*(k _ * ) & -id & d_{h(f),k }    \end{array } \right]\ ] ]    where @xmath144    consider the pasting diagram consisting of three categories @xmath42 , @xmath109 and @xmath52 , four functors @xmath136 , @xmath145 and @xmath146 , and three natural transformations @xmath147 , @xmath148 and @xmath149 , with the obvious 0- , 1- and 2- cells and a single 3-cell enforcing the condition that @xmath150 . 
   the deformation complex of the pasting diagram is given by    @xmath125 @xmath151    with coboundary given by    @xmath152 _ * & \sigma^*(g^ * ) & -id & d_{f , k(g ) }    \end{array } \right]\ ] ]    where @xmath153 
   first , observing that the map @xmath154 is simply the identity in the case where the composable pasting diagram @xmath3 is simply the 2-cell @xmath28 ( with its sources and targets ) , and that the ( dual ) mapping cylinder on a map is the cone on the direct sum of the map and the identity on its target we have    if @xmath3 is a pasting diagram with a single 3-cell the boundary of which has domain ( resp . 
   codomain ) which is a composable pasting diagram @xmath29 and codomain ( resp . 
   domain ) which is a single 2-cell , then @xmath155 is quasi - isomorphic to the ( dual ) mapping cylinder of the map of proposition 8.3 of @xcite induced by the composition . 
   observe this is a generalization of proposition [ babyqi ] and in special instances relates the ( dual ) mapping cylinder of the map in proposition [ chainmapfrom1postcomp ] ( resp . [ chainmapfrom1precomp ] ) to the complex of example [ equalspost1comp ] ( resp . 
   [ equalspre1comp ] ) . 
   the following is a trivial observation about the deformation complex constructed in proposition 8.3 of @xcite and the remarks following :    [ uniontopushout ] if @xmath156 is a pasting diagram which is the union of pasting diagrams @xmath157 and @xmath158 , then the deformation complex @xmath155 is the pushout of the induced inclusions of deformation complexes @xmath159 for @xmath160 @xmath161 . 
   in @xcite shrestha developed a method of using polygons with edges labeled by arrow - valued operations to simultaneously encode cocycle conditions , the formulas for obstructions , and the condition that the next term of a deformation cobound the obstruction . 
   suitable cell - decompositions of the surface of a 2-sphere into such polygons and `` trivial '' polygons which encode tautologous equalities in place of cocycle conditions and zero ( as a difference of identical sums ) in place of an obstruction , then provide a convenient method of proof for standard obstructions - are - cocycle results .    as an example , consider the cocycle conditions , formulas for obstructions , and cobounding conditions in the case of the deformation of composition , writing the undeformed composition as @xmath162 to match the notation for the degree @xmath163 deformation term as @xmath164 :    the cocycle condition is , of course ,    @xmath165    the formula for the degree @xmath163 obstruction is    @xmath166 .    and the condition that @xmath167 cobounds @xmath168 is    @xmath169    now consider the square , with oriented edges labeled by the compositions occuring in the expression of the associativity of composition :    @xmath170^{a(bc ) } & \circ\\ .\ar[r]_{ab}\ar[u]^{bc } & \ar[u]_{(ab)c.}}.\ ] ] + each of the above formulas can be obtained from ( or represented by ) the square by adding all possible terms obtained by assigning degrees to the edges in such a way that the degrees are chosen from a particular set and the sum of degrees along each of the oriented parts of the boundary has a specified value , and taking the difference of the expressions thus obtained for each of the oriented parts of the boundary . 
   gerstenhaber s proof @xcite that obstructions to the deformation of an associative algebra are hochschild cocycles can then be described in terms of the cube below by first noting that the formula for the coboundary of the degree @xmath163 obstruction can be written as a signed sum of the obstruction - type expressions ( with the edge - degrees ranging from @xmath171 to @xmath172 ) , five faces representing the usual terms of the coboundary , and one representing @xmath171 , `` prolonged '' by pre- or post - composing with the degree @xmath171 label on another edge so that all of the expressions represent compositions of terms from the deformation of an iterated composition of all four maps , and thus , all representing parallel maps , can all be added . 
   @xmath173_{(\overline{ab}c)d } & & & \odot \\ .\ar[ru]_{\overline{ab}c}\ar[rrr]^{cd } & & & .\ar[ru]^{\overline{ab}(cd ) } & \\ & & & & \\ & .\ar@{-->}[uuu]_{a(bc)}\ar@{-->}[rrr]^{(bc)d } &   & & .\ar[uuu]^{a(b\overline{cd})}\\ \circ\ar[uuu]_{ab}\ar@{-->}[ru]_{bc}\ar[rrr]^{cd } & & & .\ar[ru]^{b\overline{cd}}\ar[uuu]^{ab } & } \ ] ] + the proof then proceeds by iteratively `` clearing '' edges shared by two squares with the same sign by rewriting terms involving that edge s label contributed by one square using the cobounding and cocycle condition from the other square sharing the edge until the expression is reduced to two copies of the obstruction - type expression on an equatorial hexagon , with opposite signs . 
   this observation , that gerstenhaber s proof can be encoded by such a figure , then motivates a sequence of definitions and lemmas that allow for similar encoding of more general and complex proofs of the same sort . 
   shrestha s technique @xcite is most quickly and rigorously described by labeling 1-cells of certain pasting schemes and computads ( cf . 
   @xcite , @xcite , @xcite ) with well - formed expressions in a particular ( essentially ) algebraic theory :    a _ directed polygon _ 
   @xmath174 is a 2-computad with a single 2-cell , all of whose 0- and 1-cells lie in the boundary of the 2-cell . 
   a _ whiskered polygon _ 
   @xmath175 is a 2-computad with a single 2-cell @xmath174 , whose underlying 1-computad is the union of two 1-dimensional pasting schemes ( directed paths ) , with the same 0-domain and 0-codomain , whose underlying cell complex is contractible . 
   we refer to the 1-dimensional pasting scheme consisting of the path from the 0-domain of @xmath175 to the 0-domain of @xmath174 as the _ domain whisker _ 
   ( note : it may be simply a 0-cell ) , and the 1-dimensional pasting scheme consisting of the path from the 0-codomain of @xmath174 to the 0-codomain of @xmath175 as the _ codomain whisker _ 
   ( again it may simply be a 0-cell ) . 
