From 0dd5ba3e3c251fa46f00b5eef998e5aead01351c Mon Sep 17 00:00:00 2001 From: zeramorphic Date: Sun, 26 May 2024 10:34:19 +0100 Subject: [PATCH] Misc edits Signed-off-by: zeramorphic --- iii/cat/02_yoneda_lemma.tex | 4 +- iii/cat/03_adjunctions.tex | 3 +- iii/cat/04_limits.tex | 1 + iii/cat/05_monads.tex | 64 +++++++++---------- iii/commalg/01_chain_conditions.tex | 2 +- iii/commalg/02_tensor_products.tex | 2 +- ...tegrality_finiteness_finite_generation.tex | 2 +- 7 files changed, 40 insertions(+), 38 deletions(-) diff --git a/iii/cat/02_yoneda_lemma.tex b/iii/cat/02_yoneda_lemma.tex index 18369b9..3c4571b 100644 --- a/iii/cat/02_yoneda_lemma.tex +++ b/iii/cat/02_yoneda_lemma.tex @@ -11,7 +11,7 @@ \subsection{Statement and proof} We can also define \[ \mathcal C(-, A) : \mathcal C^\cop \to \mathbf{Set} \] by -\[ B \mapsto \mathcal C(B, A);\quad (B \xrightarrow f C) \mapsto ((B \xrightarrow g A) \mapsto g f) \] +\[ B \mapsto \mathcal C(B, A);\quad (B \xrightarrow f C) \mapsto ((C \xrightarrow g A) \mapsto g f) \] \begin{lemma}[Yoneda lemma] Let \( \mathcal C \) be a locally small category. Let \( A \in \ob \mathcal C \), and let \( F : \mathcal C \to \mathbf{Set} \) be a functor. @@ -127,7 +127,7 @@ \subsection{Statement and proof} Then we have two functors \[ \mathcal C \times [\mathcal C, \mathbf{Set}] \to \mathbf{Set} \] The first is the evaluation functor -\[ (A, F) = FA \] +\[ (A, F) \mapsto FA \] The second is the composite \[ \mathcal C \times [\mathcal C, \mathbf{Set}] \xrightarrow{Y \times 1} [\mathcal C, \mathbf{Set}]^\cop \times [\mathcal C, \mathbf{Set}] \xrightarrow{[\mathcal C, \mathbf{Set}](-, -)} \mathbf{Set} \] The naturality condition is that \( \Phi \) and \( \Psi \) are natural transformations between these two functors, and thus are natural isomorphisms. diff --git a/iii/cat/03_adjunctions.tex b/iii/cat/03_adjunctions.tex index f505bd1..04a8c70 100644 --- a/iii/cat/03_adjunctions.tex +++ b/iii/cat/03_adjunctions.tex @@ -328,7 +328,8 @@ \subsection{Reflections} If \( \mathcal D \) is coreflective, there is a best possible way to get \emph{out of} \( \mathcal D \) to some object in \( \mathcal C \). \begin{example} \begin{enumerate} - \item \( \mathbf{AbGp} \) is reflective in \( \mathbf{Gp} \); the left adjoint to the inclusion map sends a group \( G \) to its abelianisation \( G^{\mathrm{ab}} = \faktor{G}{H} \), the quotient of \( G \) by its commutator subgroup \( H = \qty{aba^{-1}b^{-1} \mid a, b \in G} \trianglelefteq G \). + \item \( \mathbf{AbGp} \) is reflective in \( \mathbf{Gp} \); the left adjoint to the inclusion map sends a group \( G \) to its abelianisation \( G^{\mathrm{ab}} = \faktor{G}{H} \), the quotient of \( G \) by its commutator subgroup + \[ H = \qty{aba^{-1}b^{-1} \mid a, b \in G} \trianglelefteq G \] Note that any homomorphism \( G \to A \) where \( A \) is abelian factors uniquely through the quotient map \( G \to G^{\mathrm{ab}} \), giving the adjunction as required. \item Recall that an abelian group is called \emph{torsion} if all of its elements have finite order, and \emph{torsion-free} if all of its nonzero elements have infinite order. For an abelian group \( A \), its set of torsion elements forms a subgroup \( A_t \), which is a torsion group. diff --git a/iii/cat/04_limits.tex b/iii/cat/04_limits.tex index ea815b3..1e3c516 100644 --- a/iii/cat/04_limits.tex +++ b/iii/cat/04_limits.tex @@ -313,6 +313,7 @@ \subsection{Preservation and creation} \end{tikzcd}\] is a pullback square. Thus, if \( \mathcal D \) has pullbacks, any monomorphism in \( [\mathcal C, \mathcal D] \) is a pointwise monomorphism, because the pullback in \( [\mathcal C, \mathcal D] \) is constructed pointwise by the previous lemma. + In particular, the monomorphisms and epimorphisms in \( [\mathcal C, \mathbf{Set}] \) are precisely the pointwise monomorphisms and pointwise epimorphisms respectively. \end{remark} \subsection{Interaction with adjunctions} diff --git a/iii/cat/05_monads.tex b/iii/cat/05_monads.tex index 9198c5e..45a48eb 100644 --- a/iii/cat/05_monads.tex +++ b/iii/cat/05_monads.tex @@ -150,41 +150,41 @@ \subsection{Kleisli categories} \arrow["Tg", from=1-2, to=1-3] \arrow["{\mu_C}", from=1-3, to=1-4] \end{tikzcd}\] - These satisfy the unit and associativity laws. - % https://q.uiver.app/#q=WzAsNCxbMCwwLCJBIl0sWzEsMCwiVEIiXSxbMiwwLCJUVEIiXSxbMiwxLCJUQiJdLFswLDEsImYiXSxbMSwyLCJUXFxldGFfQiJdLFsyLDMsIlxcbXVfQiJdLFsxLDMsIjFfe1RCfSJdXQ== -\[\begin{tikzcd} - A & TB & T^2B \\ - && TB - \arrow["f", from=1-1, to=1-2] - \arrow["{T\eta_B}", from=1-2, to=1-3] - \arrow["{\mu_B}", from=1-3, to=2-3] - \arrow["{1_{TB}}"', from=1-2, to=2-3] +\end{definition} +These satisfy the unit and associativity laws. +% https://q.uiver.app/#q=WzAsNCxbMCwwLCJBIl0sWzEsMCwiVEIiXSxbMiwwLCJUVEIiXSxbMiwxLCJUQiJdLFswLDEsImYiXSxbMSwyLCJUXFxldGFfQiJdLFsyLDMsIlxcbXVfQiJdLFsxLDMsIjFfe1RCfSJdXQ== +\[\begin{tikzcd} +A & TB & T^2B \\ +&& TB +\arrow["f", from=1-1, to=1-2] +\arrow["{T\eta_B}", from=1-2, to=1-3] +\arrow["{\mu_B}", from=1-3, to=2-3] +\arrow["{1_{TB}}"', from=1-2, to=2-3] \end{tikzcd}\quad\quad\begin{tikzcd} - A & TA \\ - TB & T^2B \\ - & TB - \arrow["{\eta_A}", from=1-1, to=1-2] - \arrow["Tf", from=1-2, to=2-2] - \arrow["{\mu_B}", from=2-2, to=3-2] - \arrow["f"', from=1-1, to=2-1] - \arrow["{\eta_{TB}}", from=2-1, to=2-2] - \arrow["{1_{TB}}"', from=2-1, to=3-2] -\end{tikzcd}\] -\[\begin{tikzcd} - A & TB & {T^2C} & {T^3D} & {T^2D} \\ - && TC & {T^2D} & TD - \arrow["f", from=1-1, to=1-2] - \arrow["Tg", from=1-2, to=1-3] - \arrow["{T^2h}", from=1-3, to=1-4] - \arrow["{T\mu_D}", from=1-4, to=1-5] - \arrow["{\mu_D}", from=1-5, to=2-5] - \arrow["{\mu_{TD}}", from=1-4, to=2-4] - \arrow["{\mu_D}"', from=2-4, to=2-5] - \arrow["{\mu_C}", from=1-3, to=2-3] - \arrow["Th"', from=2-3, to=2-4] +A & TA \\ +TB & T^2B \\ +& TB +\arrow["{\eta_A}", from=1-1, to=1-2] +\arrow["Tf", from=1-2, to=2-2] +\arrow["{\mu_B}", from=2-2, to=3-2] +\arrow["f"', from=1-1, to=2-1] +\arrow["{\eta_{TB}}", from=2-1, to=2-2] +\arrow["{1_{TB}}"', from=2-1, to=3-2] +\end{tikzcd}\] +\[\begin{tikzcd} +A & TB & {T^2C} & {T^3D} & {T^2D} \\ +&& TC & {T^2D} & TD +\arrow["f", from=1-1, to=1-2] +\arrow["Tg", from=1-2, to=1-3] +\arrow["{T^2h}", from=1-3, to=1-4] +\arrow["{T\mu_D}", from=1-4, to=1-5] +\arrow["{\mu_D}", from=1-5, to=2-5] +\arrow["{\mu_{TD}}", from=1-4, to=2-4] +\arrow["{\mu_D}"', from=2-4, to=2-5] +\arrow["{\mu_C}", from=1-3, to=2-3] +\arrow["Th"', from=2-3, to=2-4] \end{tikzcd}\] where in the last diagram, the upper composite is \( (hg)f \) and the lower composite is \( h(gf) \) in \( \mathcal C_{\mathbb T} \). -\end{definition} \begin{proposition} There is an adjunction \( F_{\mathbb T} \dashv G_{\mathbb T} \) where \( F_{\mathbb T} : \mathcal C \to \mathcal C_{\mathbb T} \) and \( G_{\mathbb T} : \mathcal C_{\mathbb T} \to \mathcal C \) that induces the monad \( \mathbb T \). \end{proposition} diff --git a/iii/commalg/01_chain_conditions.tex b/iii/commalg/01_chain_conditions.tex index b4c9411..7d634f8 100644 --- a/iii/commalg/01_chain_conditions.tex +++ b/iii/commalg/01_chain_conditions.tex @@ -86,7 +86,7 @@ \subsection{Noetherian and Artinian modules} Note that every Noetherian module is finitely generated. Let \( R = \mathbb Z[T_1, T_2, \dots] \), and let \( M = R \) as an \( R \)-module. \( M \) is generated by \( 1_R \), so in particular it is finitely generated. -But it has a submodule \( \langle T_1, T_2, \dots \rangle \) that is not finitely generated. +But it has a \( \mathbb Z \)-submodule \( \langle T_1, T_2, \dots \rangle \) that is not finitely generated. So in the above lemma we indeed must check every submodule. \begin{definition} A ring \( R \) is Noetherian (respectively Artinian) if \( R \) is Noetherian (resp.\ Artinian) as an \( R \)-module. diff --git a/iii/commalg/02_tensor_products.tex b/iii/commalg/02_tensor_products.tex index cb8a5a3..7fdb140 100644 --- a/iii/commalg/02_tensor_products.tex +++ b/iii/commalg/02_tensor_products.tex @@ -18,7 +18,7 @@ \subsection{Introduction} \end{example} \begin{example} Now consider \( {\mathbb R}^n \otimes_{\mathbb R} {\mathbb R}^\ell \). - We will show later that this is isomorphic to \( \mathbb R^{n+\ell} \). + We will show later that this is isomorphic to \( \mathbb R^{n\ell} \). \end{example} \subsection{Definition and universal property} diff --git a/iii/commalg/04_integrality_finiteness_finite_generation.tex b/iii/commalg/04_integrality_finiteness_finite_generation.tex index 241834c..53e2502 100644 --- a/iii/commalg/04_integrality_finiteness_finite_generation.tex +++ b/iii/commalg/04_integrality_finiteness_finite_generation.tex @@ -243,7 +243,7 @@ \subsection{Integral closure} \begin{enumerate} \item \( \mathbb Z\qty[\sqrt{5}] \) is not integrally closed, because \( \alpha = \frac{1 + \sqrt{5}}{2} \in FF\qty(\mathbb Z\qty[\sqrt{5}]) = \mathbb Q\qty[\sqrt{5}] \), and \( \alpha^2 - \alpha - 1 = 0 \) so it is \( \mathbb Z\qty[\sqrt{5}] \)-integral. \item \( \mathbb Z \) is integrally closed. - \item If \( k \) is a field, \( k[T_1, \dots, T_n] \) are integrally closed. + \item If \( k \) is a field, \( k[T_1, \dots, T_n] \) is integrally closed. \end{enumerate} \end{example} Examples (ii) and (iii) are special cases of the following result.