Olga

Exercises/Exercises.aux

@@ -23,17 +23,17 @@
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.1}{}}{2}{subsection.2.1}}
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.2}{}}{2}{subsection.2.2}}
 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.3}{}}{2}{subsection.2.3}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.4}{}}{2}{subsection.2.4}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.5}{}}{2}{subsection.2.5}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.4}{}}{3}{subsection.2.4}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.5}{}}{3}{subsection.2.5}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.6}{}}{3}{subsection.2.6}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.7}{}}{3}{subsection.2.7}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.8}{}}{3}{subsection.2.8}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.9}{}}{3}{subsection.2.9}}
 \newlabel{tocindent-1}{0pt}
 \newlabel{tocindent0}{0pt}
 \newlabel{tocindent1}{17.5pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.6}{}}{3}{subsection.2.6}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.7}{}}{3}{subsection.2.7}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.8}{}}{3}{subsection.2.8}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.9}{}}{3}{subsection.2.9}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.10}{}}{3}{subsection.2.10}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.11}{}}{3}{subsection.2.11}}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Discussion:}}{3}{section.3}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.10}{}}{4}{subsection.2.10}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.11}{}}{4}{subsection.2.11}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Discussion:}}{4}{section.3}}

Exercises/Exercises.log

-This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=pdflatex 2019.2.25)  19 SEP 2020 17:59
+This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=pdflatex 2019.2.25)  19 SEP 2020 18:04
 entering extended mode
  restricted \write18 enabled.
  %&-line parsing enabled.
@@ -1201,77 +1201,78 @@ Underfull \hbox (badness 10000) in paragraph at lines 243--244
 
  []
 
-
-Underfull \hbox (badness 10000) in paragraph at lines 246--247
+[2]
+Underfull \hbox (badness 10000) in paragraph at lines 276--277
 
  []
 
 
-Underfull \hbox (badness 10000) in paragraph at lines 259--260
+Underfull \hbox (badness 10000) in paragraph at lines 289--290
 
  []
 
 
-Underfull \hbox (badness 10000) in paragraph at lines 261--262
+Underfull \hbox (badness 10000) in paragraph at lines 291--292
 
  []
 
-[2]
-Underfull \hbox (badness 10000) in paragraph at lines 269--270
+
+Underfull \hbox (badness 10000) in paragraph at lines 299--300
 
  []
 
 
-Underfull \hbox (badness 10000) in paragraph at lines 271--272
+Underfull \hbox (badness 10000) in paragraph at lines 301--302
 
  []
 
 
-Underfull \hbox (badness 10000) in paragraph at lines 281--282
+Underfull \hbox (badness 10000) in paragraph at lines 311--312
 
  []
 
 
-Underfull \hbox (badness 10000) in paragraph at lines 283--284
+Underfull \hbox (badness 10000) in paragraph at lines 313--314
 
  []
 
-
-Underfull \hbox (badness 10000) in paragraph at lines 285--286
+[3]
+Underfull \hbox (badness 10000) in paragraph at lines 315--316
 
  []
 
-[3]
-Package atveryend Info: Empty hook `BeforeClearDocument' on input line 294.
-Package atveryend Info: Empty hook `AfterLastShipout' on input line 294.
+[4] [5]
+Package atveryend Info: Empty hook `BeforeClearDocument' on input line 335.
+Package atveryend Info: Empty hook `AfterLastShipout' on input line 335.
  (./Exercises.aux)
-Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 294.
-Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 294.
+Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 335.
+Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 335.
 Package rerunfilecheck Info: File `Exercises.out' has not changed.
 (rerunfilecheck)             Checksum: E92B3DFA7A600DB1C5E1DD495D1DCD17;145.
-Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 294.
+Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 335.
  ) 
 Here is how much of TeX's memory you used:
- 25610 strings out of 492649
- 503883 string characters out of 6129622
- 640267 words of memory out of 5000000
+ 25617 strings out of 492649
+ 503924 string characters out of 6129622
+ 641267 words of memory out of 5000000
  28769 multiletter control sequences out of 15000+600000
- 34604 words of font info for 98 fonts, out of 8000000 for 9000
+ 35954 words of font info for 100 fonts, out of 8000000 for 9000
  1302 hyphenation exceptions out of 8191
  55i,16n,98p,3246b,936s stack positions out of 5000i,500n,10000p,200000b,80000s
 {/usr/local/texlive/2018/texmf-dist/fonts/enc/dvips/base/8r.enc}</usr/local/t
 exlive/2018/texmf-dist/fonts/type1/public/txfonts/rtxmi.pfb></usr/local/texlive
 /2018/texmf-dist/fonts/type1/public/txfonts/rtxr.pfb></usr/local/texlive/2018/t
 exmf-dist/fonts/type1/public/txfonts/rtxsc.pfb></usr/local/texlive/2018/texmf-d
-ist/fonts/type1/public/txfonts/txsy.pfb></usr/local/texlive/2018/texmf-dist/fon
-ts/type1/public/txfonts/txsya.pfb></usr/local/texlive/2018/texmf-dist/fonts/typ
-e1/public/txfonts/txsyc.pfb></usr/local/texlive/2018/texmf-dist/fonts/type1/urw
-/times/utmb8a.pfb></usr/local/texlive/2018/texmf-dist/fonts/type1/urw/times/utm
-r8a.pfb></usr/local/texlive/2018/texmf-dist/fonts/type1/urw/times/utmri8a.pfb>
-Output written on Exercises.pdf (3 pages, 67073 bytes).
+ist/fonts/type1/public/txfonts/txex.pfb></usr/local/texlive/2018/texmf-dist/fon
+ts/type1/public/txfonts/txsy.pfb></usr/local/texlive/2018/texmf-dist/fonts/type
+1/public/txfonts/txsya.pfb></usr/local/texlive/2018/texmf-dist/fonts/type1/publ
+ic/txfonts/txsyc.pfb></usr/local/texlive/2018/texmf-dist/fonts/type1/urw/times/
+utmb8a.pfb></usr/local/texlive/2018/texmf-dist/fonts/type1/urw/times/utmr8a.pfb
+></usr/local/texlive/2018/texmf-dist/fonts/type1/urw/times/utmri8a.pfb>
+Output written on Exercises.pdf (5 pages, 77107 bytes).
 PDF statistics:
- 98 PDF objects out of 1000 (max. 8388607)
- 83 compressed objects within 1 object stream
- 24 named destinations out of 1000 (max. 500000)
+ 114 PDF objects out of 1000 (max. 8388607)
+ 96 compressed objects within 1 object stream
+ 29 named destinations out of 1000 (max. 500000)
  37 words of extra memory for PDF output out of 10000 (max. 10000000)
 

