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.
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.
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.
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:AB 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
n
A
f
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 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:
f
X

g
S
h
B
s.t. hf ≠ hg ⇒ h∈M
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 {(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.
Def. An M- well ordered chain is an M- chain which is indexed by a
totally well ordered set I.
X
m0
m1
m2
mn
B0
f01
B1
f12
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∈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 i0∈I and h:BB i0 with m i0=hn.
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.
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 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 Minjectives when M=Mono.
Def. A has M-transferability property if for every pair f, u of
morphisms with M-morphism f one has a commutative square
f
A
u
f
B
A
⇒
C
B
u
v
C
g
D
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.
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:xY there exist C∈C and a morphism
f:YC s.t. fm≠fn.
m
XY
n
f
C
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.
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.
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Coro.
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