   a _ tiled sphere _ is a 3-computad with a single 3-cell , all of whose 0- , 1- and 2-cells lie in the boundary of the 3-cell .    in applying the method in a given circumstance 
   , one needs to label the edges of tiled spheres with arrow - valued operations from a theory associated to the pasting diagram ( or more general structure ) . 
   for the present application the following suffice :    [ theory_of_diagram ] to any pasting diagram @xmath3 , _ theory of _ 
   @xmath3 , @xmath176 , is the essentially algebraic theory with types @xmath177 and @xmath178 for each category @xmath52 in the diagram ( the objects of @xmath52 and the arrows of @xmath52 respectively ) and operations    * @xmath179 of arity @xmath177 and type @xmath178 , * @xmath180 and @xmath181 of arity @xmath178 and type @xmath177 , and * @xmath182 of arity @xmath183 and type @xmath178    for each category @xmath52 in @xmath3 ;    * @xmath184 of arity @xmath177 and type @xmath185 and * @xmath186 of arity @xmath178 and type @xmath187    for each functor @xmath188 in @xmath3 ; and    * @xmath28 of arity @xmath177 and type @xmath187    for each natural transformation @xmath60 for @xmath189 in @xmath3 .    and , with axioms the equations expressing the axioms of categories for each 6-tuple @xmath190 , the functoriality of each @xmath191 and the naturality of each @xmath28 . 
   we term operations of type @xmath178 for any @xmath52 `` arrow - valued '' and those of type @xmath177 for any @xmath52 `` object - valued '' , and refer to the model of @xmath176 given by @xmath3 as the `` tautologous model '' . in other applications of the method , as in @xcite @xmath176 may be replaced with an extension of the theory ( for instance including operations of arity @xmath192 and type @xmath193 encoding monoidal product of two arrows ) . 
   we conjecture that broader generalizations , for instance to deformations of @xmath163-tuple categories and ( weak ) @xmath163- categories with a @xmath0-linear structure on their @xmath163-arrows , and to suitable pasting diagrams of these , or even to models of other sorts of essentially algebraic theories ( cf . 
   @xcite ) with appropriate linearizations of parts of the structure , can be described , but do not pursue this possibility here . 
   notice a subtlety in the description of the axioms of @xmath176 in definition [ theory_of_diagram ] : the only axioms are those inherited from the axioms of categories , functors and natural transformations . in the definition [ parallels ] , on the contrary , 
   all of the equations which hold in the diagram @xmath3 are enforced . 
   [ parallels ]    the _ theory of parallels _ @xmath194 for a pasting diagram @xmath3 is the essentially algebraic theory with the same types and object valued operations as @xmath176 and with arrow - valued operations given by all set functions @xmath195 with the same domain and codomain as the instantiation of an arrow - valued operation @xmath196 of @xmath176 in the tautologous model and satisfying @xmath197 and @xmath198 ( we call @xmath195 _ a parallel _ for @xmath196 ) , and all equations that hold among ( iterated generalized ) compositions of these functions as axioms 
   .    notice in neither definition did we include any addition or scalar multiplication operations induced by the @xmath0-linear structure on the categories , even though we are only applying these constructions to pasting diagrams of @xmath0-linear categories , @xmath0-linear functors and natural transformations . 
   however , when applied to such a pasting diagram , the vector space structure on the hom - sets induces a vector space structure on the set of parallels for any arrow - valued operation @xmath196 of @xmath176 . 
   given a ( countably infinite ) stock of variables of each type in the theory it is evident what is meant by a well - formed formula ( wff ) of the theory . 
   we now make some technical definitions :    an _ equivalent _ of a wff is any wff of the same type in the same variables such that for every instantiation of the variables in the two formulas , the values are equal .    in what follows , for brevity we will refer to well - formed subformulas of a wff as wfss . 
   when considered as wfss of a fixed wff @xmath175 , repetitions of equal wffs are considered distinct wfss . with this convention , 
   the following ( a generality about wfss of a wff in any formal system ) is immediate :    the wfss of a wff @xmath175 form a partially ordered set @xmath199 under @xmath200 when @xmath201 is a wfs of @xmath202 , and the incidence diagram of @xmath199 is a tree . 
   thus @xmath199 admits a natural number valued depth function @xmath203 for which @xmath204 and whenever @xmath202 covers @xmath201 , @xmath205 . 
   the minimal elements ( in the order - theoretic sense , not just those of maximum depth ) for @xmath199 are instances of variables . 
   we denote the subposet of non - minimal wfss of @xmath175 by @xmath206    a _ well - formed labeling _ of a whiskered polygon ( resp . 
   tiled sphere ) is an assignment @xmath207 to each directed 1-cell of arrow - valued wff in @xmath176 for some pasting diagram ( or an appropriate extension of this theory ) with the properties    * edges are not labeled with variables ( i.e. all labels involve an operation in the theory ) . * for each 1-pasting scheme ( directed path ) @xmath208 from the 0-domain of the whiskered polygon ( resp . 
   tiled sphere ) the labels @xmath209 can be iteratively replaced , beginning at the 0-codomain , with labels @xmath210 such that * * if the only proper wfss of @xmath211 are variables then @xmath212 . 
   * * for all @xmath213 @xmath214 is equivalent to @xmath211 . 
   * * in the list of replacement labels @xmath210 , for each @xmath214 , if @xmath215 is a wff occuring @xmath216 times as a non - minimal wfs of @xmath214 , then there exist @xmath216 distinct @xmath217 s with @xmath218 such that @xmath219 . 
   * if @xmath208 is a maximal directed path in a whiskered polygon ( resp . 
   tiled sphere ) , the replacement labels @xmath210 are precisely the non - minimal wfss of @xmath220 . 
   * for every 2-cell , and every maximal extension of the 2-cell to a whiskered polygon , the labels of the final edges of the two paths from the 0-domain to the 0-codomain of the whiskered polygon are equivalents . 
   ( note this condition is vacuously true unless the 0-codomain of the 2-cell is the 0-codomain of the whiskered polygon . ) 
   the following is then immediate :    for any maximal path @xmath208 in a whiskered polygon ( resp . 
   tiled sphere ) the totally ordered set of wff s @xmath221 is a totalization of the partially ordered set @xmath222 . 
