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Abstract Some of the so called smallness conditions in algebra as well as in Category Theory, important and interesting for their own and also tightly related to injectivity, Essential Bounded , Cogenerating set, and Residual Smallness. Here we want to see the relationships between these notions and to study these notions in the class mod(∑ ,E). That is all of objects in the Grothendick Topos E which satisfy ∑, ∑ is a class of equations. Introduction In the whole of this talk A is an arbitrary category and M is an arbitrary subclass of its morphisms. Def. Let A be an arbitrary category andM be an arbitrary subclass of it’s morphisms, also A and B are two objects inA. We say that A is anM- subobject of B provided that there exists an M- morphism m:AB.
In this case we say that (A,m) is anM- subobject of B, or (m,B) is an M-extension for A. The class of all M- subobjects of an object X is denoted by M/X, and the class ofall M-extensions of Xis denoted byX/M. We like the class of M-subobjects M/Xto behaveproper. This holds, if M has good properties. Def.One says that Mhas“good properties” with respect to composition if it is: (1) Isomorphism closed; that is, contains all isomorphism and is closed under composition with isomorphism. (2) Left regular; that is, for f in M with fg=f we have g is an isomorphism. (3)Composition closed; that is, for f:AB and g:BC in M, gf is also in M. (4) Left cancellable; that is, gf is inM, implies fis in M. (5)Right cancellable; that is, gf is in M, implies g is in M. We can define a binary relation ≤ onM/X as follows:
If (A,m) and (B,n) are two M- subobjets of X, we say that (A,m)≤(B,n) whenever there exists a morphism f:AB such that nf=m. That is m A X f n B One can easily see that ≤ is a reflective and transitive relation, but it isn’t antisymmetric. But if M is left regular, ≤ is antisymmetric, too up to isomorphism. Similarly, one defines a relation ≤ on X/M as follows: If (m,A) and (n,B) belong to X/M, we say (m,A)≤(n,B) whenever, there exists a morphism f:A B such that fm = n. That is: m X A f n B Also it is easily seen that (X/M , ≤) forms a partially ordered class up to the relation ∼. Where (m,A)∼ (n,B) iff (m,A)≤ (n,B) and (n,B)≤ (m,A) . So from now on, we consider (X/M ,≤ ) up to ∼.
Def. In Universal Algebra we say that A is subdirectly irreducible if for any morphism f:A ∏i in I Aiwith allPi fepimorphisms, there exists an index i0 in I for which pi0 f is an isomorphism. The following definition generalizes the above definition and it is seen that these are equivalent for equational categories of algebras. Def. An object S in a category is called M-subdirectly irreducible if there are an object X with two deferent morphisms f,g:X S s.t. any morphism h with domain S and hf≠hg,belongs toM. See the following: f h X S B s.t.hf ≠ hg ⇒ h∈M g Also when the class of M-subdirectly irreducible objects in a category A forms just only a set we say that A is M-residually small Def.An M-chain is a family of X/M say {(mi,B i)}i in Iwhich is indexed by a totally ordered set I such that if i ≤jin I then there exists aij:Bi Bj with aij mi i =mj. Also we have aii= id Bi and for i ≤j ≤k, a ik=ajk aij.
Def.An M- well ordered chain is an M- chain which is indexed by a totally well ordered set I. m0 X B0 m1 f01 m2 B1 f12 mn B2 mn+1 Bn fn n+1 Bn+1
Essential Boundedness and Residul Smallness Def.A is said to be M-essentially bounded if for every object A∈Athere isaset{mi:AB i:i∈I } ⊆ Ms.t. for anyM-essential extensionn:ABthere exists i0∈I and h:BBi0 with mi0=hn. Def. An M-morphism f:AB is called an M-essential extension of A if anymorphism g:BC is in M whenever g f belongs to M. the class of all M- essential extension (of A) is denoted by M* (M*A). Note. A category A is called M-cowell powered whenever for any object A in A the class ofall M-extensions of A forms a set. The.M*-cowell poweredness implies M-essential boundedness. Conversely, if M=Mono, A is M-well powered and M-essentially bounded, then A is M*-cowell powered.
The. If A has enough M- injectives then A is M- essentially bounded. The. Let M=Mono and E be another class of morphisms of A s.t. A has (E,M)-factorization diagonalization. Also, let A be E-cowell powered and have a generating set G s.t. for all G∈G, G பG ∈A. Then, M*-cowell poweredness implies M-residual smallness. Coro. Under the hypothesis of the former theorems, we can see that residual smallness is a necessary condition to having enough M- injectives when M=Mono.
Def.A hasM-transferability property if for every pair f, u of morphisms with M-morphism f one has a commutative square f f A B A B u u v ⇒ C C D g with M-morphism g. Lem. If A has enough M- injectives then, A fulfills M- transferability property. Def.We say that A has M-bounds if for any small family {hi:ABi : i∈ I }≤M there exists an M-morphism h:AB which factors over all hi,s. The. Let A satisfy the M-transferability and M-chain condition, and let M be closed under composition. Then, A has M-bounds.
Cogenerating set and Residual Smallness Def. A set C of objects of a category is a cogenerating set if for every parallel different morphisms m,n:xY there exist C∈C and a morphism f:YC s.t. fm≠fn. m f XY C n The. Let A have a cogenerating set C and A be M-well powered. Then A is M-residually small. Prop. For any equationalclass A , the following conditions are equivalent: (i) Injectivity is well behaved. (ii) A has enough injectives. (iii) Every subdirectly irreducible algebra in A has an injective extension. (iv) A satisfies M-transferability and M*- cowell poweredness.
The. Let M=Mono and A be well powered with products and a generating set G. Then, having an M-cogenerating set implies M*-cowell poweredness. Corollary. Let M=Mono and A be well powered and have products and (E,M)-factorization diagonalization for a class E of morphisms for which A is E-well powered. Then T. F. S. A. E. ( i) A is M-essentially bounded. ( ii) A is M*-cowell powered. (iii) A is M-residually small. (iv) A has a cogenerating set.
Injectivity of Algebras in a Grothendieck Topos Now we are going to see and investigate these notions and theorems in mod(∑,E). To do this we compare the two categories mod (∑,E) and mod(∑). Def. Note.
Def. Note. Def. Coro.
Prop. Lem. Res. Coro. Coro. The.
Lem. Prop. The. Prop.
The. The.
