Generalised higher Hopf formulae for homology Mathieu Duckerts-Antoine Université catholique de Louvain Workshop on Category Theory Coimbra, July 2012
Outline A description of the fundamental group in the semi-abelian context 1 A wider context 2 Generalized Higher Hopf formulae 3
� � General idea Given a “good” adjunction I A , B K H one can associate with any object B of A an invariant : 1 With a NORMAL EXTENSION p : E � B of B , one associates an object of B : Gal ( E , p , 0) . 2 If p has a kind of UNIVERSAL PROPERTY , Gal ( E , p , 0) is an invariant of B : π 1 ( B ) the abstract fundamental group of B .
� � General idea Given a “good” adjunction I A , B K H one can associate with any object B of A an invariant : 1 With a NORMAL EXTENSION p : E � B of B , one associates an object of B : Gal ( E , p , 0) . 2 If p has a kind of UNIVERSAL PROPERTY , Gal ( E , p , 0) is an invariant of B : π 1 ( B ) the abstract fundamental group of B .
� � General idea Given a “good” adjunction I A , B K H one can associate with any object B of A an invariant : 1 With a NORMAL EXTENSION p : E � B of B , one associates an object of B : Gal ( E , p , 0) . 2 If p has a kind of UNIVERSAL PROPERTY , Gal ( E , p , 0) is an invariant of B : π 1 ( B ) the abstract fundamental group of B .
� � General idea For the adjunction ab Grp Ab K � π 1 ( B ) � K X [ P , P ] � H 2 ( B , Z ) [ K , P ] (for K � P � B is a projective presentation of B ) .
� � Galois structure [G. Janelidze] Definition A Galois structure is given by : 1 B a full replete reflective subcategory of A I A ; B K � 2 E a class of morphisms in A which contains the isomorphisms of A and has some stability properties : I ( E ) � E ; 1 2 A has pullback along morphisms in E ; 3 E is closed under composition and pullback stable.
� � Galois structure [G. Janelidze] Definition A Galois structure is given by : 1 B a full replete reflective subcategory of A I A ; B K � 2 E a class of morphisms in A which contains the isomorphisms of A and has some stability properties : I ( E ) � E ; 1 2 A has pullback along morphisms in E ; 3 E is closed under composition and pullback stable.
� � � � Extensions For a given Galois structure. Definition f : E � B in E is a ( B -)trivial extension if η E I ( E ) E I ( f ) f � I ( B ) B η B is a pullback.
� � Extensions Definition An extension f : A � B is a ( B -)normal extension if in π 2 � A � B A A π 1 f � B A f π 1 and π 2 are trivial.
� � � � � Extensions For ab Grp Ab K � A regular epimorphism f : A � B is Ab -trivial iff [ A , A ] � � A f � � � B . [ B , B ] � � A regular epimorphism f : A � B is Ab -normal iff it is central, i.e. if Ker f � Z ( A ) .
� � � � � Extensions For ab Grp Ab K � A regular epimorphism f : A � B is Ab -trivial iff [ A , A ] � � A f � � � B . [ B , B ] � � A regular epimorphism f : A � B is Ab -normal iff it is central, i.e. if Ker f � Z ( A ) .
� � � � � Extensions For ab Grp Ab K � A regular epimorphism f : A � B is Ab -trivial iff [ A , A ] � � A f � � � B . [ B , B ] � � A regular epimorphism f : A � B is Ab -normal iff it is central, i.e. if Ker f � Z ( A ) .
� � � Extensions Definition A normal extension p : E � B is weakly universal if it factors through every other normal extension with the same codomain : p E B D u � p 1 normal E 1
� � � � � � The abstract fundamental group [G. Janelidze, 1984] For a Galois structure with A semi-abelian I B A K � and p : E � B a normal extension. Definition The Galois groupoid of p is : I ( σ ) I ( π 1 ) I ( τ ) � I ( E � B E ) I (( E � B E ) � E ( E � B E )) I ( E ) . I ( δ ) I ( π 2 )
� � � The abstract fundamental group Definition The Galois group of p is defined via the following pullback : Gal ( E , p , 0) � I ( E � B E ) x I ( π 1 ) , I ( π 2 ) y � I ( E ) � I ( E ) . 0 �
The abstract fundamental group Definition The abstract fundamental group of an object B of B of A is the Galois group of any weakly universal normal extension of B .
� � � � � A composite adjunction One works with an adjunction F G ( A ) F B A K K � � where 1 A is semi-abelian ; 2 B is a Birkhoff subcategory of A ; 3 F is a regular epi-reflective subcategory of B ; 4 F is protoadditive [T. Everaert and M. Gran, 2010] : F preserves split short exact sequences s � K � � k � A � � B � 0; 0 f and with E = RegEpi ( A ) .
