From 74080a7904a294a10d1bc333c3576efbeb7455d0 Mon Sep 17 00:00:00 2001
From: Muhammed Guelen <muhammedguelen@macbook-pro.speedport_w_723v_1_47_000>
Date: Fri, 6 Nov 2020 19:24:56 +0100
Subject: [PATCH] Typos corrected

---
 Exercises/Exercises.aux        |   2 +-
 Exercises/Exercises.log        |  46 +++++++++++++++------------------
 Exercises/Exercises.pdf        | Bin 256586 -> 250663 bytes
 Exercises/Exercises.synctex.gz | Bin 433136 -> 432921 bytes
 Exercises/Exercises.tex        |  12 +++------
 Exercises/Exercises.toc        |   2 +-
 6 files changed, 27 insertions(+), 35 deletions(-)

diff --git a/Exercises/Exercises.aux b/Exercises/Exercises.aux
index a111b38..45450ae 100644
--- a/Exercises/Exercises.aux
+++ b/Exercises/Exercises.aux
@@ -93,7 +93,7 @@
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{12.2}{}}{35}{subsection.12.2}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{12.3}{}}{35}{subsection.12.3}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{12.4}{}}{36}{subsection.12.4}\protected@file@percent }
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{12.5}{}}{37}{subsection.12.5}\protected@file@percent }
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{12.5}{}}{36}{subsection.12.5}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{12.6}{}}{37}{subsection.12.6}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{12.7}{}}{37}{subsection.12.7}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{12.8}{}}{37}{subsection.12.8}\protected@file@percent }
diff --git a/Exercises/Exercises.log b/Exercises/Exercises.log
index 3197717..d05f8d8 100644
--- a/Exercises/Exercises.log
+++ b/Exercises/Exercises.log
@@ -1,4 +1,4 @@
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+This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=pdflatex 2020.9.18)  6 NOV 2020 19:24
 entering extended mode
  restricted \write18 enabled.
  %&-line parsing enabled.
@@ -1550,55 +1550,52 @@ Underfull \hbox (badness 10000) in paragraph at lines 1680--1681
  []
 
 [35]
-Underfull \hbox (badness 10000) in paragraph at lines 1743--1745
+Underfull \hbox (badness 10000) in paragraph at lines 1739--1741
 
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-(Font)              Font shape `OT1/txr/bx/it' tried instead on input line 1748
-.
 
-Underfull \hbox (badness 10000) in paragraph at lines 1758--1759
+Underfull \hbox (badness 10000) in paragraph at lines 1754--1755
 
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-Underfull \hbox (badness 10000) in paragraph at lines 1760--1761
+Underfull \hbox (badness 10000) in paragraph at lines 1756--1757
 
  []
 
 [36]
-Underfull \hbox (badness 10000) in paragraph at lines 1769--1771
+Underfull \hbox (badness 10000) in paragraph at lines 1765--1767
 
  []
 
 [37]
-Underfull \hbox (badness 10000) in paragraph at lines 1819--1820
+Underfull \hbox (badness 10000) in paragraph at lines 1815--1816
 
  []
 
 
-Underfull \hbox (badness 10000) in paragraph at lines 1835--1837
+Underfull \hbox (badness 10000) in paragraph at lines 1831--1833
 
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 [38]
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-Package atveryend Info: Empty hook `AfterLastShipout' on input line 1855.
+Package atveryend Info: Empty hook `BeforeClearDocument' on input line 1851.
+Package atveryend Info: Empty hook `AfterLastShipout' on input line 1851.
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-Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 1855.
-Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 1855.
+Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 1851.
+Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 1851.
 