   # 1#2#3#4#5 @font    ( 3885,7414)(1816,-7874 ) ( 2206,-5356 ) ( 0,-1)1275 ( 2206,-6646 ) ( 4,-3)1112 ( 3316,-7456 ) ( 1 , 0)1530 ( 4426,-4186 ) ( 4,-3)1080 ( 5536,-5041 ) ( 0,-1)1605 ( 5558,-6669)(-1,-1)735 ( 2926,-4111)(-2 , 3)750 ( 2206,-2956 ) ( 0 , 1)1275 ( 2206,-1666 ) ( 4 , 3)1112 ( 3316,-856 ) ( 1 , 0)1530 ( 4426,-4126 ) ( 4 , 3)1080 ( 5536,-3271 ) ( 0 , 1)1605 ( 5558,-1643)(-1 , 1)735 ( 3016,-4156 ) ( 1 , 0)1350 ( 3001,-4261 ) ( 0,-1)1350 ( 2963,-5657)(-3,-4)642 ( 3031,-4231 ) ( 1,-1)765 ( 3841,-5011 ) ( 1 , 0)1560 ( 3061,-5641 ) ( 3,-2)675 ( 3751,-5056 ) ( 0,-1)975 ( 3796,-6091 ) ( 3,-4)945 ( 2251,-2986 ) ( 1 , 0)1320 ( 3946,-1696 ) ( 1 , 0)1485 ( 3871,-1726)(-2 , 3)550 ( 3608,-2913 ) ( 1 , 4)285 ( 4321,-4111)(-2 , 3)750 ( 1831,-2326)(0,0)[lb ] ( 2176,-1126)(0,0)[lb ] ( 2641,-2806)(0,0)[lb ] ( 3871,-2371)(0,0)[lb ] ( 3706,-1246)(0,0)[lb ] ( 3661,-631)(0,0)[lb ] ( 2161,-3706)(0,0)[lb ] ( 5296,-1096)(0,0)[lb ] ( 3991,-3361)(0,0)[lb ] ( 3241,-3976)(0,0)[lb ] ( 5041,-3886)(0,0)[lb ] ( 5686,-2566)(0,0)[lb ] ( 2071,-4621)(0,0)[lb ] ( 5071,-4501)(0,0)[lb ] ( 5671,-6061)(0,0)[lb ] ( 1831,-6046)(0,0)[lb ] ( 2161,-7411)(0,0)[lb ] ( 3541,-7801)(0,0)[lb ] ( 5356,-7261)(0,0)[lb ] ( 2746,-5131)(0,0)[lb ] ( 4426,-5281)(0,0)[lb ] ( 2581,-6391)(0,0)[lb ] ( 3496,-4516)(0,0)[lb ] ( 3841,-5581)(0,0)[lb ] ( 2971,-6121)(0,0)[lb ] ( 4201,-6406)(0,0)[lb ] ( 2416,-5491)(0,0)[lb ] ( 2956,-3421)(0,0)[lb ] ( 4111,-4606)(0,0)[lb ] ( 4351,-1981)(0,0)[lb ]    an example of a well - formed labeling of a tiled sphere is given in figure [ functor_obstructions_are_cocycles ] . 
   ( the boundaries of the two octogons bordering the unbounded region should be identified along the edges with the same labels , and the 3-cell inserted to fill the resulting topological sphere . ) 
   this particular example arises in applying the method to show that the obstructions to deforming ( the arrow part of ) a functor , when the compositions in its source and target categories are also being deformed , are cocycles . 
   note that in the labeling , at one point notation has been abused and @xmath223 has been used in place of either of the equivalent wffs @xmath224 or @xmath225 . 
   it is also easy to see along which paths the label - replacement axiom wf1 results in non - trival replacements of labels ( as , for instance along any path ending in the top right - most edge in which @xmath226 occurs , rather than @xmath227 : the label of the top right - most edge must be replaced with @xmath228 ) . 
   now each arrow - valued operation is a part of the algebraic structure which is subject to infinitesimal deformation , by being replaced by a formal power - series ( or polynomial in @xmath229 with @xmath230 for some @xmath163 ) whose coefficients are arrow - valued operations of the same type and arity ( e.g. the arrow part of a functor @xmath188 is replaced with @xmath231 where for each @xmath232 , @xmath233 is a map from @xmath234 to @xmath235 ) . from this 
   , we abstract    consider any set of arrow - valued operations @xmath236 in @xmath176 for some @xmath3 . 
   fix @xmath237 . 
   to each arrow - valued operation @xmath238 associate a sequence of parallels @xmath239 in the sense of definition [ parallels ] ( truncated at @xmath240 if @xmath241 ) with @xmath242 . 
   we call @xmath239 _ the degree @xmath0 parallel of @xmath243 _ , and a choice @xmath244 of such a sequence for every operation in @xmath236 , _ a degree @xmath163 family of parallels for @xmath236 _    now , for any well - formed labeling of a whiskered polygon or tiled sphere with , let @xmath236 be the set of all arrow - valued operations occuring in the labels of the edges , closed under equivalence . every family of parallels for @xmath236 , then gives rise to many labelings of the maximal paths of the whiskered polygon or tiled sphere by wffs from @xmath194 by replacing the labels along the path using wf1 , then for each edge , chosing a degree @xmath0 and replacing the last - applied operation @xmath243 in the label on that edge with @xmath239 in the label for that edge and all edges later in the path for which the label on the given edge as a wfs .    in particular , any such choice of degrees for each edge along a path from the 0-domain to the 0-codomain creates a new wff in @xmath194 . 
   suppose the sequence of labels on a maximal path was @xmath245 , where each @xmath246 denotes a well - formed expression all of whose iterated inputs have an equivalent among the labels earlier in the sequence , and by abuse of notion also that well - formed expression s last - applied operation . 
   a choice of degrees @xmath247 then produces a new sequence of well - formed expression @xmath248 in which each instance of an operation has been replaced with its parallel of the chosen degree in that and all later edge - labels . 
   note : in general these new labelings are not well - formed labelings .    at this point , 
   recall that all of this is taking place in a linear setting , so that parallel arrows ( and thus parallels ) can be added . 
   we will now define several different expressions which a system of parallels associates to a whiskered polygon . 