� � � � � Induced adjuncion Theorem One has an induced adjunction : F 1 � G 1 NExt F ( A ) Ext ( A ) . K � The reflection is given by A � A � [ f ] 1 , F � � � � � 0 0 [ f ] 1 , F � F 1 G 1 ( f ) α � � Ker ( f ) � � � A � � B � 0 . 0 ker f f
� � � � � � � Fröhlich construction Construction The Fröhlich construction is : ˆ π 1 � � ker (ˆ π 1 ) � Ker ( η A � B A ) � � [ f ] 1 , F � � � � Ker ( η A ) � � � π 2 ˆ α π 1 � � f � A � B A � � B Ker ( f ) � � � � A ker π 1 π 2 where η is the unit.
� � � � Fröhlich construction For ab Grp Ab K � and f : A � B a regular epimorphism, one has [ f ] 1 , Ab = [ Ker ( f ) , A ] Ab = x kak � 1 a � 1 | k P Ker f , a P A y = [ Ker f , A ] . For G CRng Rng K � and f : A � B a regular epimorphism, one has [ f ] 1 , CRng = [ ker ( f ) , A ] CRng = x ak � ka | k P Ker f , a P A y .
� � � � Fröhlich construction For ab Grp Ab K � and f : A � B a regular epimorphism, one has [ f ] 1 , Ab = [ Ker ( f ) , A ] Ab = x kak � 1 a � 1 | k P Ker f , a P A y = [ Ker f , A ] . For G CRng Rng K � and f : A � B a regular epimorphism, one has [ f ] 1 , CRng = [ ker ( f ) , A ] CRng = x ak � ka | k P Ker f , a P A y .
� � � Construction of weakly universal normal extensions Lemma If A has enough projective objects w.r.t. E , then for all B in A one can construct a weakly universal normal extension of B . Proof : If f : P � B is a projective presentation of B : f � � P B � � � � � � � � � � � � � � � � � F 1 G 1 ( f ) � � � � � � � ˜ P � p F -normal � � α � � � β � � E
� � � Construction of weakly universal normal extensions Lemma If A has enough projective objects w.r.t. E , then for all B in A one can construct a weakly universal normal extension of B . Proof : If f : P � B is a projective presentation of B : f � � P B � � � � � � � � � � � � � � � � � F 1 G 1 ( f ) � � � � � � � ˜ P � p F -normal � � α � � � β � � E
The generalized Hopf formula Theorem For f : P � B a projective presentation of B F π 1 ( B ) � ([ P , P ] B ) P X Ker ( f ) . F ([ Ker f , P ] B ) Ker f F is a homological closure operator [D. Bourn and M. Gran, 2006]. �
� � � � Groups with coefficients in torsion free abelian groups For F ab Grp , Ab t . f . Ab K K � � and a projective presentation K � P � B of a group B , π 1 ( B ) � t p P K | D n P N 0 : p n P [ P , P ] u t p P K | D n P N 0 : p n P [ K , P ] u .
� � � � Rings with coefficients in reduced commutative rings For F G RedCRng CRng Rng , K K � � and a projective presentation K � P � B of a ring B , a [ P , P ] CRng ( P ) X K π 1 ( B ) � . a [ K , P ] CRng ( K )
� � � A wider context Grp ( Top ) is regular but not exact. But in some way, it is “almost exact” : X � Q X R � � � � � � � �������� � � � ( r 1 , r 2 ) � � ( π 1 ,π 2 ) X � X where q = coeq ( r 1 , r 2 ): X � � Q .
� � � A wider context Grp ( Top ) is regular but not exact. But in some way, it is “almost exact” : X � Q X R � � � � � � � �������� � � � ( r 1 , r 2 ) � � ( π 1 ,π 2 ) X � X where q = coeq ( r 1 , r 2 ): X � � Q .
� � � � A wider context For A with ( E , M ) a proper stable factorization system, finite limits and coequalizers of effective equivalence relations : Definition A is E -exact if for every internal equivalence relation R there exist an effective equivalence relation S such that R � E S , i.e. i RS R � S � � � � � � � � � � � � � ( s 1 , s 2 ) ( r 1 , r 2 ) � � � � X � X where i RS is in E . ( E -exact efficiently regular [D. Bourn, 2007] � almost Barr exact [G. Janelidze and M. Sobral, 2011]) �
A wider context For our purpose, a good context to work in is the one of E -exact homological categories . Examples 1 All semi-abelian categories ( E = RegEpi ) ; 2 All topological semi-abelian varieties ( E = Epi ) ; 3 All integral almost abelian categories ( E = Epi ) � Raïkov semi-abelian categories.
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