From 85d9968331807a2dc5008a7fe76c8bff1f943029 Mon Sep 17 00:00:00 2001 From: zeramorphic Date: Wed, 24 Apr 2024 16:07:13 +0100 Subject: [PATCH] Minor edits Signed-off-by: zeramorphic --- iii/cat/01_definitions_and_examples.tex | 6 +++--- iii/cat/02_yoneda_lemma.tex | 4 ++-- iii/cat/04_limits.tex | 7 +++---- 3 files changed, 8 insertions(+), 9 deletions(-) diff --git a/iii/cat/01_definitions_and_examples.tex b/iii/cat/01_definitions_and_examples.tex index 84079fd..83cd4e2 100644 --- a/iii/cat/01_definitions_and_examples.tex +++ b/iii/cat/01_definitions_and_examples.tex @@ -178,7 +178,7 @@ \subsection{Natural transformations} \end{tikzcd}\] In particular, the existence of \( \alpha_x \) proves that \( fx \leq gx \). Thus a natural transformation \( f \to g \) exists if and only if \( fx \leq gx \) pointwise for all \( x \in P \). - Note that every commutative square in a poset commutes. + Note that every square of morphisms in a poset commutes. \item Let \( u, v : G \rightrightarrows H \) be group homomorphisms. For \( g \in G \), the naturality square is \[\begin{tikzcd} @@ -260,7 +260,7 @@ \subsection{Equivalence of categories} and \[ G : \mathbf{Part} \to \mathbf{Set}_\star;\quad G(A) = A \cup \qty{A};\quad G(A \xrightarrow f B \text{ partial})(x) = \begin{cases} f(x) & \text{if } f \text{ is defined at } x \\ - V & \text{otherwise} + B & \text{otherwise} \end{cases} \] Note that \( FG = 1_{\mathbf{Part}} \), but \( GF \) is not equal to \( 1_{\mathbf{Set}_\star} \). It is not possible for these two categories to be isomorphic, because there is an isomorphism class of \( \mathbf{Part} \) that has only one member, namely \( \qty{\varnothing} \), but this cannot occur in \( \mathbf{Set}_\star \). @@ -359,7 +359,7 @@ \subsection{Equivalence of categories} \end{proof} We call a subcategory full if its inclusion functor is full. \begin{definition} - A category of \emph{skeletal} if every isomorphism class has a single member. + A category is called \emph{skeletal} if every isomorphism class has a single member. A \emph{skeleton} of \( \mathcal C \) is a full subcategory \( \mathcal C' \) containing exactly one object for each isomorphism class. \end{definition} Note that an equivalence of skeletal categories is bijective on objects, and hence is an isomorphism of categories. diff --git a/iii/cat/02_yoneda_lemma.tex b/iii/cat/02_yoneda_lemma.tex index f9a843d..18369b9 100644 --- a/iii/cat/02_yoneda_lemma.tex +++ b/iii/cat/02_yoneda_lemma.tex @@ -209,7 +209,7 @@ \subsection{Representable functors} Hence, \( e \) is a monomorphism. Monomorphisms that occur in this way are called \emph{regular}. \item Dually, there is also a notion of coequaliser, giving rise to an epimorphism. - We again epimorphisms \emph{regular} if they arise in this way. + We again call epimorphisms \emph{regular} if they arise in this way. \end{enumerate} In \( \mathbf{Set} \), the categorical product is the Cartesian product, and the categorical coproduct is the disjoint union. The equaliser of \( f, g : A \rightrightarrows B \) is the set @@ -361,5 +361,5 @@ \subsection{Projectivity} % any coproduct of projectives is projective Then \( P \) is pointwise projective, since the \( \mathcal C(A, -) \) are. There is a natural transformation \( \alpha : P \to F \) where the \( (A, x) \)-indexed term is \( \Psi(x) : \mathcal C(A, -) \to F \). - This is pointwise epic, since any \( x \in FA \) is in the image of \( \Psi(x) \). + This is pointwise epic, since any \( x \in FA \) is in the image of \( \Psi(x) \). \end{proof} diff --git a/iii/cat/04_limits.tex b/iii/cat/04_limits.tex index f7ae3d8..ea815b3 100644 --- a/iii/cat/04_limits.tex +++ b/iii/cat/04_limits.tex @@ -37,7 +37,7 @@ \subsection{Cones over diagrams} \end{example} \begin{definition} Let \( D \) be a diagram of shape \( J \) in \( \mathcal C \). - A \emph{cone over \( D \)} consists of an object \( C \in \ob \mathcal C \) called the \emph{apex} of the cone, together with morphisms \( \lambda_j : A \to D(j) \) called the \emph{legs} of the cone, such that all triangles of the following form commute. + A \emph{cone over \( D \)} consists of an object \( A \in \ob \mathcal C \) called the \emph{apex} of the cone, together with morphisms \( \lambda_j : A \to D(j) \) called the \emph{legs} of the cone, such that all triangles of the following form commute. % https://q.uiver.app/#q=WzAsMyxbMSwwLCJBIl0sWzAsMSwiRChqKSJdLFsyLDEsIkQoaicpIl0sWzAsMSwiXFxsYW1iZGFfaiIsMl0sWzEsMiwiRChcXGFscGhhKSIsMl0sWzAsMiwiXFxsYW1iZGFfe2onfSJdXQ== \[\begin{tikzcd} & A \\ @@ -400,6 +400,7 @@ \subsection{General adjoint functor theorem} for some \( i \in I \) and \( g : B_i \to B \). This set \( I \) is called a solution-set at \( A \). \end{theorem} +The solution-set condition can be equivalently phrased as the assertion that the categories \( (A \downarrow G) \) all have \emph{weakly initial} sets of objects: every object of \( (A \downarrow G) \) admits a morphism from a member of the solution set. \begin{proof} If \( F \dashv G \), then \( G \) preserves all limits that exist in its domain, so in particular it preserves small limits, and \( \qty{\eta_A : A \to GFA} \) is a solution-set at \( A \) for any \( A \). Now suppose \( A \in \ob \mathcal C \). @@ -441,7 +442,7 @@ \subsection{General adjoint functor theorem} \end{enumerate} \end{example} -\subsection{Well-poweredness} +\subsection{Special adjoint functor theorem} \begin{definition} Let \( A \in \ob \mathcal C \). A \emph{subobject} of \( A \) is a monomorphism with codomain \( A \); dually, a \emph{quotient} of \( A \) is an epimorphism with domain \( A \). @@ -485,8 +486,6 @@ \subsection{Well-poweredness} \end{tikzcd}\] So \( \ell \) and \( m \) are both factorisations of \( (h\ell, k\ell) \) through the pullback, so \( \ell = m \). \end{proof} - -\subsection{Special adjoint functor theorem} \begin{theorem} Let \( \mathcal C, \mathcal D \) be locally small, and suppose that \( \mathcal D \) is complete, well-powered, and has a coseparating set. Then a functor \( G : \mathcal D \to \mathcal C \) preserves all small limits if and only if it has a left adjoint.