Fixes for the degree function (#3298)

* Remove :task from sub methods for modules.

* Introduce new function submodule.

* Remove all deprecated uses of sub.

* Export submodule.

* Fix tests.

* Fix tests again.

* Register morphisms in general.

* Rebase network of natural maps on WeakKeyIdDicts.

* Clean up the morphisms network and dont store the actual morphisms.

* Some small fix for Wolfram.

* Some fixes and marking the test as broken again.

* Fix documentation.

* Another small fix for Wolfram.

* Fix the fix.

* Fix method selection.

* Introduce kw argument for caching morphisms.

* Redirect previous usages.

* Adjust tests.

* submodule -> sub_object.

* Fix docstrings and signatures for quo.

* Fix methods for quo.

* Adjust documentation.

* Fix up usages of quo.

* Export new functions.

* Some fixes.

* Fix tests.

* Fix tests.

* Fix tests.

* Fix tests.

* Fix tests.

* Fix tests.

* Squashed changes.

* Fix tests.

* Fix up the truncation.

* New rebase.

* Fix tests.

* WIP with debug messages.

* Progress in debugging.

* WIP on debugging.

* Write truely generic presentation and kernel methods [no ci].

* Repair graded_map to accept zero entries.

* Add tensor decomposition function to return value.

* Add some assertions.

* Repair kernel routine.

* Rewrite hom methods for modules.

* Add generic method for simplification of SubquoModuleElems.

* Add missing check argument.

* Fix or disable brittle tests.

* Fix faulty merge.

* Go back to old hom.

* Switch tests to old hom.

* Clean up some deprecated assertions.

* Repair truncate.

* Disable internal checks.

* Update docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/free_modules.md

* Revert "Go back to old hom."

This reverts commit a6b7c15.

* Revert "Switch tests to old hom."

This reverts commit 178eaf3.

* Disable duplicate method.

* Adjust printing.

* Fix tests.

* Fix keyword argument for duals.

* Fix doctests.

* Fix tests.

* Fix doctests.

* Delete some falsely added files.

* Remove deprecated code.

* Another round of disabling internal checks.

* Disable various internal checks.

* Disable further internal checks.

* Set random seeds in tests.

* Adjust use of keyword argument.

* Fix more random seeds.

* Fix the generic simplify method for SubquoModuleElems.

* Revert "Remove deprecated code."

This reverts commit 190256b.

* Repair revert to old hom.

* Repair preimage function.

* Repair degree function.

* Adjust tests to use of repaired old hom.

* Add dummy simplify function for FreeModuleElems.

* Fix doctests.

* Repair simplify.

* Fix documentation.

* Fix doctests.

* Some tuning.

* Fix tests.

* Bugfix from running book code run.

* Remove some debugging artifacts.

* Fix faulty merge.

* Restrict signature.

* Update flattenings of modules over towers of polynomial rings to make the tests run again.

* Small fix.

* Disable the superfluous kernel routine.

* Restrict some further signatures to cases Singular can handle.

* Add tests for flattenings of modules.

* Readd methods which are not superfluous, yet.

---------

Co-authored-by: Lars Göttgens <[email protected]>
Co-authored-by: Lars Göttgens <[email protected]>

docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/free_modules.md

@@ -174,7 +174,7 @@ is_homogeneous(f::FreeModElem)
 ```
 
 ```@docs
-degree(f::FreeModElem)
+degree(f::FreeModElem{T}) where {T<:Union{<:MPolyDecRingElem, <:MPolyQuoRingElem{<:MPolyDecRingElem}}}
 ```
 
 

docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/module_operations.md

@@ -37,7 +37,7 @@ direct_product(M::ModuleFP{T}...; task::Symbol = :prod) where T
 ## Truncation
 
 ```@docs
-truncate(M::ModuleFP, g::FinGenAbGroupElem, task::Symbol = :with_morphism)
+truncate(M::ModuleFP, g::FinGenAbGroupElem, task::Symbol=:with_morphism)
 ```
 
 ## Twists

docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/subquotients.md

@@ -359,11 +359,11 @@ intersect(M::SubquoModule{T}, N::SubquoModule{T}) where T
 ## Submodules and Quotients
 
