1. Introduction
Before we get into the main text, let’s talk about homological algebra. This work originated from the generalization of some simple topological relations.
This framework is sketched in Chapter 2, we’ll start with simplicial sets, while discussing simple categories. And in this chapter we would like to talk more about simplicial complexes.
We’ll start with module in Chapter 3, and from an algebraic point of view, discuss the projective and injective module with the free module, whose theme is the same as that of [1].
In Chapter 4, we’ll show more detail about Projective and Injective, and knowledge about Yoneda Lemma. Here we can refer to [1] [2] more details can be referred to [3]-[5].
We will abstract the concept of simple sets from algebraic topology, and give a more fundamental understanding and thinking of homology theory. This will provide a transcendental reflection on many of our subsequent study directions which is to obtain global information from local information (Although the method may still be categorical). For example, we can think of sheaf theory as a kind of singular homology which I’ll talk a lot about that in Chapter 1. The article can be thought as a simple one of Sur quelques points d’algèbre homologique [TOHOKU] [1957] without some derived category and with some abstract algebraic topology. More detail can be referred to [6]-[8].
2. Simplicial Sets and Cohomology
2.1. Simplicial Sets
In this section, we will present some brief observations of simplicial sets.
Definition 1.1. Before defining other definitions, we need a bit of the analytic geometry of Euclidean space. Given a set
of points of
, this set is said to be geometrically independent if for any scalars
, the equations
and
(1)
imply that
.
Definition 1.2. Let
be a geometrically independent set in
. We define the n-simplex
spanned by
to be the set of all points x of
such that
, where
(2)
and
for all i.
Definition 1.3. The points
that span
are called the vertices of
; the number n is called the dimension of
. Any simplex spanned by a subset of
is called a face of
. The face of
spanned by
is called the face opposite
. Their union is called the boundary of
, which we denote as
. The interior of
is defined by the equation
.
Definition 1.4. A simplicial set is a family of sets
, elements of
are n-simplices, and of map
, one for each nondecreasing map
such the following conditions are satisfied:
(3)
For any nondecreasiong map
we define the “f-th face”
as the linear map
that maps any vertex
into the vertex
. The geometric realization of the simplicial set
is the p-dimensional simplex
. We can describe
as the simplicial set of all singular simplices of
that are compatible with the standard triangulation of
.
A topological space
with the underlying set
, where R is the weakest equivalence relation that identified
and
with
(4)
for some increasing mapping
. We denote the situation as in (4) by
. The canonical topology on
is the weakest topology for which the canonical mapping
is continuous.
Definition 1.5. Let
be a simplex. Define two orderings of its vertex set to be equivalent if they differ from one another by an even permutation. Each of these classes is called an orientation of
. The oriented simplex [
] is indicated in the figure by drawing an arrow in the direction from
to
to
. We can check that [
] and [
] are indicated by the same arrow. In particular, 0-simplex has only one element and only one orientation.
Definition 1.6. Let X be a simplicial set. An n-simplex
is said to be degenerate if there exists a surjective nondecreasing map
,
, and an element
such that
. One can easily check that if x is nondegenerate and
for some f and y, then f is an injection.
Example 1.7. (Triangulation of the Product of Two Simplices) The product of two segments
is not a triangle but a square; it can be naturally divided into two triangles by a diagonal, of two diagonals one is singled out by the fact that its vertices are naturally ordered:
.
Generalizing this construction, we define the canonical triangulation
of the product
. One element of
is a set of
different pairs of integers
, where they are a directed set. Any increasing mapping
define
as follows:
(5)
where
,
.
Define a mapping
(6)
as follows: with the x-th simplex,
(7)
associates the simplex
in
spanned by the points
,
, where
(resp.
) is the i-th vertex of
(resp.
). Or, more formally,
is a linear order-preserving mapping with the image
.
We claim that there exists a commutative diagram.
where
is a bijection.
Definition 1.8. Let G be a group. Let
(8)
and for
let
(9)
where
(10)
Taking a example: the mapping
with
,
,
, we can see
;
;
The geometric realization
of
is called the classifying space of G.
The structure of the geometric realization of a simplicial set X can be clarified like that:
(11)
where X(n) = the set of nondegenerate n-simplices of X.
