File:Interchange-Law.png

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File:2-category horizontal composition lower.svg is a vector version of this file. It should be used in place of this PNG file when not inferior. File:Interchange-Law.png → File:2-category horizontal composition lower.svg For more information, see Help:SVG. In other languages Alemannisch ∙ Bahasa Indonesia ∙ Bahasa Melayu ∙ British English ∙ català ∙ čeština ∙ dansk ∙ Deutsch ∙ eesti ∙ English ∙ español ∙ Esperanto ∙ euskara ∙ français ∙ Frysk ∙ galego ∙ hrvatski ∙ Ido ∙ italiano ∙ lietuvių ∙ magyar ∙ Nederlands ∙ norsk bokmål ∙ norsk nynorsk ∙ occitan ∙ Plattdüütsch ∙ polski ∙ português ∙ português do Brasil ∙ română ∙ Scots ∙ sicilianu ∙ slovenčina ∙ slovenščina ∙ suomi ∙ svenska ∙ Tiếng Việt ∙ Türkçe ∙ vèneto ∙ Ελληνικά ∙ беларуская (тарашкевіца) ∙ български ∙ македонски ∙ нохчийн ∙ русский ∙ српски / srpski ∙ татарча/tatarça ∙ українська ∙ ქართული ∙ հայերեն ∙ বাংলা ∙ தமிழ் ∙ മലയാളം ∙ ไทย ∙ 한국어 ∙ 日本語 ∙ 简体中文 ∙ 繁體中文 ∙ עברית ∙ العربية ∙ فارسی ∙ +/−

 

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File:2-category vertical composition.svg is a vector version of this file. It should be used in place of this PNG file when not inferior. File:Interchange-Law.png → File:2-category vertical composition.svg For more information, see Help:SVG. In other languages Alemannisch ∙ Bahasa Indonesia ∙ Bahasa Melayu ∙ British English ∙ català ∙ čeština ∙ dansk ∙ Deutsch ∙ eesti ∙ English ∙ español ∙ Esperanto ∙ euskara ∙ français ∙ Frysk ∙ galego ∙ hrvatski ∙ Ido ∙ italiano ∙ lietuvių ∙ magyar ∙ Nederlands ∙ norsk bokmål ∙ norsk nynorsk ∙ occitan ∙ Plattdüütsch ∙ polski ∙ português ∙ português do Brasil ∙ română ∙ Scots ∙ sicilianu ∙ slovenčina ∙ slovenščina ∙ suomi ∙ svenska ∙ Tiếng Việt ∙ Türkçe ∙ vèneto ∙ Ελληνικά ∙ беларуская (тарашкевіца) ∙ български ∙ македонски ∙ нохчийн ∙ русский ∙ српски / srpski ∙ татарча/tatarça ∙ українська ∙ ქართული ∙ հայերեն ∙ বাংলা ∙ தமிழ் ∙ മലയാളം ∙ ไทย ∙ 한국어 ∙ 日本語 ∙ 简体中文 ∙ 繁體中文 ∙ עברית ∙ العربية ∙ فارسی ∙ +/−

 

w: 2-category interchange law

	=		=		∘0
∘1
	PNG version:

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\documentclass[a4paper]{article}
\usepackage[all]{xy}

\begin{document}
\thispagestyle{empty}
\[
\vcenter{
\xymatrix@R=0.5pc{
  {\bullet} \ar[r]^{}="mt1" \ar@/^2pc/[r]_{}="t1" \ar@{=>}"t1";"mt1" &
  {\bullet} \ar[r]^{}="mt2" \ar@/^2pc/[r]_{}="t2" \ar@{=>}"t2";"mt2" &
  {\bullet} \\
  & \circ_1 & \\
  {\bullet} \ar[r]_{}="mb1" \ar@/_2pc/[r]_{}="b1" \ar@{=>}"mb1";"b1" &
  {\bullet} \ar[r]_{}="mb2" \ar@/_2pc/[r]_{}="b2" \ar@{=>}"mb2";"b2" &
  {\bullet}
} } \qquad = \qquad
\xymatrix{
  {\bullet} \ar[r]^{}="ml" \ar@/^2pc/[r]_{}="tl" \ar@/_2pc/[r]^{}="bl"
  \ar@{=>}"tl";"ml" \ar@{=>}"ml";"bl" & 
  {\bullet} \ar[r]^{}="ml" \ar@/^2pc/[r]_{}="tl" \ar@/_2pc/[r]^{}="bl"
  \ar@{=>}"tl";"ml" \ar@{=>}"ml";"bl" & {\bullet}
} \qquad = \qquad
\xymatrix{
  {\bullet} \ar[r]^{}="ml" \ar@/^2pc/[r]_{}="tl" \ar@/_2pc/[r]^{}="bl"
  \ar@{=>}"tl";"ml" \ar@{=>}"ml";"bl" & {\bullet} \ar@{}[r]|{\textstyle\circ_0}
  &j
  {\bullet} \ar[r]^{}="ml" \ar@/^2pc/[r]_{}="tl" \ar@/_2pc/[r]^{}="bl"
  \ar@{=>}"tl";"ml" \ar@{=>}"ml";"bl" & {\bullet}
}
\]
\end{document}