   [ conditions_and_obstructions ]    for any well - formed labeling of a whiskered polygon @xmath175 , @xmath236 the set of operations occuring in the replacement labelings of both paths , and @xmath244 a family of parallels for @xmath236 , let @xmath249 ( resp . 
   @xmath250 ) be the replacement labeling on the maximal path which traverses domain ( resp . 
   codomain ) of the 2-cell . 
   the _ cocycle - type condition _ associated to @xmath175 is the equation    @xmath251    the _ 
   @xmath252 order obstruction - type expression _ associated to @xmath175 is the expression    @xmath253    and , the _ 
   @xmath252 order cobounding - type condition _ associated to @xmath175 is the expression    @xmath254    in each case , the string of parallels should be understood as naming the well - formed experssion obtained by the iterated substitution of the named parallel for the corresponding operation in the well - formed expression labeling the last edge of the path . 
   the following then provides the basis for shrestha s method :    [ elementary ] for any well - formed labeling of a tiled sphere @xmath255 with 3-cell @xmath256 , @xmath236 the set of operations occuring in the replacement labelings of all paths , and @xmath244 a family of parallels for @xmath236 , the domain ( resp . 
   codomain ) of @xmath256 can be expressed as the union of whiskered polygons with the same 0-domain and 0-codomain as @xmath255 , one with each 2-cell in @xmath257 ( resp . 
   @xmath258 ) as its 2-cell in such a way that the sum of the @xmath252 order obstruction - type expressions for the whiskered polygons is the @xmath252 order obstruction - type expression for the ( whiskered ) polygon consisting of a single 2-cell and the union of the 1-pasting schemes @xmath259 and @xmath260 .    once the combinatorial structure of power s proof of the the uniqueness of pasting compositions @xcite is recalled , the result is immediate 
   sums cancel in pairs leaving only the difference giving the desired @xmath252 order obstruction - type expression . 
   unfortunately , as it stands , the result is not immediately applicable . recall the cube encoding gerstenhaber s proof . 
   any face is part of a whiskered polygon ( with one whisker having one edge , and the other having none ) . 
   the @xmath252 order obstruction is given by    @xmath261    the terms in its coboundary are _ not _ @xmath252 order obstruction - type expressions associated to the faces of their cubes with their whiskers . the @xmath252 order obstruction type expressions are , instead , instances of @xmath262 with the obstruction and a variable as inputs , or instances of the obstruction with one of its variables replaced with @xmath262 applied to two variables . 
   what then is the relationship between actual @xmath252 order obstructions and @xmath252 order obstruction - type expressions ?    if one considers one of the whiskered polygons in the example , it is easy to see that the terms in the @xmath252 order obstruction - type expression which do not correspond to terms from the coboundary of the obstruction have a label of positive degree on the edge of the whisker . if we fix the label on the whisker edge to be of degree @xmath263 , the terms with this label are then the condition that @xmath264 cobound the degree @xmath265 obstruction ( or the cocycle condition if @xmath266 ) with @xmath267 of two variables as argument ( resp . used as input to an instance of @xmath267 ) when the non - trivial whisker is the domain ( resp . 
   codomain ) whisker .    in this case , provided the system of parallels is describing the terms of an associative deformation , the @xmath252 order obstruction - type expression is thus equal to the corresponding term in the coboundary of the @xmath252 order obstruction , since its extra terms all vanish by the cocycle and cobounding conditions .    to imitate this in general , we need conditions depending only on the labels on the domain and codomain of the 2-cell in a whiskered polygon which ensures that the cocycle- and cobounding - type conditions in all lower degrees hold .    as in definition [ conditions_and_obstructions ] 
   consider a whiskered polygon @xmath175 with 2-cell @xmath174 and let @xmath249 ( resp . 
   @xmath250 ) be the replacement labeling on the maximal path which traverses domain ( resp . 
   codomain ) of @xmath174 . 
   suppose the first @xmath268 edges lie in the domain whisker . in this case , for @xmath269 , @xmath246 and @xmath270 
   are equivalents , and the orderings on the edge labels of the domain whisker induced by restricting the partial orderings on @xmath271 and @xmath272 coincide . 
   let @xmath273 be the resulting partially ordered set of equivalence classes of wffs . 
   [ strong_vanishing ]    given a well - formed labeling of a whiskered polygon @xmath175 with 2-cell @xmath174 , let @xmath236 be the set of operations occuring in the replacement labelings of both paths , and @xmath244 a family of parallels for @xmath236 . 
   recall that @xmath274 is the theory of parallels for @xmath175 ( and thus includes the elements of @xmath244 . 
   let @xmath268 and @xmath273 be as in the discussion above . and 
   suppose the last @xmath275 edges of each path lie in the codomain whisker . 
   the _ @xmath252 order strong vanishing condition _ associated to @xmath175 ( for @xmath276 ) is the condition that    @xmath277    where the hatting of wffs indicates the result of the following process : for each maximal element of @xmath273 select a variable of the same type which does not occur in among the variables used in the labeling ; the hatted wff is the result of replacing all wfss in that equivalence class with the corresponding variable ; and the universal quantification ranges over all parallels in @xmath274 to the labels on the corresponding edge of the codomain whisker . 
   we then have    [ strong_works ] for any well - formed labeling of a whiskered polygon @xmath175 with operations @xmath278 and a family of parallels @xmath279 , the @xmath280 order strong vanishing condition implies the cocycle - type condition , the @xmath281 order strong vanishing condition implies the @xmath281 order cobounding - type condition , and , moreover , if the @xmath281 order vanishing conditions hold for all @xmath282 , then the @xmath252 order coboundary - type expression is equal to    @xmath283    where @xmath268 and @xmath275 are as in the previous definition and the meanings of the sequences of wffs are as in definition [ conditions_and_obstructions ] . 
   the cocycle - type condition simply an instantiation of the @xmath280 order strong vanishing condition , while the @xmath281 order cobounding - type condition is the sum of all instantiations of the @xmath281 order strong vanishing condition ranging over all choices of parallels for the labels of edges in the whiskers .    for the last statement , 
   notice that the terms in the @xmath252 order coboundary - type expression which are not represented in the expression of the proposition all have at least one label on an edge of one of the whiskers which is of positive degree . 
   these terms can be partitioned into subsets which according to the degrees of the labels in the whiskers . for each choice of degrees for the labels in the whiskers , the terms are an instatiation of the strong cocycle - type condition , and thus add to zero . 