2.2. Category I
In this section we will briefly introduce some category knowledge, the main of which is the addition category and abelian categories. More details can be referred to [7] [8]
Definition 1.9. The aim is to fix some notations and to recall the axioms of universes. We do not intend neither to enter Set Theory, nor to say more about universes than what we need.
For a set u, we denote by
the set of subsets of
. For
, we denote by
the set whose elements are
.
A universe U is a set satisfying the following properties:
1)
,
2)
implies
, (equivalently,
and
implies
, or else
),
3)
implies
,
4)
implies
,
5) if
and
for all
, then
,
,
6)
Following Grothendieck, we shall add an axiom to the Zermelo-Fraenkel theory, asking that for any set X there exists a universe U such that
.
Let U be a universe.
1) A set is called a U-set if it belongs to U.
2) A set is called U-small if it is isomorphic to a set belonging to U.
We can also define order. But that’s enough for what we’ll talk about below.
Definition 1.10. A category
consists of the following data.
1) A family
, whose members are called the objects of
.
2) for all pairs
of
, a set
, whose elements are called morphisms from X to Y.
3) for any triple
of
, a map from
to
, called the composition map, and denoted
.
These data satisfying: the composition of morphisms is associative, for any
there exists
such that
and
for an
and any
.
Definition 1.11 A subcategory
of
is a category
such that
and for any pair (
) of
,
, with the induced composition law, and
.
If moreover
, then
is called a full subcategory of
.
Definition 1.12. Let
be a category. The opposite category, denoted
, is defined by:
Definition 1.13. Let
be a morphism in
. One says that f is a monomorphism if for any
and any pair
of
such that
, one has
. In the dual way, we can define epimorphism.
In a category
an object P is called initial if
has exactly one element for any
. Similarly an object Q is called final if
has only one element, that is, if Q is initial in
. Note that two initial (resp.final) object are naturally isomorphic.
Definition 1.14. Let
and
be two categories. A functor
consists of a map
and of maps
for all
, such that
A contravariant functor from
to
is a functor from
to
. In other words, it satisfies
.
Definition 1.15. A diagram in a category
is a family of symbols representing objects of
and a family of arrows between these symbols representing morphisms of these objects. One defines in an obvious way the notion of a commutative diagram.
Definition 1.16. Let
be a functor. We say that F is faithful (resp. full, fully faithful) if
, is injective (resp. surjective, bijective) for any
in
.
Let F1 and F2 be two functors from
to
. A morphism
from F1 to F2 consists of the follow data.
(12)
These data satisfying: for any
the following diagram commutes:
We gets a new category whose objects are functors from
to
and morphisms are morphism of such functors.
Definition 1.17. An additive category
is a category
such that:
1) for any pair
of
,
has a structure of additive (i.e. abelian) group, and the composition law is bilinear.
2) there exists an object 0 such that
.
An additive functor
between Ab-category
and
is a functor such that each
is a group homomorphism.
In any additive category
, a kernel of a morphism
is defined to be a map
such that
and that is universal with respect to this property. Dually, a cokernel of f is a map
, which is universal with respect to having
. In
, am map
is monic if
implies
for every map
, and a map
is an epi if
implies
for every map
(The definition of monic and epi in a non-abelian category is slightly different). It’s easy to see that every kernel is monic and that every cokernel is an epi.
2.3. Homology and Cohomology I
The boundary of the geometric simplex
is the difference of its vertices (1)-(0). In such a form the boundary appears in the Leibniz formula
. Similarly, the boundary of
is the alternating sum of its faces.
To make these definitions precise we need the following notions.
An n-dimensional chain (or simply n-chain) of a simplicial set X is an element of the free abelian group
generated by all n-simplices of X. So an n-dimensional chain is a formal linear combination
, where
and
for a finite number of simplices x. Let
be the unique strictly increasing mapping whose image does not contain
.
Definition 1.18. The boundary of an n-chain
is the (n-1)-chain
defined by the following formula:
(13)
The so defined boundary operator
is clearly a group homomorphism. For
, we set
. There exists an obvious generalization of this construction, namely, chains with coefficients in an abelian group A. Such a chain is a formal linear combination
,
, we set
. The boundary operator
is again defined by before. Dually, one can define cochains with coefficients in
is the group of functions on
with values in A. The coboundary
is given by the formula:
(14)
Formally, chains can be considered as special cases of cochains: there exists an inclusion
that maps a chain
into the function
. However, this inclusion is incompatible with the action of
and
(they act in opposite directions) and, even more important, it is incompatible with the behaviour of
and
under simplicial maps
.
