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:AB.

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:AB and g:BC 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:

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

A X

B

m

nf

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: X A

B

m

nf

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 Ai with all Pi f epimorphisms, 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 to M. See the following:

SX f

gB

hs.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 aij:Bi Bj

with aij mi i =m j. Also we have aii = id Bi and for i ≤j ≤k, a ik=ajk aij.

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- well ordered chain is an M- chain which is indexed by a totally well ordered set I.

X B0

B1

B2

Bn

Bn+1

m1

m2

mn

mn+1

f01

f12

fn n+1

m0

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:AB i : i∈ I } ⊆ M s.t. for any M-essential extension n:AB there exists i0∈I and h:BB i0 with m i0=hn.

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.

Def. An M-morphism f:AB is called an M-essential extension of A if any morphism 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 of all M-extensions of A forms a set.

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 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=Mono .

The. If A has enough M- injectives then A is M- essentially bounded .

Def. A has M-transferability property if for every pair f, u of morphisms with M-morphism f one has a commutative square

A B

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:AB i : i∈ I }≤ M there exists an M-morphism h:AB 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

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.

XYm

nC

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 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.

( i) A is M-essentially bounded.

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.

( 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.

Coro.

Def.

Lem.

Res.

Coro.

Coro.

The.

Prop.

Prop.

The.

Prop.

Lem.

The.

The.