   two sorts of whiskered polygons satisfying the strong vanishing conditions arise in practice : `` non - trivial '' whiskered polygons in which the strong vanishing conditions follow from the well - formedness of the labeling and deformation theoretic cocycle and cobounding conditions satisfied by the parallels of the labels in the boundary of the 2-cell , and `` trivial '' whiskered polygons in which the sets of edge labels on the domain and codomain of the 2-cell differ only by changing the choice of totalization of the partial order on wfss of the final label in the path and substitution of equivalent wffs . 
   we formalize this in the following propositions :    [ nontrivial ] given a well - formed labeling of a whiskered polygon @xmath175 with 2-cell @xmath174 , let @xmath236 be the set of operations occuring in the replacement labelings of both paths , and @xmath244 a family of parallels for @xmath236 . 
   let @xmath268 , @xmath275 , @xmath0 , @xmath284 and @xmath273 and the hatting of wffs be as definition [ strong_vanishing ] . 
   if @xmath285 and @xmath286 are a well - formed labeling of the directed polygon @xmath174 , then @xmath175 satisfies the @xmath287 order strong vanishing condition . 
   if , moreover , this labeling of @xmath174 and the system of parallels satisfy the cocycle - type condition ( resp . 
   the @xmath252 order cobounding - type condition ) then the well - formed labeling of @xmath175 and system of parallels satisfies the @xmath280 order ( resp . 
   @xmath252 order strong vanishing condition ) . 
   notice first that the condition that the labeling of @xmath174 be well - formed means that the wff labeling the two edges incident with the 0-codomain of @xmath174 are equivalents , and thus their difference ( as an arrow - valued operation ) vanishes . 
   the well - formedness may thus be viewed as a zeroth order analogue of the cocycle - type and cobounding - type relations . 
   the proposition is immediate once it is observed that at each order the strong vanishing condition on @xmath175 is simply the result of applying the parallels to labels on the codomain whisker to the terms of the expression of that order which vanishes on @xmath174 .    in practice 
   the labels on @xmath174 come from an equational condition ( associativity , functoriality , or naturality in the present work ) and the cocycle - type and cobounding - type conditions are actual cocycle and cobounding conditions derived from the requirement that deformations preserve the equational condition , while the whiskers arise in taking coboundaries of the corresponding obstruction .    [ trivial ] given a well - formed labeling of a whiskered polygon @xmath175 with 2-cell @xmath174 , let @xmath268 , @xmath275 , @xmath0 , @xmath284 , and @xmath273 and the hatting of wffs be as definition [ strong_vanishing ] . 
   if @xmath288 and @xmath285 is obtained from @xmath286 by changing the totalization of the partial ordering on the wfss of the label on the edge(s ) incident with the 0-codomain of @xmath175 and replacing wffs with equivalents , then the well - formed label of @xmath175 satisfies the strong vanishing conditions of all orders for any system of parallels .    in this case for any choice of parallels labels on edges of the codomain whisker , and of degrees in a system of parallels for the corresponding @xmath289 and @xmath290 , the resulting expressions for the two paths are equivalent and thus their difference is zero , and summing over all such choices of total order @xmath291 gives the @xmath281 order strong vanishing condition for @xmath175 . 
   we refer to whiskered polygons equipped with a labeling satifying the hypotheses of proposition [ trivial ] as trivial whiskered polygons . 
   finally , we need to describe in general the relationship between the cohomological description of infinitesimal deformations and the evident expression of the same data in terms of systems of parallels and the various expressions given by labeling of ( whiskered ) polygons and tiled spheres .    if @xmath3 is a @xmath0-linear pasting diagram , @xmath176 its theory , or an extension thereof with the same types , its deformation theory is _ polygonizable _ if the cochain group in which cocycles specify first order deformations admits a direct sum decomposition indexed a family of equational axioms of the theory each of which can be expressed as the vanishing of the difference of the values of the paths in a well - formed labeling of a directed polygon ( we call such a polygon equipped with its well - formed labeling an _ axiomatic polygon _ ) , and which , moreover satisfy    * the vanishing of each direct summand of the cocycle is precisely the cocycle - type condition associated to the axiomatic polygon indexing the direct summand . * each direct summand of the @xmath252 order obstruction is the obsturuction - type expression associated to the axiomatic polygon indexing the direct summand . 
   * the cobounding condition for the extension of a deformation to the next degree is the direct sum of the cobounding - type conditions associated to the axiomatic polygons . * and the coboundary of an obstruction admits a direct sum decomposition in which each direct summand is a signed sum of expressions specified by labeling whiskered polygons with the polygon labeled by the obstruction , and the whiskers labeled by operations of @xmath176 ( degree 0 labels in the system of parallels naming the deformation terms ) . 
   we are now in a position to state a theorem which encapsulates shrestha s polygonal method for our purposes :    if @xmath176 is the theory of a pasting diagram , or an extension thereof , and admits a polygonizable deformation theory , then all obstructions are cocycles , provided for each direct summand of the coboundary of the obstruction , there exists a tiled sphere , and a well - formed labeling of the tiled sphere with the properties    * each whiskered polygon naming a summand in the signed sum of p4 occurs exactly once in the tiled sphere  either in the domain ( resp . 
   codomain ) if its sign is positive ( resp . 
   negative ) , or with its domain and codomain swapped and in the codomain ( resp . 
   domain ) if its sign is positive ( resp . 
   negative ) . 
   * every 2-cell which is not part of the whiskered polygons of s1 is trivial in the sense of proposition [ trivial ]    this follows immediately from propositions [ elementary ] , [ strong_works ] , [ nontrivial ] and [ trivial ] . 
   figures 1 through 6 then establish the following :    [ singlecomposition ] if @xmath3 is a pasting diagram with a single instance of composing two natural transformations or of composing a natural transformation with a functor , then all obstructions to its deformation are cocycles .    in principal , it appears , one could apply the polygonal method to directly show that obstructions to the deformation of any given ( arbitrarily complicated ) pasting diagram are cocycles  however a metatheorem to this effect has proved to be beyond the authors capabilities . instead , we will approach the general problem indirectly by reducing the deformations of any pasting diagram to deformations of a related pasting diagram all of whose cells are one of a small finite set of forms , provided one can specify that certain cells are deformed trivially : the composition - free pasting diagrams for which the result was established in @xcite , those of [ singlecomposition ] and a short list of diagrams in which specified cells are deformed trivially : triangles and solid tetrahedra all of whose faces are identity natural transformations and are deformed trivially , and two diagrams derived from those of examples [ equalspost1comp ] and [ equalspre1comp ] by replacing the ( degenerate ) square labeled by @xmath13 with a bigon whose edged are labeled by the composite functors and two triangles labeled with the identity arrows of the composite functors , which must be deformed trivially . 