 ```@docs
-sub(M::ModuleFP{T}, V::Vector{<:ModuleFPElem{T}}, task::Symbol = :with_morphism) where T
+sub(M::ModuleFP{T}, V::Vector{<:ModuleFPElem{T}}) where T
 ```
 
 ```@docs
-quo(M::ModuleFP{T}, V::Vector{<:ModuleFPElem{T}}, task::Symbol = :with_morphism) where T
+quo(M::ModuleFP{T}, V::Vector{<:ModuleFPElem{T}}; cache_morphism::Bool=false) where T
 ```
 
 ## Homomorphisms From Subqotients

docs/src/CommutativeAlgebra/homological_algebra.md

@@ -104,7 +104,7 @@ homology(C::ComplexOfMorphisms{<:ModuleFP}, i::Int)
 ## Hom and Ext
 
 ```@docs
-hom(M::ModuleFP, N::ModuleFP, algorithm::Symbol=:maps)
+hom(M::ModuleFP, N::ModuleFP; algorithm::Symbol=:maps)
 ```
 
 ```@docs

experimental/DoubleAndHyperComplexes/src/Morphisms/ext.jl

@@ -56,18 +56,18 @@ function (fac::HomMapFactory)(hc::AbsHyperComplex, p::Int, i::Tuple)
   cod = hc[i_inc]
 
   if iszero(dom) || iszero(cod)
-    return hom(dom, cod, elem_type(cod)[zero(cod) for i in 1:ngens(dom)])
+    return hom(dom, cod, elem_type(cod)[zero(cod) for i in 1:ngens(dom)]; check=false)
   end
 
   if p <= dim(d)
     # contravariant induced map on first argument
     i1_inc = Tuple(-i1 - [k == p ? inc : 0 for k in 1:dim(d)])
     img_gens = [homomorphism_to_element(cod, compose(map(d, p, i1_inc), element_to_homomorphism(g))) for g in gens(dom)]
-    return hom(dom, cod, img_gens)
+    return hom(dom, cod, img_gens; check=false)
   else
     # covariant induced map on second argument
     img_gens = [homomorphism_to_element(cod, compose(element_to_homomorphism(g), map(c, p - dim(d), Tuple(i2)))) for g in gens(dom)]
-    return hom(dom, cod, img_gens)
+    return hom(dom, cod, img_gens; check=false)
   end
 end
 
@@ -214,7 +214,7 @@ function (fac::InterpretationMorphismFactory)(self::AbsHyperComplexMorphism, I::
   projections_tot = projections_for_summand(tot, offset)
   M = dom[i]
   N = cod[j]
-  result = hom(M, N, elem_type(N)[zero(N) for i in 1:ngens(M)])
+  result = hom(M, N, elem_type(N)[zero(N) for i in 1:ngens(M)]; check=false)
   for (k, K) in enumerate(indices_dom)
     pr = projections_dom[k]
     for (l, L) in enumerate(indices_cod)
@@ -341,7 +341,7 @@ function (fac::HomComplexMorphismFactory)(self::AbsHyperComplexMorphism, I::Tupl
     img_gens = elem_type(cod_mod)[homomorphism_to_element(cod_mod, compose(element_to_homomorphism(v), fac.codomain_morphism[I_out])) for v in g]
   end
 
-  return hom(dom_mod, cod_mod, img_gens) # Set to false eventually
+  return hom(dom_mod, cod_mod, img_gens; check=false) # Set to false eventually
 end
 
 function can_compute(fac::HomComplexMorphismFactory, self::AbsHyperComplexMorphism, I::Tuple)

experimental/DoubleAndHyperComplexes/src/Morphisms/free_resolutions.jl

@@ -22,10 +22,11 @@ function (fac::ResolutionModuleFactory{ChainType})(c::AbsHyperComplex, I::Tuple)
   end
 