Exercises/Exercises.tex

@@ -242,6 +242,36 @@ where we have used the fact that identity arrow exists for each object A and B.\
 
 \subsection{}  Check that functors compose in an associative way, proving that $\mathbf{Cat}$ is a category. \hfill \break
 
+<<<<<<< HEAD
+=======
+\iam{Muhammed}  Let $\mathbf{C_1}, \mathbf{C_2}, \mathbf{C_3}, \mathbf{C_4}$ be  categories and $F: \mathbf{C_1} \to \mathbf{C_2}, \, G: \mathbf{C_2} \to \mathbf{C_3}, \, H: \mathbf{C_3} \to \mathbf{C_4}$ functors. We consider 
+$$K:= G \circ F:  \mathbf{C_1} \to  \mathbf{C_3},$$
+where for any object $A$ and any arrow $f$ in $\mathbf{C_1}$, $$
+K(A) := G(F(A))  \quad \text{and} \quad K(f) := G(F(f)) \quad \quad (2.3.1)$$
+We show that $K$ is a functor. \\ Let $A,B$ be objects and $f: A \to B, \, g: B \to C$ arrows in $\mathbf{C_1}$. \begin{enumerate}
+\item Since $F,G$ are functors, $F(f): F(A) \to F(B)$ and $G(F(f)): G(F(A)) \to G(F(B))$. Thus, $K(f): (G \circ F)(A) \to (G \circ F)(B)$.
+\item $(G \circ F)(f \circ g) := G(F(f \circ g)) = G(F(f) \circ F(g))= G(F(f)) \circ G(F(g))$.
+\item $K(1_A) := G(F(1_A)) = G(1_{F(A)}) = 1_{G(F(A))} =: 1_{(G \circ F)(A)}$
+\end{enumerate}
+Also, clearly, $K(A)$ is an object in $\mathbf{C_3}$. This proves the assertion that $K$ is a functor.\\
+To show associativity of the composition, we require
+$$
+H \circ (G \circ F) = (H \circ G) \circ F \quad (2.3.2) 
+$$ 
+where both sides are functors from $\mathbf{C_1}$ to $\mathbf{C_4}$ by previous considerations.
+Now, by (2.3.1), it readily follows that, 
+$$
+(H \circ (G \circ F))(A) = H((G \circ F)(A)) = H(G(F(A))) = (H \circ G)(F(A)) = ((H \circ G) \circ F)(A) 
+$$
+and
+$$(H \circ (G \circ F))(f) = H((G \circ F)(f)) = H(G(F(f))) = (H \circ G)(F(f)) = ((H \circ G) \circ F)(f)$$
+for every any object $A$ and any arrow $f$ in $\mathbf{C_1}$. This yields the desired equality stated in (2.3.2).
+
+
+
+
+
+>>>>>>> ee2c8e437d66a66aa46fdcd6f59bda9fddb9c8fc
 
