From 48626d20f19ea4feb10461514fd834d9fec21a40 Mon Sep 17 00:00:00 2001 From: niyoushanajmaei Date: Tue, 2 Jan 2024 14:07:42 +0100 Subject: [PATCH 1/2] Fix Typos in categories.tex --- categories.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/categories.tex b/categories.tex index 567c397a2..a7c212f7a 100644 --- a/categories.tex +++ b/categories.tex @@ -339,7 +339,7 @@ \section{Functors and transformations} \end{lem} \begin{proof} If $\gamma$ is an isomorphism, then we have $\delta:G\to F$ that is its inverse. - By definition of composition in $B^A$, $(\delta\gamma)_a\jdeq \delta_a\gamma_a$ and similarly. + By definition of composition in $B^A$, $(\delta\gamma)_a\jdeq \delta_a\gamma_a$ and similarly $(\gamma\delta)_a\jdeq \gamma_a\delta_a$. Thus, $\id{\delta\gamma}{1_F}$ and $\id{\gamma\delta}{1_G}$ imply $\id{\delta_a\gamma_a}{1_{Fa}}$ and $\id{\gamma_a\delta_a}{1_{Ga}}$, so $\gamma_a$ is an isomorphism. Conversely, suppose each $\gamma_a$ is an isomorphism, with inverse called $\delta_a$, say. @@ -453,7 +453,7 @@ \section{Functors and transformations} \cref{ct:functor-assoc} is coherent, i.e.\ the following pentagon\index{pentagon, Mac Lane} of equalities commutes: \[ \xymatrix{ & K(H(GF)) \ar@{=}[dl] \ar@{=}[dr]\\ (KH)(GF) \ar@{=}[d] && K((HG)F) \ar@{=}[d]\\ - ((KH)G)F && (K(HG))F \ar@{=}[ll] } + ((KH)G)F && (K(HG))F \ar@{=}[ll].} \] \end{lem} \begin{proof} @@ -468,7 +468,7 @@ \section{Functors and transformations} \[ \xymatrix{ G\circ (1_B \circ F) \ar@{=}[rr] \ar@{=}[dr] && (G\circ 1_B)\circ F \ar@{=}[dl] \\ - & G \circ F.} + & G \circ F} \] \end{lem} @@ -551,7 +551,7 @@ \section{Equivalences} \indexdef{precategory!equivalence of}% \index{functor!equivalence}% if it is a left adjoint for which $\eta$ and $\epsilon$ are isomorphisms. - We write $\cteqv A B$ for the type of equivalences of categories from $A$ to $B$. + We write $\cteqv A B$ for the type of equivalences of (pre)categories from $A$ to $B$. \end{defn} By \cref{ct:adjprop,ct:isoprop}, if $A$ is a category, then the type ``$F$ is an equivalence of precategories'' is a mere proposition. From fd00fe5cda07fd562bd68b57bf8b9e4e717c13d1 Mon Sep 17 00:00:00 2001 From: niyoushanajmaei Date: Thu, 4 Jan 2024 23:00:14 +0100 Subject: [PATCH 2/2] more consistent punctuation --- categories.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/categories.tex b/categories.tex index a7c212f7a..28d5400f4 100644 --- a/categories.tex +++ b/categories.tex @@ -464,11 +464,11 @@ \section{Functors and transformations} We have a similar coherence result for units. \begin{lem}\label{ct:units} - For a functor $F:A\to B$, we have equalities $\id{(1_B\circ F)}{F}$ and $\id{(F\circ 1_A)}{F}$, such that given also $G:B\to C$, the following triangle of equalities commutes. + For a functor $F:A\to B$, we have equalities $\id{(1_B\circ F)}{F}$ and $\id{(F\circ 1_A)}{F}$, such that given also $G:B\to C$, the following triangle of equalities commutes: \[ \xymatrix{ G\circ (1_B \circ F) \ar@{=}[rr] \ar@{=}[dr] && (G\circ 1_B)\circ F \ar@{=}[dl] \\ - & G \circ F} + & G \circ F.} \] \end{lem}