A bicategory of reduced orbifolds from the point of view of differential geometry

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• 作者：Matteo Tommasini ^{matteo.tommasini2@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14D20 ; 14H51 ; 14H60 ; 14F45
• 刊名：Journal of Geometry and Physics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：108
• 期：Complete
• 页码：117-137
• 全文大小：932 K

文摘

We describe a bicategory an id="mmlsi1" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0393044016300699&_mathId=si1.gif&_user=111111111&_pii=S0393044016300699&_rdoc=1&_issn=03930440&md5=a63bfe5a3435771ec352e32b6e6c4c05">g class="imgLazyJSB inlineImage" height="15" width="73" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0393044016300699-si1.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">(athvariant="script">Rathvariant="bold">edace width="0.16667em">ace>athvariant="script">Oathvariant="bold">rb)ath>an>an>an> of reduced orbifolds in the framework of classical differential geometry (i.e. without any explicit reference to the notions of Lie groupoids or differentiable stacks, but only using orbifold atlases, local lifts and changes of charts). In order to construct such a bicategory, we firstly define a 2-category an id="mmlsi2" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0393044016300699&_mathId=si2.gif&_user=111111111&_pii=S0393044016300699&_rdoc=1&_issn=03930440&md5=d0f3804a9511e5b125e5522ae328cfe7">g class="imgLazyJSB inlineImage" height="15" width="69" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0393044016300699-si2.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">(athvariant="script">Rathvariant="bold">edace width="0.16667em">ace>athvariant="script">Aathvariant="bold">tl)ath>an>an>an> whose objects are reduced orbifold atlases (on any paracompact, second countable, Hausdorff topological space). The definition of morphisms is obtained as a slight modification of a definition by A. Pohl, while the definitions of 2-morphisms and compositions of them are new in this setup. Using the bicalculus of fractions described by D. Pronk, we are able to construct the bicategory an id="mmlsi1" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0393044016300699&_mathId=si1.gif&_user=111111111&_pii=S0393044016300699&_rdoc=1&_issn=03930440&md5=a63bfe5a3435771ec352e32b6e6c4c05">g class="imgLazyJSB inlineImage" height="15" width="73" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0393044016300699-si1.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">(athvariant="script">Rathvariant="bold">edace width="0.16667em">ace>athvariant="script">Oathvariant="bold">rb)ath>an>an>an> from the 2-category an id="mmlsi2" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0393044016300699&_mathId=si2.gif&_user=111111111&_pii=S0393044016300699&_rdoc=1&_issn=03930440&md5=d0f3804a9511e5b125e5522ae328cfe7">g class="imgLazyJSB inlineImage" height="15" width="69" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0393044016300699-si2.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">(athvariant="script">Rathvariant="bold">edace width="0.16667em">ace>athvariant="script">Aathvariant="bold">tl)ath>an>an>an>. We prove that an id="mmlsi1" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0393044016300699&_mathId=si1.gif&_user=111111111&_pii=S0393044016300699&_rdoc=1&_issn=03930440&md5=a63bfe5a3435771ec352e32b6e6c4c05">g class="imgLazyJSB inlineImage" height="15" width="73" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0393044016300699-si1.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">(athvariant="script">Rathvariant="bold">edace width="0.16667em">ace>athvariant="script">Oathvariant="bold">rb)ath>an>an>an> is equivalent to the bicategory of reduced orbifolds described in terms of proper, effective, &eacute;tale Lie groupoids by D. Pronk and I. Moerdijk and to the well-known 2-category of reduced orbifolds constructed from a suitable class of differentiable Deligne&ndash;Mumford stacks.