   if isone(i)
-    aug = hom(c[0], fac.orig_mod, gens(fac.orig_mod))
+    aug = hom(c[0], fac.orig_mod, gens(fac.orig_mod); check=false)
+    aug.generators_map_to_generators = true
     K, inc = kernel(aug)
     next = _make_free_module(K, gens(K))
-    phi = hom(next, c[0], ambient_representatives_generators(K))
+    phi = hom(next, c[0], ambient_representatives_generators(K); check=false)
     push!(fac.map_cache, phi)
     return next
   end
@@ -45,7 +46,7 @@ function (fac::ResolutionModuleFactory{ChainType})(c::AbsHyperComplex, I::Tuple)
     return next
   end
   next = _make_free_module(K, gens(K))
-  phi = hom(next, c[i-1], ambient_representatives_generators(K))
+  phi = hom(next, c[i-1], ambient_representatives_generators(K); check=false)
   push!(fac.map_cache, phi)
 
   return next
@@ -54,10 +55,10 @@ end
 function zero_object(M::ModuleFP)
   if is_graded(M)
     result = graded_free_module(base_ring(M), [])
-    return result, hom(result, M, elem_type(M)[])
+    return result, hom(result, M, elem_type(M)[]; check=false)
   else
     result = FreeMod(base_ring(M), 0)
-    return result, hom(result, M, elem_type(M)[])
+    return result, hom(result, M, elem_type(M)[]; check=false)
   end
 end
 
@@ -99,7 +100,8 @@ function free_resolution(::Type{T}, M::SubquoModule{RET}) where {T<:SimpleFreeRe
                                  )
   result = SimpleFreeResolution(M, internal_complex)
   MC = ZeroDimensionalComplex(M)[0:0] # Wrap MC as a 1-dimensional complex concentrated in degree 0
-  aug_map = hom(result[(0,)], M, gens(M)) # The actual augmentation map
+  aug_map = hom(result[(0,)], M, gens(M); check=false) # The actual augmentation map
+  aug_map.generators_map_to_generators = true
   aug_map_comp = MorphismFromDict(result, MC, Dict{Tuple, typeof(aug_map)}([(0,)=>aug_map]))
   result.augmentation_map = aug_map_comp
   return result, aug_map_comp

experimental/DoubleAndHyperComplexes/src/Morphisms/simplified_complexes.jl

@@ -80,7 +80,7 @@ function (fac::SimplifiedChainFactory)(d::AbsHyperComplex, Ind::Tuple)
     # Create the maps to the old complex
     img_gens_dom = elem_type(M)[sum(c*M[j] for (j, c) in S[i]; init=zero(M)) for i in I]
     new_dom = _make_free_module(M, img_gens_dom)
-    dom_map = hom(new_dom, M, img_gens_dom)
+    dom_map = hom(new_dom, M, img_gens_dom; check=false)
 
     if haskey(fac.maps_to_original, i)
       # This means that for the next map a partial or 
@@ -93,7 +93,7 @@ function (fac::SimplifiedChainFactory)(d::AbsHyperComplex, Ind::Tuple)
 
     img_gens_cod = elem_type(N)[sum(c*N[i] for (i, c) in T[j]; init=zero(N)) for j in J]
     new_cod = _make_free_module(N, img_gens_cod)
-    cod_map = hom(new_cod, N, img_gens_cod)
+    cod_map = hom(new_cod, N, img_gens_cod; check=false)
 
     if haskey(fac.maps_to_original, next)
       fac.maps_to_original[next] = compose(cod_map, fac.maps_to_original[next])
@@ -122,7 +122,7 @@ function (fac::SimplifiedChainFactory)(d::AbsHyperComplex, Ind::Tuple)
       # end
       push!(img_gens_dom, v)
     end
-    dom_map_inv = hom(M, new_dom, img_gens_dom)
+    dom_map_inv = hom(M, new_dom, img_gens_dom; check=false)
 
     if haskey(fac.maps_from_original, i)
       fac.maps_from_original[i] = compose(fac.maps_from_original[i], dom_map_inv)
@@ -144,7 +144,7 @@ function (fac::SimplifiedChainFactory)(d::AbsHyperComplex, Ind::Tuple)
       w_new = sparse_row(base_ring(w), new_entries)
       push!(img_gens_cod, FreeModElem(w_new, new_cod))
     end
-    cod_map_inv = hom(N, new_cod, img_gens_cod)
+    cod_map_inv = hom(N, new_cod, img_gens_cod; check=false)
 