Lemma 1.19. We have
for
,
for
.
Proof: Note first that for any
we have
indeed, both sides of the equality give unique increasing mapping of
into
not taking values i and j.
To prove the lemma it suffices to check that
for any
. Compositions
for different
all yield increasing maps of
into
, and the map whose image does not contain i and j appears exactly twice: the first time as
with the sign
and the second time as
with the opposite sign
. Hence
.
Similarly one proves another.
Definition 1.20. Let us define several algebraic notions. A chain complex is a sequence of abelian groups and homomorphisms.
(15)
with the property
for all n. Homomorphisms
are called boundary maps or boundary operators.
A cochain complex is a similar sequence
(16)
Any chain complex can be transformed into a cochain complex by setting
,
. So we will usually consider only cochain complexes.
Definition 1.21. Homology groups of a chain complex
are
Cohomology groups of a cochain complex
are
A substantial part of homological algebra can be considered as a collection of methods for computing (co)homology of various complexes, which we’ll talk about that later. For a simplicial set X we will use the following notations:
,
.
Elements of the group
are called homology classes, and those of
are called cohomology classes (of the simplicial set X with coefficients)
Definition 1.22. Each homology (resp. cohomology) class is represented by an n-chain c (resp. cochain f) such that
(resp.
). Such chains (resp. cochains) are called cycles (resp. cocycles). A cycle c in a given homology class is defined up to a summand of the form
; such chains are called boundaries. Similarly, cochains of the form
are called coboundaries. Two chains whose difference is a boundary are said to be homological.
2.4. Geometry of Chains and Cochains with Coefficient System
Example 1.23. Look at the picture of a tetrahedron
.
On any edge two adjacent faces induce opposite orientation. Therefore the corresponding terms in
have opposite signs.
Example 1.24. Zero-dimensional homology yields somewhat different information. Since,
, any 0-chain is a cycle. Let us show that there exists a natural isomorphism.
= free abelian group generated by piecewise connected components of
. Denote for a moment the group on the right-hand side by
. Define a map
by associating with a component of
the class of a chain consisting of one (arbitrary) point in this component.
Example 1.25. An important role in geometry is played by various modifications of topological spaces that eliminate some (co)homology classes or generate new ones.
Let X be a triangulated space. The cone CX over X is a triangulated space obtained from X in the following manner:
and for
so that
is the vertex of the cone.
We claim that any hole in X is filled in CX (by the cone over the boundary of the hole) and no new holes appear.
Indeed, let us define the complex of chains
of a triangulated space X as follows:
= the free abelian group generated by
the boundary operator on
is same as before, we claim that
,
and
,
. We have
. Later we define the cone of any complex.
Definition 1.26. We can construct chains and cochains of a simplicial set using as coefficients something more involved than just abelian groups. There are two types of coefficient systems: for homology and for cohomology. A homological coefficient system A on a simplicial set X is a family of abelian groups
, one for each simplex
, and a family of homomorphisms
, one for each pair
,
, such that the following conditions are satisfied:
The second equality means that the following diagram is commutative:
A cohomological coefficient system B on a simplicial set X is a family of abelian groups
, one for each simplex
, and a family of ho momorphisms
, one for each pair
, such that the following conditions are satisfied:
The second equality is equivalent to the commutativity of a diagram similar to the one above.
Definition 1.27. Let A be a homological coefficient system on a simplicial set X. An n-dimensional chain of X with coefficients in A is a formal linear combination
,
. Such chains form an abelian group (under addition) which is denoted by
. The boundary of an n-dimensional chain
is an
-dimensional chain
defined by:
As before we can easily see that
is a chain complex, i.e.
.
Homology groups of the complex
are called the homology groups of the simplicial set X with coefficients in A; they are denoted by
.
Similarly and dually, we can define cohomology groups.
2.5. The Exact Sequence I
We defined groups
and
, where X is a simplicial set, and A and B are coefficient systems. In some simple cases these groups can be computed directly. But the main technique consists in the study of the behaviour of these groups under the change of X or the change of A.
In this subsection we fix X and study the dependence of homology and cohomology on coefficients. The main tool here is the theorem about the exact sequence.
