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Author |
hbghlyj
Posted 2025-5-7 08:32
初学者可以阅读 Emily Riehl 的著作《Category Theory in Context 情境中的范畴论》。以下是摘录
In linear algebra, elementary row operations on a matrix with n rows ultimately arise from doing those row operations on the nxn identity matrix. This is because a special functor called the Yoneda Embedding has very nice properties. (It is "full" and "faithful")
$$
\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right] \xrightarrow[1\leftrightarrow2]{\text { switch rows }}\left[\begin{array}{lll}
4 & 5 & 6 \\
1 & 2 & 3 \\
7 & 8 & 9
\end{array}\right]=\left[\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]
$$
same as multiplying on the left by identity matrix usth rows #1 & #2 switched.
every group is a category with one object (the morphisms correspond to group elements)
$$\bullet\rightarrow C:\text{any category}$$
a functor from that group defines a group action on its image
and a natural transformation between two such functors G-equivariant map (where G = your group)
the Axiom of Choice is the statement that a certain construction called a limit (more specifically, a product) exists in the category of sets.
$\{Χ_α\}_{α\in A}$ = nonempty collection of sets
A functor called $π_1$, associates a group to every topological space $X$ the fundamental group of $X$:
Top$\to$Group
$X\mapsto π_1(Χ)$
You can use this functor to prove
①the fundamental theorem of algebra
every nonconstant polynomial in $\mathbb C[z]$ has a root in $\mathbb C$.
②Brouwer's Fixed Point Theorem
every homeomorphism of the disc has a fixed point
③ the Perron-Frobenius theorem
every 3x3 matrix with positive real entries has a positive eigenvalue
to name a few!
The universal property of a free group $F(S)$ on a set S arises as a special relationship between the categories Group and Set. (i.e. there is a pair of "adjoint functors" between them).
A free group on a set $S$ comes with a set map
F(S)
↑
S
so that for any group and any set map...
$$\xymatrix{
F(S) \ar@{-->}[dr] & \\
S \ar[u] \ar[r] & G
}$$
...there is a unique homomorphism here causing this triangle to commute.
Viewing both $\mathbb Z$ and $\mathbb R$ as posets (and hence categories), the definitions of the floor $\lfloor\ \rfloor:\mathbb R→\mathbb Z$ and ceiling functions $\lceil\ \rceil:\mathbb R→\mathbb Z$ also arise from a special relationship (a pair of adjoint functors) between $\mathbb R$ and $\mathbb Z$.
The fundamental theorem of Galois Theory says that a certain functor is an isomorphism of categories":
$$\theta^\text{op}_G→\text{Field}_F^K$$
$\theta^\text{op}_G$ = orbit category of G
objects = quotients G/H by Subgroups H of G morphisms = G-equivariant maps
$\text{Field}_F^K$
objects = intermediate fields $F\subset E\subset K$ morphisms = field homomorphisms that fix $F$
K/F = finite Galois extension
G = corresponding Galois group
The Riesz Representation Theorem is the statement that a natural transformation between certain functors is quite special. (It's a "natural isomorphism")
\[
\xymatrix@C+2pc{
\mathsf{cHaus} \rtwocell& \mathsf{Ban}
}
\]
cHaus
objects = compact Hausdorff spaces
morphisms = continuous functions
Ban
objects = real Banach spaces
morphisms = continuous linear maps maps |
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写这本书的人其实都是专家，但他们的读者大多是彼此。