     if haskey(fac.maps_from_original, next)
       fac.maps_from_original[next] = compose(fac.maps_from_original[next], cod_map_inv)
@@ -184,7 +184,7 @@ function (fac::SimplifiedChainFactory)(d::AbsHyperComplex, Ind::Tuple)
     # Create the maps to the old complex
     img_gens_dom = elem_type(M)[sum(c*M[j] for (j, c) in S[i]; init=zero(M)) for i in I]
     new_dom = _make_free_module(M, img_gens_dom)
-    dom_map = hom(new_dom, M, img_gens_dom)
+    dom_map = hom(new_dom, M, img_gens_dom; check=false)
 
     if haskey(fac.maps_to_original, prev)
       fac.maps_to_original[prev] = compose(dom_map, fac.maps_to_original[prev])
@@ -195,7 +195,7 @@ function (fac::SimplifiedChainFactory)(d::AbsHyperComplex, Ind::Tuple)
 
     img_gens_cod = elem_type(N)[sum(c*N[i] for (i, c) in T[j]; init=zero(N)) for j in J]
     new_cod = _make_free_module(N, img_gens_cod)
-    cod_map = hom(new_cod, N, img_gens_cod)
+    cod_map = hom(new_cod, N, img_gens_cod; check=false)
 
     if haskey(fac.maps_to_original, i)
       fac.maps_to_original[i] = compose(cod_map, fac.maps_to_original[i])
@@ -214,7 +214,7 @@ function (fac::SimplifiedChainFactory)(d::AbsHyperComplex, Ind::Tuple)
       end
       push!(img_gens_dom, v)
     end
-    dom_map_inv = hom(M, new_dom, img_gens_dom)
+    dom_map_inv = hom(M, new_dom, img_gens_dom; check=false)
 
     if haskey(fac.maps_from_original, prev)
       fac.maps_from_original[prev] = compose(fac.maps_from_original[prev], dom_map_inv)
@@ -235,7 +235,7 @@ function (fac::SimplifiedChainFactory)(d::AbsHyperComplex, Ind::Tuple)
       w_new = sparse_row(base_ring(w), new_entries)
       push!(img_gens_cod, FreeModElem(w_new, new_cod))
     end
-    cod_map_inv = hom(N, new_cod, img_gens_cod)
+    cod_map_inv = hom(N, new_cod, img_gens_cod; check=false)
 
     if haskey(fac.maps_from_original, i)
       fac.maps_from_original[i] = compose(fac.maps_from_original[i], cod_map_inv)
@@ -357,7 +357,7 @@ end
 ### Helper functions
 function _make_free_module(M::ModuleFP, g::Vector{T}) where {T<:ModuleFPElem}
   if is_graded(M)
-    w = degree.(g)
+    w = _degree_fast.(g)
     return graded_free_module(base_ring(M), w)
   else
     return FreeMod(base_ring(M), length(g))
@@ -598,9 +598,9 @@ function _alt_simplify(M::SubquoModule)
   Z0, inc_Z0 = kernel(simp, 0)
 
   result_to_M = hom(result, M, 
-                    elem_type(M)[aug[0](simp_to_orig[0](inc_Z0(preimage(Z0_to_result, x)))) for x in gens(result)])
+                    elem_type(M)[aug[0](simp_to_orig[0](inc_Z0(preimage(Z0_to_result, x)))) for x in gens(result)]; check=false)
   M_to_result = hom(M, result,
-                    elem_type(result)[Z0_to_result(preimage(inc_Z0, orig_to_simp[0](preimage(aug[0], y)))) for y in gens(M)])
+                    elem_type(result)[Z0_to_result(preimage(inc_Z0, orig_to_simp[0](preimage(aug[0], y)))) for y in gens(M)]; check=false)
   return result, M_to_result, result_to_M
 end
 

experimental/DoubleAndHyperComplexes/src/Objects/koszul_complexes.jl

@@ -36,7 +36,7 @@ function (fac::KoszulMorphismFactory)(c::AbsHyperComplex, p::Int, i::Tuple)
   dom = c[i]
   cod = c[first(i) - 1]
   if first(i) == 0 || first(i) == r + 1
-    return hom(dom, cod, elem_type(cod)[zero(cod) for i in 1:ngens(dom)])
+    return hom(dom, cod, elem_type(cod)[zero(cod) for i in 1:ngens(dom)]; check=false)
   end
   return wedge_multiplication_map(dom, cod, fac.v)
 end