Definition 1.28. An exact sequence of abelian groups is a complex
with all cohomology groups vanishing (for chain complexes the definition is the same). This means that
for all n. Usually such a sequence is written as
(17)
To give such a triple is the same as to give an abelian group B and its subgroup A. The homomorphism theorem says that
implies
(18)
Theorem 1.29. Let X be a simplicial set. Any exact triple of abelian groups canonically determines a cohomology exact sequence:
(19)
and a similar homology sequence.
To proof the theorem we must define the morphisms of complexes and construct the boundary homomorphism.
Theorem 1.30. Let
be two complexes. A morphism
is a family of homomorphisms
commuting with differentials:
(20)
Given
, let us construct a family of homomorphisms
(21)
as follows. Let
be represented by a cocycle
, Then
, and we define
to be the class of
in
. this class does not depend on the choice of a representative of b modulo
.
It is clear also that if
is another morphism of complexes then
. Let
,
, so
and
are complexes.
Lemma 1.31. Let
be an exact triple of abelian groups. Then the sequences of groups of chains and of cochains
are exact.
Proof: An element of
is a formal linear combination
,
. The image of this element under the mapping
is
and since i is an injection,
is also an injection. Similarly one proves that
is a surjection. Further,
for
. Let now
and
. Then
for all
, i.e.,
,
and
for
. The second sequence is treated similarly.
Theorem 1.32 Proof of 1.29: 1) Exactness at
. First of all,
, because
. Next, let
and
. We construct
with
as follows. Let
be a representative of b so that
. Since
, we have
for some
and,
being a surjection,
for some
. It is clear that
so that by the exactness of before,
for some
; moreover,
. As
is injective,
. Now one can easily check that
mod
satisfies the required property.
2)
. Let
for some
and let
,
be representatives of
respectively, so that
. Then the definition of
shows that
.
3)
. Let
and let
be a cocycle representing the cohomology class c. Let
and
for
,
. Then
implies
for some
. Let
. Then
and
. Hence
, where
mod
.
It’s the same on the other side.
3. Module: With View of Free Module in Projective and
Injective
Modules over a ring are a generalization of abelian groups (which are modules over
). In the section, we’ll cover projective and injective with view of the free module.
3.1. Module and the Exact Sequence II
Definition 2.1. Let R be a ring. A (left) R-module is an additive abelian group A together with a function
(the image of
being denoted by
) such that for all
and
:
1)
2)
3)
If R has an identity element
and
4)
for all
then A is said to be a unitary R-module. If R is a division ring, then a unitary R-module is called a vector space.
Definition 2.2. Let A and B be modules over a ring R. A function
is an R-module homomorphism provided that for all
and
:
and
(22)
If R is a division ring, then an R-module homomorphism is called a linear transformation.
Definition 2.3. Let R be a ring, A an R-module and B a nonempty subset of A. B is a submodule of A provided that B is an additive subgroup of A and
for all
,
. A submodule of a vector space over a division ring is called a subspace.
If X is a subset of a module A over a ring R, then the intersection of all submodules of A containing X is called the submodule generated by X (or. spanned by X). If X is finite, and X generates the module B, B is said to be finitely generated. Let B be a submodule of a module A over a ring R. Then the quotient group A/B is an R-module with the action of R on A/B given by:
for all
Definition 2.4. A pair of module homomorphisms,
, is said to be exact at B provided
. A finite sequence of module homomorphisms,
, is exact provided
for
. At the same time we can define the exact of an infinite sequence.
Lemma 2.5. Let R be a ring and
a commutative diagram of R-modules and R-module homomorphisms such that each row is a short exact sequence. Then
1)
monomorphisms
is a monomorphisms;
2)
epimorphisms
is a epimorphisms;
3)
isomorphisms
is a isomorphisms.
Proof: 1) Let
and suppose
; we must show that
. By commutativity we have
. This implies
, since
is a monomorphism. By exactness of the top row at B, we have
, say
. By commutativity,
. By exactness of the bottom row at
,
is a monomorphism, hence
. But
is a monomorphism; therefore
and hence
. Thus
is a monomorphism.
2) Let
. Then
; since
is an epimorphism
for some
. By exactness of the top row at C, g is an epimorphism; hence
for some
. By commutativity,
. Thus
and
by exactness, say
,
. Since
is an epimorphism,
for some
. Consider
:
, by commutativity,
, hence
and
is an epimorphism.
3) is an immediate consequence of 1) and 2).