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   in @xcite deformations of a functor @xmath41 ( resp . a natural transformation @xmath19 ) in which the source and target are left undeformed ( or to say the same thing differently deformed trivially in the strong sense ) were shown to be governed by the hochschild complex @xmath292 ( resp . 
   @xmath1 ) .    for the desired reduction we will need to consider deformations of pasting diagrams in which a natural transformation ( in particular the identity natural transformation ) is deformed trivially in the strong sense while its domains and codomains are deformed , possibly nontrivially .    as a warm - up and for potential use in other applications , let us consider first the problem of deforming the ( pasting ) diagram @xmath3 consisting of two categories @xmath42 and @xmath109 and a functor @xmath38 , subject to the requirement that @xmath41 be deformed trivially .    without the restriction that @xmath41 
   be deformed trivially , a(n @xmath293 order ) deformation would be determined by a family of parallels for the set of operations @xmath294 , the compositions in @xmath42 and @xmath109 and the arrow - part of @xmath41 satisfying the usual cocycle and cobounding conditions . 
   the restriction that @xmath41 be deformed trivially requires that all of the positive order parallels to @xmath41 be zero . 
   cohomologically , the triples of degree @xmath0 parallels @xmath295 ( in the absence of the triviality requirement ) lie in @xmath155 , the mapping cone on @xmath296 and satisfy cocycle ( @xmath297 ) and cobounding ( @xmath298 ) conditions . 
   the requirement that @xmath299 together with the coboundary condition thus implies that @xmath300 lies in @xmath301 . 
   in fact we have    first order deformations of the ( pasting ) diagram @xmath3 consisting of two categories @xmath42 and @xmath109 and a functor @xmath38 , subject to the requirement that @xmath41 be deformed trivially are classified by the second cohomology of @xmath301 . 
   moreover all obstructions to deforming @xmath3 with @xmath41 deformed trivially are cocycles in the third cochain group of @xmath301 and an @xmath293 order deformation with @xmath41 deformed trivially can be extended to an @xmath302 order deformation with @xmath41 deformed trivially if and only if there is a degree 2 cochain in @xmath301 cobounding the degree @xmath163 obstruction . 
   the classification of first order deformations follows from the remarks above . 
   the rest of the proposition follows from the corresponding result in @xcite without the triviality restriction once it is shown that all obstructions , which _ a priori _ lie in the cone on @xmath303 lie in the kernel ( considered as a subcomplex under the obvious inclusion ) .    as observed above 
   , @xmath300 lies in @xmath301 . 
   so suppose as an induction hypothesis that for all @xmath304 @xmath305 lies in @xmath301 ( or equivalently and more usefully @xmath306 ) . 
   the degree @xmath163 obstruction @xmath168 in the cone then has coordinates    @xmath307,\ ] ]    @xmath308,\ ] ]    and @xmath309    the last vanishes since each term involves @xmath310 for some @xmath311 ( and in the latter sum all the @xmath312 are bilinear ) . 
   it thus remains only to show that @xmath313 of the first coordinate equals @xmath314 of the second . 
   computing    @xmath315)\ ] ]    @xmath316\ ] ]    @xmath317\ ] ]    in each case the equality holds by the induction hypothesis , and in the second case the bilinearity @xmath312 for each @xmath0 . 
   thus the proposition holds . 
   a similar result holds for deformations of a diagram consisting of two parallel functors @xmath49 and a natural transformation @xmath19 between them which is to be deformed trivially : without the triviality restriction , the deformation complex is the cone on @xmath318 . with the restriction , @xmath319 becomes the deformation complex :    first order deformations of the pasting diagram @xmath3 consisting of two categories a pair of parallel functors @xmath49 , their sources and target and a natural transformation @xmath19 subject to the requirement that @xmath28 be deformed trivially are classified by the second cohomology of @xmath319 . 
   moreover all obstructions to deforming @xmath3 with @xmath28 deformed trivially are cocycles in the third cochain group of @xmath319 and an @xmath293 order deformation with @xmath28 deformed trivially can be extended to an @xmath302 order deformation with @xmath28 deformed trivially if and only if there is a degree 2 cochain in @xmath319 cobounding the degree @xmath163 obstruction . 
   the argument is essentially identical to that for the previous proposition , except for the calculation showing that obstructions , which _ a priori _ lie in the full deformation complex of @xmath3 lie in @xmath319 .    to see that all obstructions lie in @xmath319 , consider figure [ naturality_obstructions_are_cocycles ] . 
   the condition that degree @xmath163 obstruction lies in @xmath319 is precisely the condition that the signed sum of the degree @xmath163 obstruction - type expressions associated to the non - trivial faces other than the three hexagons vanish , when all higher order parallels of @xmath28 are instantiated as 0 . but 
   observe that the other faces are all trivial or evaluate to 0 , in all degrees greater than 0 , trivially under the hypothesis that all higher order parallels of @xmath28 are 0 , so the result follows from the general principles of the polygonal method . 
   applying the same argument _ mutatis mutandis _ to figure [ nat_from_composition_obstructions_are_cocycles ] shows    [ trivialtriangle ] first order deformations of the pasting diagram @xmath3 consisting of three categories @xmath320 and @xmath321 three functors @xmath38 , @xmath322 and @xmath323 and a natural transformation @xmath324 subject to the requirement that @xmath28 be deformed trivially are classified by the second cohomology of @xmath325 , where    @xmath326:{\frak c}^\bullet(\partial d ) \rightarrow c^\bullet(g(f),h ) .\ ] ]    where @xmath327 is the diagram obtained by omitting @xmath28 . 