experimental/DoubleAndHyperComplexes/src/Objects/total_complexes.jl

@@ -119,7 +119,7 @@ function (fac::TotalComplexMapFactory)(c::AbsHyperComplex, p::Int, I::Tuple)
   next = d + inc
   dom = c[d]
   cod = c[next]
-  result = hom(dom, cod, elem_type(cod)[zero(cod) for i in 1:ngens(dom)])
+  result = hom(dom, cod, elem_type(cod)[zero(cod) for i in 1:ngens(dom)]; check=false)
   for (ind, J) in enumerate(index_cache(chain_fac)[d])
     for k in 1:dim(orig)
       target = collect(J) + (direction(orig, k) == :chain ? -1 : 1)*[(l == k ? 1 : 0) for l in 1:dim(orig)]

experimental/Schemes/DerivedPushforward.jl

@@ -79,7 +79,7 @@ function rank(phi::FreeModuleHom{FreeMod{T}, FreeMod{T}, Nothing}) where {T<:Fie
 end
 
 
-_regularity_bound(F::FreeMod) = maximum(Int(degree(a)[1]) for a in gens(F))
+_regularity_bound(F::FreeMod) = maximum(Int(degree(a; check=false)[1]) for a in gens(F))
 
 @doc raw"""
     simplify(c::ComplexOfMorphisms{ChainType}) where {ChainType<:ModuleFP}

src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Constructors.jl

@@ -186,7 +186,7 @@ function ambient_embedding(X::AbsProjectiveScheme)
   S = homogeneous_coordinate_ring(IP)
   T = homogeneous_coordinate_ring(X)
   I = defining_ideal(X)
-  pb = hom(S, T, gens(T))
+  pb = hom(S, T, gens(T); check=false)
   inc_sub = ProjectiveSchemeMor(X, IP, pb, check=false)
   return ProjectiveClosedEmbedding(inc_sub, I, check=false)
 end

src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Morphisms/Methods.jl

@@ -74,7 +74,8 @@ with default covering
       u = inv(OO(U_ij)(denom))
       mor_dict[U_ij] = morphism(U_ij, V_j, 
                                hom(OO(V_j), OO(U_ij), 
-                                   [OO(U_ij)(dehom(pbf(gen(SY, k))))*u for k in 1:ngens(SY) if k != j]
+                                   [OO(U_ij)(dehom(pbf(gen(SY, k))))*u for k in 1:ngens(SY) if k != j];
+                                   check=false
                                   )
                               )
     end
@@ -120,11 +121,11 @@ function pushforward(inc::ProjectiveClosedEmbedding, M::FreeMod)
   S = codomain(f)
   T = domain(f)
   S === base_ring(M) || error("rings do not match")
-  FT = graded_free_module(T, [degree(a) for a in gens(M)])
+  FT = graded_free_module(T, [_degree_fast(a) for a in gens(M)])
   I = image_ideal(inc)
   IFT, inc_IFT = I*FT
   MT = cokernel(inc_IFT)
-  id = hom(MT, M, gens(M), f)
+  id = hom(MT, M, gens(M), f; check=false)
   return MT, id
 end
 
@@ -141,7 +142,7 @@ function pushforward(inc::ProjectiveClosedEmbedding, M::SubquoModule)
   G, inc_G = sub(FT, vcat(gT, relT))
   Q, inc_Q = sub(G, gens(G)[length(gT)+1:end])
   MT = cokernel(inc_Q)
-  id = hom(MT, M, vcat(gens(M), elem_type(M)[zero(M) for i in 1:length(relT)]))
+  id = hom(MT, M, vcat(gens(M), elem_type(M)[zero(M) for i in 1:length(relT)]); check=false)
   return MT, id
 end
 

src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Attributes.jl

@@ -533,10 +533,10 @@ function relative_euler_sequence(X::AbsProjectiveScheme{<:Ring, <:MPolyRing})
   S = homogeneous_coordinate_ring(X)::MPolyDecRing
   W1 = kaehler_differentials(S)
   W0 = kaehler_differentials(S, 0)
-  theta = hom(W1, W0, [x*W0[1] for x in gens(S)])
+  theta = hom(W1, W0, [x*W0[1] for x in gens(S)]; check=false)
   W, inc = kernel(theta)
   Z = graded_free_module(S, 0)
-  inc_Z = hom(Z, W, elem_type(W)[])
+  inc_Z = hom(Z, W, elem_type(W)[]; check=false)
   comp = ComplexOfMorphisms(ModuleFP, [inc_Z, inc, theta], typ=:cochain, seed = -1)
   return comp
 end
@@ -578,10 +578,10 @@ function relative_cotangent_module(X::AbsProjectiveScheme{<:Ring, <:MPolyQuoRing
 