Definition 2.6 Let R be a ring and
. a short exact sequence of R-module homomorphisms. Then the following conditions are equivalent.
1) There is an R-module homomorphism
with
;
2) There is an R-module homomorphism
with
;
3) the given sequence is isomorphic (with identity maps on
and
) to the direct sum short exact sequence
; in particular
.
A short exact sequence that satisfies the equivalent conditions is said to be split or a split exact sequence.
3.2. Free Module
In this subsection we show free modules, the most important examples of which are vector spaces over a division ring. We’ll show the objects without proof.
Definition 2.7. A subset X of an R-module A is said to be linearly independent provided that for distinct
and
.
for every i. A set that is not linearly independent is said to be linearly dependent. If A is generated as an R-module by a set Y, then we say that Y spans A. If R has an identity and A is unitary, Y spans A if and only if every element of A may be written as a linear combination:
; A linearly independent subset of A that spans A is called a basis of A. Observe that the empty set is linearly independent and is a basis of the zero module.
Definition 2.8. A unitary module F over a ring R with identity, which satisfies F has a nonempty basis is called a free R-module.
Definition 2.9. A maximal linearly independent subset X of a vector space V over a division ring D is a basis of V. Every vector space V over a division ring D has a basis and is therefore a free D-module. More generally every linearly independent subset of V is contained in a basis of V. If V is a vector space over a division ring D and X is a subset that spans V, then X contains a basis of V.
Definition 2.10. Let R be a ring with identity and F a free R-module with an infinite basis X. Then every basis of F has the same cardinality as X. If V is a vector space over a division ring D, then any two bases of V have the same cardinality.
3.3. Projective and Injective Module
Every free module is projective and arbitrary projective modules (which need not be free) have some of the same properties as free modules. Injectivity, which is also studied here, is the dual notion to projectivity. Before we start our section we consider such example when expressed in modern language, the Riemann-Roch theorem give a formula for the difference of the dimensions of two vector spaces attached to algebraic line bundle over a non-singular projective curve. Thus, we can see easily where the projective or injective module come from (we don’t expect its historic origin which comes from Homological Algebra written by Cartan). At the same time, in order to fit in with the category we mentioned above, we will make sacrifices to use its abstract algebraic language.
Definition 2.11. A module P over a ring R is said to be projective if given any diagram of R-module homomorphisms
with bottom row exact (that is, g an epimorphism), there exists an R-module homomorphism
such that the diagram
is commutative (that is,
).
Theorem 2.12 Every free module F over a ring R with identity is projective.
Proof: We are given a diagram of homomorphisms of unitary R-modules:
with g an epimorphism and F a free R-module on the set X (
). For each
,
. Since g is an epimorphism, there exists
with
. Since F is free, the map
given by
induces an R-module homomorphism
such that
for all
. Consequently,
for all
so that
, we have
. Therefore F is projective. We also can see every module A over a ring R is the homomorphic image of a projective R-module.
Theorem 2.13 Let R be a ring. The following conditions on an R-module P are equivalent.
1) P is projective;
2) there is a free module F and an R-module K such that
;
3) every short exact sequence
is split exact.
Proof: 1)
3) Consider the diagram
with bottom row exact by hypothesis. Since P is projective there is an R-module homomorphism
such that
. Therefore, the short exact sequence
is split exact.
3)
2) There is a free R-module F and an epimorphism
. If
, then
. By hypothesis the sequence splits.
2)
1) Let
be the composition
where the second map is the canonical projection. Similarly let
be the composition
with the first map the canonical injection. Given a diagram of R-module homomorphisms
with exact bottom row, consider the diagram
Since F is projective, there is an R-module homomorphism
such that
. Let
. Then
. Therefore, P is projective.
Definition 2.14 A module J over a ring R is said to be injective if given any diagram of R-module homomorphisms
with top row exact (that is, g a monomorphism), there exists an R-module homomorphism
such that the diagram
is commutative (that is,
).
Theorem 2.15. A right R-module E is injective if and only if for every right ideal J of R, every map
can be extended to a map
.
Proof: The “only if” direction is a special case of the definition of injective. Conversely, suppose given an R-module B, a submodule A and a map
. Let
be the poset of all extensions
of
to an intermediate submodule
; the partial order is that
if
extends
. By Zorn’s lemma there is a maximal extension
in
; we have to show that
. Suppose there is some
not in
. The set
is a right ideal of R. By assumption, the map
extends to a map
. Let
be the submodule
of B and define
by
,
and
. This is well defined because
for
in
, and
extends
, contradicting the existence of b. Hence
.
