2. 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 and M be an arbitrary subclass of it’s morphisms, also A and B are two objects in A. We say that A is an M - subobject of B provided that there exists an M - morphism m:A  B.

3. Def. One says that M has “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:A  B and g:B  C in M, gf is also in M. (4) Left cancellable; that is, gf is in M, implies f is in M. (5) Right cancellable; that is, gf is in M, implies g is in M. In this case we say that (A,m) is an M - 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 of all M -extensions of X is denoted by X/ M. We like the class of M -subobjects M /X to behave proper. This holds, if M has good properties. We can define a binary relation ≤ on M/ X as follows:

4. 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:A  B such that nf=m. That is AX B m n f 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: XA B m n f 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 ∼.

5. Def. In Universal Algebra we say that A is subdirectly irreducible if for any morphism f:A ∏ i in I A i with all P i f epimorphisms, there exists an index i 0 in I for which p i0 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 to M. See the following: SX  f g B h s.t. hf ≠ hg ⇒ h ∈ M Def. An M- chain is a family of X/ M say {(m i,B i )} i in I which is indexed by a totally ordered set I such that if i ≤j in I then there exists a ij :B i  B j with a ij m i i =m j. Also we have a ii = id Bi and for i ≤j ≤k, a ik =a jk a ij. 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

6. Def. An M- well ordered chain is an M- chain which is indexed by a totally well ordered set I. X B0B0 B1B1 B2B2 BnBn B n+1 m1m1 m2m2 mnmn m n+1 f 01 f 12 f n n+1 m0m0

7. Essential Boundedness and Residul Smallness Def. A is said to be M- essentially bounded if for every object A ∈ A there is a set {m i: A  B i : i ∈ I } ⊆ M s.t. for any M- essential extension n:A  B there exists i 0 ∈ I and h:B  B i0 with m i0 =hn. The. M *-cowell poweredness implies M -essential boundedness. Conversely, if M=M ono, A is M -well powered and M -essentially bounded, then A is M *-cowell powered. Def. An M -morphism f:A  B is called an M -essential extension of A if any morphism g:B  C 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 of all M -extensions of A forms a set.

8. The. Let M=M ono 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 ப G ∈ 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=M ono. The. If A has enough M - injectives then A is M - essentially bounded.

9. Def. A has M -transferability property if for every pair f, u of morphisms with M -morphism f one has a commutative square AB C ⇒ A C B D f u f v 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 {h i :A  B i : i ∈ I }≤ M there exists an M- morphism h:A  B which factors over all h i,s. g The. Let A satisfy the M- transferability and M- chain condition, and let M be closed under composition. Then, A has M -bounds. u

10. 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:x  Y there exist C ∈ C and a morphism f:Y  C s.t. fm≠fn. XYXY m n C f The. Let A have a cogenerating set C and A be M- well powered. Then A is M- residually small. Prop. For any equational class 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 s atisfies M -transferability and M *- cowell poweredness.

11. The. Let M=M ono and A be well powered with products and a generating set G. Then, having an M- cogenerating set implies M*- cowell poweredness. ( i) A is M- essentially bounded. Corollary. Let M=M ono 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. ( ii) A i s M *-cowell powered. (iii) A is M -residually small. (iv) A has a cogenerating set.

12. 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.

13. Def. Note. Coro. Def.

14. Lem. Res. Coro. The. Prop.

15. The. Prop. Lem.

16. The.