   SP = homogeneous_coordinate_ring(P)
   F = graded_free_module(SP, degree.(f))
-  jac = hom(F, eu[1], df)
+  jac = hom(F, eu[1], df; check=false)
   jac_res = _change_base_ring_and_preserve_gradings(phi, jac, codomain_change = res_Omega1)
   img_gens = [preimage(inc_W1X, jac_res(x)) for x in gens(domain(jac_res))]
-  psi = hom(domain(jac_res), W1X, img_gens)
+  psi = hom(domain(jac_res), W1X, img_gens; check=false)
   return cokernel(psi)
 end
 

src/Modules/ExteriorPowers/FreeModules.jl

@@ -25,7 +25,7 @@ function exterior_power(F::FreeMod, p::Int; cached::Bool=true)
     G = grading_group(F)
     weights = elem_type(G)[]
     for ind in OrderedMultiIndexSet(p, n)
-      push!(weights, sum(degree(F[i]) for i in indices(ind); init=zero(G)))
+      push!(weights, sum(_degree_fast(F[i]) for i in indices(ind); init=zero(G)))
     end
     grade(result_, weights)
   else

src/Modules/ExteriorPowers/Generic.jl

@@ -63,15 +63,15 @@ function wedge_multiplication_map(F::ModuleFP, G::ModuleFP, v::ModuleFPElem)
   success, orig_mod, p = _is_exterior_power(F)
   if !success 
     Fwedge1, _ = exterior_power(F, 1)
-    id = hom(F, Fwedge1, gens(Fwedge1))
+    id = hom(F, Fwedge1, gens(Fwedge1); check=false)
     tmp = wedge_multiplication_map(Fwedge1, G, v)
     return compose(id, tmp)
   end
 
   success, orig_mod_2, q = _is_exterior_power(G)
   if !success
     Gwedge1, _ = exterior_power(G, 1)
-    id = hom(Gwedge1, G, gens(G))
+    id = hom(Gwedge1, G, gens(G); check=false)
     tmp = wedge_multiplication_map(F, Gwedge1, v)
     return compose(tmp, id)
   end
@@ -93,7 +93,7 @@ function wedge_multiplication_map(F::ModuleFP, G::ModuleFP, v::ModuleFPElem)
 
   # map the generators
   img_gens = [wedge(v, e, parent=G) for e in gens(F)]
-  return hom(F, G, img_gens)
+  return hom(F, G, img_gens; check=false)
 end
 
 # The wedge product of two or more elements.
@@ -177,7 +177,7 @@ function induced_map_on_exterior_power(phi::FreeModuleHom{<:FreeMod, <:FreeMod,
 
   imgs = phi.(gens(F))
   img_gens = [wedge(imgs[indices(ind)], parent=codomain) for ind in OrderedMultiIndexSet(p, m)]
-  return hom(domain, codomain, img_gens)
+  return hom(domain, codomain, img_gens; check=false)
 end
 
 # The induced map on exterior powers

src/Modules/FreeModules-graded.jl

@@ -1100,7 +1100,7 @@ function hom(M::ModuleFP_dec, N::ModuleFP_dec)
   psi = kDelta[2]*pro[1]
   psi = hom(kDelta[1], H_s0_t0, [psi(g) for g = gens(kDelta[1])])
 
-  H = quo(sub(D, kDelta[1]), image(rho)[1])
+  H = quo_object(sub(D, kDelta[1]), image(rho)[1])
   set_attribute!(H, :show => Hecke.show_hom, :hom => (M, N))
 
   #x in ker delta: mH_s0_t0(pro[1](x)) should be a hom from M to N