memo
This post may be updated continuously.
Category theory
|
Limit
Inverse limit Projective limit |
Colimit
Direct limit Inductive limit |
|
|---|---|---|
| origin of arrows | target of arrows | |
| product | coprodut | |
|
product (inverse limit) in
the 2nd variable |
coproduct (direct limit) in
the 1st var |
$\operatorname{Hom}$ sends … to product |
| cartesian product | disjoint union | in $\textsf{Set}$ |
| direct product = direct sum $\oplus$ | in $\textsf{Ab}$, $R$-$\textsf{Mod}$, $K$-$\textsf{VectSpace}$, etc. | |
| direct product | free product | in $\textsf{Grp}$ |
|
product topology
coarsest among conti. projections $X\to X_i$ |
disjoint union topology
finest among conti. injections $X_i\to X$ |
in $\textsf{Top}$ |
|
pullback
limit of \(*\rightarrow*\leftarrow*\) |
pushout
colimit of \(*\leftarrow*\rightarrow*\) |
|
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equalizer
$\text{coeq}(f,g) \to X\stackrel{f,g}{\rightrightarrows}Y$ |
coequalizer
$X\stackrel{f,g}{\rightrightarrows} Y \to \text{coeq}(f,g)$ |
|
| kernel | cokernel (quotient) | |
| terminal object | initial object | |
|
$p$-adic integers $\mathbb{Z}_p$
$\mathbb{Z}/p^{n+1}\mathbb{Z}\stackrel{\text{proj}}{\to}\mathbb{Z}/p^{n}\mathbb{Z}$ |
Prüfer group $\mathbb{Z}(p^\infty)$
$\mathbb{Z}/p^{n}\mathbb{Z}\to\mathbb{Z}/p^{n+1}\mathbb{Z}$; $a\mapsto pa$ |
|
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profinite integer $\hat{\mathbb{Z}}\cong \prod_{p<\infty} \mathbb{Z}_p$
$\mathbb{Z}/mn\mathbb{Z}\stackrel{\text{proj}}{\to}\mathbb{Z}/n\mathbb{Z}$ |
$\mathbb{Q}/\mathbb{Z}$
$\mathbb{Z}/n\mathbb{Z}\to\mathbb{Z}/mn\mathbb{Z}$; $a \mapsto ma$ |
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Stalk of sheaf $\mathcal{F}_x$
$\mathcal{F}(U)\to\mathcal{F}(V)$; $s \mapsto s|_V$ |
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Congruence subgroups
(From Diamond & Shurman)
\[\begin{matrix} \operatorname{SL}_2(\mathbb{Z}) & & M_k(\operatorname{SL}_2(\mathbb{Z}))\\ \cup & & \cap \\ \Gamma_0(N) & = \left\{ \begin{pmatrix} * & * \\ 0 & *\end{pmatrix}\pmod{N}\right\} & M_k(\Gamma_0(N))\\ \cup & & \cap \\ \Gamma_1(N) & = \left\{ \begin{pmatrix} 1 & * \\ 0 & 1\end{pmatrix}\pmod{N}\right\} & M_k(\Gamma_1(N)) & = \oplus M_k(N, \chi)\\ \cup & & \cap \\ \Gamma(N) & = \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}\pmod{N}\right\} & M_k(\Gamma(N)) \end{matrix}\]$M_k(\Gamma_1(N)) = \oplus M_k(N, \chi)$ is the direct sum over all Dirichlet characters $\chi$ modulo $N$ and is respected by the Hecke operators, where $M_k(N, \chi) = \left\{ f \in M_k(\Gamma_1(N)) : f[\gamma]_k = \chi(\gamma) f\ \forall \gamma \in \Gamma_0(N) \right\}$