   moreover all obstructions to deforming @xmath3 with @xmath28 deformed trivially are cocycles in the third cochain group of @xmath325 and an @xmath293 order deformation with @xmath28 deformed trivially can be extended to an @xmath302 order deformation with @xmath28 deformed trivially if and only if there is a degree 2 cochain in @xmath325 cobounding the degree @xmath163 obstruction .    in the special case where @xmath328 and @xmath329 , it is intuitively clear that the deformations of @xmath3 in which @xmath28 is deformed trivially are completely determined by the deformation complex of the simpler diagram @xmath330 . 
   in fact we have    [ reducestotwoedges ] in the deformation complex of the pasting diagram @xmath3 consisting of three categories @xmath320 and @xmath321 three functors @xmath38 , @xmath322 and @xmath331 and the natural transformation @xmath332 the complex @xmath325 where @xmath75 is as in proposition [ trivialtriangle ] is isomorphic to the deformation complex of the subdiagram @xmath330 .    the obvious quotient map from the deformation complex of @xmath327 to    @xmath333    is split by the obvious inclusion . 
   the kernel of the quotient map is easily seen to be isomorphic to @xmath325 , while the quotient of the inclusion is easily seen to be the deformation complex of @xmath334 .    combining propositions [ uniontopushout ] , [ trivialtriangle ] and [ reducestotwoedges ] then gives    [ trivialthreecell ]    let @xmath3 be a pasting diagram which could arise by the following construction : begin with a computad consisting of a composable 1-pasting diagram with any number of edges , a single edge with the same domain and codomain as the 1-pasting diagram , a pair of 2-cells both with the resulting circle as boundary ( and a 3-cell , or not ) . now , triangulate each 2-cell in a way corresponding to any parenthesization of the edges of the original 1-pasting diagram . apply a map of pasting schemes to the underlying pasting scheme of @xmath335 such that the value of every 2-cell is an identity natural transformation with domain given by two composable functors and codomain given by their composition . 
   if @xmath3 is such a pasting diagram , then the single edge in the initial part of the construction is labeled with the composition of the initial 1-pasting diagram , and if , moreover , @xmath70 is the chain map from the deformations complex @xmath155 to the direct sum indexed by the triangular 2-cells of @xmath3 whose coordinate for a 2-cell @xmath336 with edges labeled by a composable pair of functors @xmath38 and @xmath322 and their composition @xmath337 is given by projection onto @xmath338 followed by the map @xmath75 of proposition [ trivialtriangle ] onto @xmath339 , then @xmath340 classifies the deformations of @xmath3 in which all of the identity natural transformations are deformed trivially , and is isomorphic to the deformation complex of the original composable 1-pasting diagram . 
   the key thing to note here is that because all triangulations of disks are shellable , the pasting diagram can be constructed out of the trivial triangles of proposition [ trivialtriangle ] by iterated pushouts with diagrams previously so constructed until a pushout of the two triangulated disks along their boundary is made , and that once the triviality condition are imposed the presence or absence of the 3-cell makes no difference to the kernel . 
   note also that a 2-dual form of proposition [ trivialthreecell ] in which all of the identity natural transformations have the composition as their domain and two composable functors as their codomain also holds with the same proof . 
   the proofs of propositions [ trivialtriangle ] and [ reducestotwoedges ] can be extended to show    [ fatpost1comp ] let @xmath3 be the pasting diagram consisting of three categories @xmath341 , functors @xmath342 , the composite functors @xmath343 and @xmath344 , natural transformations @xmath19 , @xmath345 labeling bigons , and @xmath346 and @xmath347 labeling triangles with two edges labeled by the composands as domain and one edge labeled by the composition as codomain , and a single 3-cell asserting the commutativity of the diagram . letting @xmath70 
   be the map with one coordinate for each triangle given by @xmath75 of proposition [ trivialtriangle ] for that 2-cell , @xmath340 classifies the deformations of @xmath3 in which the two identity natural transformations are deformed trivially , and is isomorphic to the complex of example [ equalspost1comp ] .    and 
   [ fatpre1comp ] let @xmath3 be the pasting diagram consisting of three categories @xmath341 , functors @xmath348 , the composite functors @xmath349 and @xmath350 , natural transformations @xmath19 , @xmath351 labeling bigons , and @xmath352 and @xmath353 labeling triangles with two edges labeled by the composands as domain and one edge labeled by the composition as codomain , and a single 3-cell asserting the commutativity of the diagram . letting @xmath70 be the map with one coordinate for each triangle given by @xmath75 of proposition [ trivialtriangle ] for that 2-cell , @xmath340 classifies the deformations of @xmath3 in which the two identity natural transformations are deformed trivially , and is isomorphic to the complex of example [ equalspre1comp ] . 
   observe , that the conclusion of both proposition [ fatpost1comp ] and [ fatpre1comp ] hold regardless of which part of the boundary ( two composable functors or their composition ) is the domain and which is the codomain of each of the triangular faces labeled with identity natural transformations . 
   we are now in a position to prove the main result : that the deformations of any pasting diagram are classified by deformation complex in which all obstructions to extending a deformation of order @xmath163 to a deformation of order @xmath354 are cocycles . to do this we will replace the diagram with a more complex diagram with simpler parts , whose deformation theory , when some of the 2-cells are required to be deformed trivially , coincides with that of the original pasting diagram . 
   a pasting diagram is _ finely divided _ 
   if every 2-cell is either a bigon or a triangle labeled with the identity natural transformation , and every 3-cell is of one of the following forms :    * a composition - free pillow with bigonal cross - section ( imposing the equality of two natural transformations between the same pair of functors ) * a diagram of the form in proposition [ trivialthreecell ] or its 2-dual * a `` triangular pillow '' as in example [ triangularpillow ] * a diagram of the form in example [ equalspost1comp ] * a diagram of the form in proposition [ fatpost1comp ] * a diagram of the form in example [ equalspre1comp ] * a diagram of the form in proposition [ fatpre1comp ]    every pasting diagram @xmath3 can be replaced with a finely divided pasting diagram @xmath355 whose underlying cell complex is a subdivision of the underlying cell complex of @xmath3 and whose deformations when all identity natural transformations labeling triangles are deformed trivially are equivalent to deformations of @xmath3 . 
   begin by subdividing each 2-cell as follows : insert a bigon with 1-cells labeled by the 1-composition of the domain and codomain of the 2-cell and labeled by the same natural transformation as the original 2-cell , making the domain ( resp . 
   codomain ) of the bigon coincide with that of the original 2-cell if the domain ( resp . 