 \subsection{}  Prove that inverses are unique. \hfill \break
 
@@ -291,4 +321,15 @@ This shows, that $f$ is indeed a bijection.
 \iam{Severin} IN Awodey, p. 22 it says "For example, all finite categories are clearly small, as is the category $\textbf{Sets}_{\text{fin}}$ of finite sets and functions." This confused me a bit, as it is not true. Indeed, we can consider the class (this is the generalized notion for stuff that is too big to be a set, just like the collection of all sets)
 $$ \{ \{M\} \ : \ M\in \textbf{Set} \}. $$
 Nicola explained me that the statement is at least morally true. The category is "roughly the same" as some category that is small. The main problem is that we allow for too many copies of the same thing. There is no big difference between $\{\lozenge\}$ and $\{ \heartsuit\}$ (as sets). The correct notion of "roughly the same" will be equivalence of categories, which we can find in Chapter $7$ in Awodey. In fact our example is explicitely treated in Example $7.23.$ (p. $146$). 
+
+\iam{Marius} Regarding, the above, I'm not sure your counterexample works. If I understand correctly, you are saying that the collection $\Theta := \{ \Psi \}$, where $\Psi := \{M : M \in \mathbf{Set}\}$, is finite, since it contains only the single element $\Psi$ (I'm unclear why you put $\{M\}$ in your example?). On the other hand since, $\Theta$ is a proper class (if it where a set, $\bigcup{\Theta} = \Psi$ would also be by the axiom of union, contradiction by Russell's paradox). In particular, the collection $\Theta$ is not a set and the category $\mathcal{C}$ with $Ob(\mathcal{C}) := \Theta$ is not small. 
+
+Assuming this is the argument, I doubt that $\Theta$ finite. In set theory we define a set to be finite if it is in bijection with an element of the natural numbers (as constructed set-theoretically from ordinals and the axiom of infinity). Since $\Theta$ is a proper class, there aren't any set-theoretic mappings between $\Theta$ and any set: Any such mapping is a subset of the cartesian product of two sets. Hence, if we assume such as map, we could again union our way to a set of all sets. Therefore, it seems we would need an extension of the concept of finiteness if we wanted to apply it to proper classes. Maybe, we could re-construct an analogue of the natural numbers at each level of the class hierarchy and then count elements with respect to these? Under the set-theoretic definition, however, $\Theta$ fails to be finite and is therefore not a counterexample.
+
+\iam{Severin} I am not sure, whether I understand, what you mean. I want to show that the collection of all finite set is a proper class (and thus $\textbf{Set}_{\text{fin}}$ is not a small category). For this I note that the collection of all singeltons is a proper class. What I am saying is that $\{M\}$ is finite (clearly $M$ is not finite in general and thus I wrote the brackets). I am not sure, why you introduce $\Psi$. Can you clarify?
+
+\iam{Marius} Okay, I see now what you were objecting to the claim that $\textbf{Set}_{\text{fin}}$ is a small category. I now agree with you. Thanks for pointing that out! I implicitly assumed we could somehow enumerate finite sets based on cardinality, but this is clearly incorrect upon reflection, as your example shows. Because of this mistaken assumption I thought you were objecting to the claim that all finite categories are small. I apologize for any confusion I created. I believe my example still shows that a category might not be small, even if it only contains a single object.
+
+What you report Nicola as saying makes more sense now. Indeed, we can consider the category whose objects are isomorphism classes of finite sets. Restricting to some choice of representatives, we obtain an equivalent subcategory of $\textbf{Set}_{\text{fin}}$ which contains no isomorphic objects (called the skeleton of $\textbf{Set}_{\text{fin}}$, see e.g. wikipedia). This skeleton will then be isomorphic as a category to $\mathbf{FinOrd}$, the category of finite ordinals, a small category.
+
 \end{document}

Exercises/Exercises.toc

@@ -5,12 +5,12 @@
 \contentsline {subsection}{\tocsubsection {}{2.1}{}}{2}{subsection.2.1}
 \contentsline {subsection}{\tocsubsection {}{2.2}{}}{2}{subsection.2.2}
 \contentsline {subsection}{\tocsubsection {}{2.3}{}}{2}{subsection.2.3}
-\contentsline {subsection}{\tocsubsection {}{2.4}{}}{2}{subsection.2.4}
-\contentsline {subsection}{\tocsubsection {}{2.5}{}}{2}{subsection.2.5}
+\contentsline {subsection}{\tocsubsection {}{2.4}{}}{3}{subsection.2.4}
+\contentsline {subsection}{\tocsubsection {}{2.5}{}}{3}{subsection.2.5}
 \contentsline {subsection}{\tocsubsection {}{2.6}{}}{3}{subsection.2.6}
 \contentsline {subsection}{\tocsubsection {}{2.7}{}}{3}{subsection.2.7}
 \contentsline {subsection}{\tocsubsection {}{2.8}{}}{3}{subsection.2.8}
 \contentsline {subsection}{\tocsubsection {}{2.9}{}}{3}{subsection.2.9}
-\contentsline {subsection}{\tocsubsection {}{2.10}{}}{3}{subsection.2.10}
-\contentsline {subsection}{\tocsubsection {}{2.11}{}}{3}{subsection.2.11}
-\contentsline {section}{\tocsection {}{3}{Discussion:}}{3}{section.3}
+\contentsline {subsection}{\tocsubsection {}{2.10}{}}{4}{subsection.2.10}
+\contentsline {subsection}{\tocsubsection {}{2.11}{}}{4}{subsection.2.11}
+\contentsline {section}{\tocsection {}{3}{Discussion:}}{4}{section.3}