 Package rerunfilecheck Info: File `Exercises.out' has not changed.
 (rerunfilecheck)             Checksum: 3241C52EA2B865C61DCD95A898875109;589.
-Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 1855.
+Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 1851.
  ) 
 Here is how much of TeX's memory you used:
- 30468 strings out of 480608
- 613573 string characters out of 5903630
+ 30463 strings out of 480608
+ 613525 string characters out of 5903630
  911221 words of memory out of 5000000
- 45393 multiletter control sequences out of 15000+600000
- 571657 words of font info for 124 fonts, out of 8000000 for 9000
+ 45391 multiletter control sequences out of 15000+600000
+ 569714 words of font info for 121 fonts, out of 8000000 for 9000
  1302 hyphenation exceptions out of 8191
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 {/usr/local/texlive/2020/texmf-dist/fonts/enc/dvips/base/8r.enc}</usr/local/t
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-ve/2020/texmf-dist/fonts/type1/urw/times/utmr8a.pfb></usr/local/texlive/2020/te
-xmf-dist/fonts/type1/urw/times/utmri8a.pfb>
-Output written on Exercises.pdf (38 pages, 256586 bytes).
+cal/texlive/2020/texmf-dist/fonts/type1/urw/times/utmr8a.pfb></usr/local/texliv
+e/2020/texmf-dist/fonts/type1/urw/times/utmri8a.pfb>
+Output written on Exercises.pdf (38 pages, 250663 bytes).
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- 396 compressed objects within 4 object streams
+ 450 PDF objects out of 1000 (max. 8388607)
+ 393 compressed objects within 4 object streams
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diff --git a/Exercises/Exercises.pdf b/Exercises/Exercises.pdf
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diff --git a/Exercises/Exercises.tex b/Exercises/Exercises.tex
index 16d1e65..077e96d 100644
--- a/Exercises/Exercises.tex
+++ b/Exercises/Exercises.tex
@@ -1723,21 +1723,17 @@ G \cong \text{Hom}_{\mathcal{D}}(D, -),
 $$
 proving representability of $G$.
 
-
-
-
-
 \begin{claim} If $G$ is representable, then $F$ is representable.
 \end{claim}
-We start noting that $H$ being an equivalence implies readily that $K$ is an equivalence as $H$ and $K$ are symmetric in the condition of being a category equivalence. The rest of the proof here follows the exact same line of thought as above with the only difference that we compose the hom-functor with $H$ instead of $K$. Hence, we forego providing every detail.\\
-Now, suppose $G$ is representable, we can find $D \in \mathcal{D}$ s.t. $$
+We start noting that $H$ being an equivalence implies readily that $K$ is an equivalence as $H$ and $K$ are symmetric in the condition of being a category equivalence. The rest of the proof here follows the exact same line of thought as above with the only difference that we compose the hom-functor with $H$ instead of $K$. \\
+Now, supposing, $G$ is representable, we can find $D \in \mathcal{D}$ s.t. $$
 \beta: G \cong \text{Hom}_{\mathcal{D}}(D, -).
 $$
 Then, 
 $$
 F \cong G \circ H  \cong \text{Hom}_{\mathcal{D}}(D, -) \circ H
 $$
-and we can find – details are skipped – a $C \in \mathcal{C}$ s.t.
+and we can find a $C \in \mathcal{C}$ such that
 $$
 F \cong \text{Hom}_{\mathcal{C}}(C, -) 
 $$
@@ -1745,7 +1741,7 @@ where $C = K(D) \in \mathcal{C}$. This completes the proof. \\
 
 We can summarize:
 
-\begin{cor} Representability of functors is \textbf{not} preserved under category equivalence.
+\begin{cor} Representability of functors is preserved under category equivalence.
 \end{cor}
 
 
diff --git a/Exercises/Exercises.toc b/Exercises/Exercises.toc
index 09c3fa1..64a9902 100644
--- a/Exercises/Exercises.toc
+++ b/Exercises/Exercises.toc
@@ -73,7 +73,7 @@
 \contentsline {subsection}{\tocsubsection {}{12.2}{}}{35}{subsection.12.2}%
 \contentsline {subsection}{\tocsubsection {}{12.3}{}}{35}{subsection.12.3}%
 \contentsline {subsection}{\tocsubsection {}{12.4}{}}{36}{subsection.12.4}%
-\contentsline {subsection}{\tocsubsection {}{12.5}{}}{37}{subsection.12.5}%
+\contentsline {subsection}{\tocsubsection {}{12.5}{}}{36}{subsection.12.5}%
 \contentsline {subsection}{\tocsubsection {}{12.6}{}}{37}{subsection.12.6}%
 \contentsline {subsection}{\tocsubsection {}{12.7}{}}{37}{subsection.12.7}%
 \contentsline {subsection}{\tocsubsection {}{12.8}{}}{37}{subsection.12.8}%
-- 
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