   codomain ) consists of a single 1-cell . 
   label the complementary cell(s ) with identity natural transformations to be deformed trivially . 
   now each of the complementary cells has a domain ( resp . 
   codomain ) which is a sequence of 1-cells labeled by functors and a codomain ( resp . 
   domain ) which is a single 1-cell labeled with their composition . 
   choose a parenthesization of the composition and subdivide the 2-cell in the usual way corresponding to a parenthesization , labeling the new 1-cells with the appropriate pairwise compositions of labels already present and all 2-cells with identity natural transformation ( to be deformed trivially ) . 
   now , for each 3-cell which is not already of one of the forms specified in the proposition , choose an order of pasting composition for the domain ( resp . 
   codomain ) , and use this to create a subdivision of the 3-cell as follows :    iteratively , for each 1-composition of a natural transformation with a functor , insert a diagram of the form in proposition [ fatpost1comp ] or [ fatpre1comp ] as appropriate and for each binary 2-composition , insert a triangular pillow . 
   when this has been done for both the domain and codomain , there will be bigons labeled with the pasting compostion of the domain and codomain of the original 3-cell . 
   identify their domains ( resp . 
   codomains ) . 
   the 3-cells of the resulting subdivision are now those explicitly inserted in the construction , a composition - free pillow with bigonal cross - section , and , we claim , 3-cells all of the form in proposition [ trivialthreecell ] . 
   verifying the claim is a matter of keeping track of the triangular faces in such a way as to organize them into the pairs of triangulated disks with appropriate boundaries : note that immediately after the subdivision of 2-cells into a bigon and triangles , the triangulated cell(s ) ( if any  the original 2-cell could have been a bigon ) are of the desired form : they have a domain ( resp . 
   codomain ) consisting of a composable 1-pasting diagram and codomain ( resp . 
   domain ) consisting of a single edge labeled with the composition . 
   call such a triangulated 2-cell a `` nicely triangulated 2-cell '' . 
   thus the remainder of the construction begins with the original 2-cells decomposed into a family of nicely triangulated 2-cells and family of bigons . throughout the rest of the construction the family of nicely triangulated 2-cells 
   will be iteratively replaced with families with fewer nicely triangulated 2-cells ( but with more triangles in the newer ones ) . 
   for addition of a 3-cell of the forms in propositions [ fatpost1comp ] or [ fatpre1comp ] , the triangles will have as two of their edges the single edge labeled with the composition from previously constructed nicely triangulated 2-cells . 
   for each of the triangles , replace these two nicely triangulated 2-cells with the union of the triangle and the two nicely triangulated 2-cells , noticing that it is nicely triangulated . when the triangular pillows and the bigonal pillow are adjoined , the family of nicely triangulated 2-cells is unchanged . 
   thus , when the construction ends , the remaining 3-cells in the decomposition must each be bounded by the union of two nicely triangulated 2-cells . 
   we can now prove the main result ( recalling the dimension conventions of @xcite which leaves the deformation cohomology of a pasting digram in dimension @xmath356 ) :    [ main ] for any pasting diagram @xmath3 of @xmath0-linear categories , @xmath0-linear functors and natural transformations , the deformations of @xmath3 are classified by the @xmath356-cohomology of the @xmath340 , where @xmath70 is the map from the deformation complex @xmath357 given by the same description as in proposition [ trivialthreecell ] , and all obstructions to extension of a degree @xmath163 deformation to a degree @xmath354 deformation are 0-cocycles .    for the classification statement , first observe that the proof of proposition [ uniontopushout ] applies equally well to taking unions of pasting diagrams in which some 2-cells have been specified as deforming trivially ( provided that cells in the intersection are specified as deforming trivially in both pasting diagrams of which the union is being taken ) . 
   the isomorphisms of propositions [ trivialthreecell ] , [ fatpost1comp ] and [ fatpre1comp ] then combine by universal property of pushouts to give an isomorphism between @xmath155 and @xmath340 in the category of chain complexes .    that all obstructions are cocycles is a condition local to each cell together with its boundary , and all cells of @xmath355 are either deformed trivially , so that the obstruction vanishes by triviality , are of the forms which were shown to have vanishing obstructions in theorem [ singlecomposition ] , or are composition - free and thus have vanishing obstructions by the results of @xcite . 
   the primary purpose of this paper , to complete the deformation theory for pasting diagrams of @xmath0-linear categories described in @xcite , was accomplished by theorem [ main ] . 
   its primary importance , however , may lie less in the result than in the techniques used . 
   it is the authors intent to apply the deformation theory of pasting diagrams to the still - open problem of providing a complete deformation theory for monoidal categories in which all arrow - valued elements are deformed simultaneously , rather than just the structure maps as in @xcite . 
   this will require using extensions of @xmath358 hinted at above , in which monoidal prolongations are included as additional operations , as the basis of the polygonal method . 
   it will also avoid the difficulties in @xcite arising from the need to intuit the correct formulas for higher differentials in a multicomplex from the scant data provided by the instances arising in the deformation theory .    likewise , although for the present purpose there is something unsatisfying about our detour though finely divided pasting diagrams with specified trivially - deformed identity cells 
   we believe the direct result , that all obstructions in the deformation complex of any pasting diagram as defined in @xcite are cocycles , though it is beyond our present ability to prove  like the polygonal technique , this detour may have other applications . 
   the difficulty with the direct result was that although 2-categories , and the pieces of them corresponding to direct summands in deformation complexes are inherently `` globular '' , pasting diagrams are `` opetopic '' . 
   the reduction of the opetopic to trivial ( mostly simplicial ) elements and globular elements might well find application in connecting the zoo of opetopic definitions of weak n - categories with the globular approach of batanin .
summary: we continue the development of the infinitesimal deformation theory of pasting diagrams of @xmath0-linear categories begun in @xcite . in @xcite 
   the standard result that all obstructions are cocycles was established only for the elementary , composition - free parts of pasting diagrams . in the present work we give a proof for pasting diagrams in general . as tools we use the method developed by shrestha @xcite of representing formulas for obstructions , along with the corresponding cocycle and cobounding conditions by suitably labeled polygons , giving a rigorous exposition of the previously heuristic method , and deformations of pasting diagrams in which some cells are required to be deformed trivially . 
   [ multiblock footnote omitted ]