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2026-05-19 15:27:49 +02:00
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Tesi.log
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This is pdfTeX, Version 3.141592653-2.6-1.40.29 (TeX Live 2026/Arch Linux) (preloaded format=pdflatex 2026.4.6) 19 MAY 2026 15:10
This is pdfTeX, Version 3.141592653-2.6-1.40.29 (TeX Live 2026/Arch Linux) (preloaded format=pdflatex 2026.4.6) 19 MAY 2026 15:27
entering extended mode
restricted \write18 enabled.
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@@ -861,7 +861,7 @@ Package logreq Info: Writing requests to 'Tesi.run.xml'.
Here is how much of TeX's memory you used:
26647 strings out of 469515
568827 string characters out of 5470808
1335039 words of memory out of 5000000
1336039 words of memory out of 5000000
54941 multiletter control sequences out of 15000+600000
640012 words of font info for 86 fonts, out of 8000000 for 9000
14 hyphenation exceptions out of 8191
@@ -885,7 +885,7 @@ exmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/share/texmf-dist/fonts
msfonts/cm/cmti12.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/symbol
s/msam10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.
pfb></usr/share/texmf-dist/fonts/type1/public/cm-super/sfrm1200.pfb>
Output written on Tesi.pdf (27 pages, 334432 bytes).
Output written on Tesi.pdf (27 pages, 334383 bytes).
PDF statistics:
211 PDF objects out of 1000 (max. 8388607)
134 compressed objects within 2 object streams
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Tesi.tex
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\begin{theorem}\label{completenesscomma}
Let $\mathcal{A}$ and $\mathcal{B}$ be two finitely cocomplete categories and let $F:\mathcal{A} \to \mathcal{C}$, $\mathcal{B} \to \mathcal{C}$ two functors. If $F$ is a continuous functor - i.e. preserves small colimits - then the comma category $(F\downarrow G)$ is finitely cocomplete; moreover the forgetful functors are continuous. \\
Let $\mathcal{A}$ and $\mathcal{B}$ be two finitely cocomplete categories and let $F:\mathcal{A} \to \mathcal{C}$, $\mathcal{B} \to \mathcal{C}$ two functors. If $F$ is a cocontinuous functor - i.e. preserves small colimits - then the comma category $(F\downarrow G)$ is finitely cocomplete; moreover the forgetful functors are continuous. \\
Analogously if $\mathcal{A}$ and $\mathcal{B}$ are finitely complete categories and $G: \mathcal{A} \to \mathcal{C}$ is continuos - i.e. preserves small limits - then $(F \downarrow G)$ is finitely complete and both forgetful functors are continuos.
\end{theorem}