Definition 2.16. An abelian group D is said to be divisible if given any
and
, there exists
such that
. For example, the additive group
is divisible, but
is not. An abelian group D is divisible if and only if D is an injective (unitary) Z-module.
Theorem 2.17. Let R be a ring. The following conditions on an R-module J are equivalent.
1) J is injective;
2) J is a direct summand of any module B of which it is a submodule;
3) every short exact sequence
is split exact.
Proof like before.
Example 2.18. The divisible abelian groups
and
are injective. Every injective abelian group is direct sum of these. In particular, the injective abelian group
is isomorphic to
.
If A is an abelian group, let
be the product of copies of the injective group
, indexed by the set
, then
is injective.
Example 2.19. Nice rings every projective module is a free module like
, fields, division rings…
4. Homological Algebra
Homological algebra is a tool used to prove nonconstructive existence theorems in algebra. It also provides obstructions to carrying out various kinds of constructions; when the obstructions are zero, the construction is possible. In the section, we will show many theorems without proof because of space. And we’ll skip content like δ-function that is interesting.
Definition 3.1. Let A be an abelian category. Then
is a left exact functor from A to Ab for every M in A. That is, given an exact sequence
, the following sequence of abelian groups is also exact:
(23)
is a left exact contravariant functor.
Theorem 3.2. Yoneda Embedding Every additive category A can be embedded in the abelian category
by the functor h sending A to
. Since each
is left exact, h is a left exact functor. Since the functors
are left exact, the Yoneda embedding actually lands in the abelian subcategory
of all left exact contravariant functors from A to Ab whenever A is an abelian category.
Lemma 3.3. The Yoneda embedding h reflects exactness. That is, a sequence
in A is exact, provided that for every M in A the following sequence is exact:
(24)
Theorem 3.4. M is projective if and only if
is an exact functor. That is, the sequence of groups:
(25)
is exact for every exact sequence
in A.
Definition 3.5. Let M be an object of A. A left resolution of M is a complex
with
for
,together with a map
so that the augmented complex:
(26)
is exact. It is a projective resolution if each
is projective.
Theorem 3.6. Every R-module M has a projective resolution. More generally, if an abelian category A has enough projectives, then every object M in A has a projective resolution.
Forming a resolution by splicing.
Theorem 3.7. Let
be a projective resolution of M and
a map in A. Then for every resolution
of N there is a chain map
lifting
in the sense that
. The chain map f is unique up to chain homotopy equivalence.
Theorem 3.8. Suppose given a commutative diagram
where the column is exact and the rows are projective resolutions. Set
Then the
assemble to form a projective resolution P of A, and the right-hand column lifts to an exact sequence of complexes
where are the natural inclusion and projection, respectively.
Definition 3.9 Let M be an object of A. A right resolution of M is a cochain complex
with
for
and a map
such that the augmented complex:
(27)
is exact. This is the same as a cochain map
, where M is considered as a complex concentrated in degree 0. It is called an injective resolution if each
is injective.
The other theorems mentioned above are all in dual form in injective.
What Is Homological Algebra?
Now we have met some injective and projective module and also found the continuity with algebraic topology. It’s appropriate to ask, what is homological algebra?
I’d like to the view that homological algerbra is a tool used to prove some theorems in algebra and it can show us how far away from the good structure. At the same time, it can also provide enough details in arithmetic. From a historical point of view, homological algebra is actually a metaphysical weapon. Kant used to say: There can be no doubt that all our knowledge begins with experience. Whether or not there is such knowledge, which does not rely on experience, or even on any sensory impressions, is at least a question that needs to be examined more carefully, and one that cannot be answered immediately and lightly. Therefore, it is easy to see that the weapon we use is actually derived from algebraic topology, but it is beyond the category of algebraic topology and has become a kind of prior knowledge. This provides us with a philosophical reflection on how to understand mathematics, Copernicus, a philosopher born 300 years ago, said that the intellectual category can only be used empirically, not transcendentally. Empirical know- ledge such as arithmetic, is difficult to be as transcendent as such knowledge. However, this kind of prior knowledge is difficult to understand and acquire, and it still requires a lot of empirical knowledge as a background refinement and the field is still breeding lilacs out of the dead land.