,9 ma!matics1 an ac;n gr|poid or a trans=ma;n gr|poid is a gr|poid t expresses a gr|p ac;n4 ,"nly1 giv5 a "<"r"> gr|p ac;n
=mula.-#a
we get ! gr|poid =mula.-#b "<"7 a category ~: morphisms >e all 9v}tible"> ":
,a gr|poid is (t5 depict$ us+ two >r{s4 ,"h ! abv c 2 writt5 z3
=mula.-#ab
": =mula.-#ac d5ote ! s|rce & ! t>get ( a morphism 9 =mula.-#b2 ?us1 =mula.-#ae is ! projec;n & =mula.-#af is ! giv5 gr|p ac;n "<"h ! set ( morphisms 9 =mula.-#b is id5tifi$ ) =mula.-#ah">4
,9 an #=-category4
,let =mula.-#ai 2 an #=-category & =mula.-#g a gr|poid object 9 x4 ,!n a gr|p ac;n or an ac;n gr|poid on an object 8;,x0 9 8;,c0 is ! simplicial diagram 
=mula.-#ba
t satisfies ! axioms simil> to an ac;n gr|poid 9 ! usual case4
,fur!r r1d+4