A Personal Glance at George’s Category Theory Walter Tholen York University, Toronto Coimbra, 2012 Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 1 / 29
George Janelidze 19 May 1952 1974 Diploma Tbilisi State University 1978 Ph.D. Tbilisi State University 1992 D.Sc. St.-Petersburg State University Georgian Academy of Sciences (since 1975) McGill, York, Milan, Chicago, Bielefeld, Sydney Hungarian Academy of Sciences, Trieste, Genova, Wales (at Bangor) Tours, Louvain-la-Neuve, Littoral (at Calais), Coimbra Insubria (at Como), Aveiro, IST Lisbon, . . . University of Cape Town (since 2004) Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 2 / 29
Major areas of work Categorical Galois Theory Descent Theory Categories for Algebra Categories for Topology Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 3 / 29
Categorical Galois Theory Galois Theory in categories with inclusions (Proc. Junior Sci. 1974) The fundamental theorem of Galois Theory (USSR Sbornik 1989) Pure Galois Theory in Categories (J. Algebra 1990) ◮ Galois Theories (Cambridge 2001, with F. Borceux) Categorical Galois Theory: Revision and some recent developments (Potsdam 2001) Descent and Galois Theory (Haute Bodeux 2007) Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 4 / 29
Central extensions – Classically A α − → B surjective ( A , α ) ∈ ( Grp ↓ B ) central extension ⇐ ⇒ ker α ⊆ centre ( A ) ( A , α ) trivial central extension ⇒ ( A , α ) ∼ ⇐ = ( K × B , K × B − → B ) with K Abelian Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 5 / 29
� � � Central extensions – Categorically ( A , α ) ∈ ( Grp ↓ B ) central extension ⇐ ⇒ ∃ p : E − → B surjective such that p ∗ ( A , α ) trivial: E × B A A α π 1 � B E p ⇐ ⇒ : ( A , α ) split over ( E , p ) Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 6 / 29
Separable extensions – Classically α ← − B in CR , B field A Example f ∈ B [ x ] , deg f � 1 , B f = B [ x ] / ( f ) ← − B Facts ⇒ B f ∼ f = g · h , ( g , h ) = 1 = = B g × B h B ( x − b ) n ∼ = B x n n � f separable ⇐ ⇒ f = a · ( x − b i ) , b i � = b j for i � = j i = 1 ⇒ B f ∼ ⇐ = B × . . . × B ⇐ ⇒ B f is a trivial B -algebra Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 7 / 29
� � � � Separable extensions – Classically (continued) If f ∈ B [ x ] does not split: ∃ E ⊇ B such that f ∈ E [ x ] splits, E f ∼ = E ⊗ B B f f separable ⇐ ⇒ E ⊗ B B f trivial E -algebra E ⊗ B B f B f trivial E � � B Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 8 / 29
� � � � � Separable extensions – Categorically A separable B -algebra ⇐ ⇒ dim B A < ∞ , ∀ a ∈ A : a separable over B ⇐ ⇒ ∃ field extension E � � : E ⊗ B A trivial E -algebra B ⇐ ⇒ : A is split over B E ⊗ B A A α E B Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 9 / 29
�� � � � Covering spaces – Classically A α − → B local homeomorphism ⇐ ⇒ ( A , α ) ´ etale space over B Very trivial example ∼ = A = � A i ⊆ A open, A i − → B A i (disjoint) i ∈ I α B Trivial example B = � B λ (disjoint) λ ∈ Λ B λ ⊆ B open, α − 1 ( B λ ) − → B λ very trivial Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 10 / 29
� � Covering spaces – Categorically ( A , α ) covering space over B ⇒ ∀ b ∈ B ∃ open V ∋ b in B : α − 1 ( V ) − ⇐ → V very trivial p ⇐ ⇒ ∃ E − → B surjective, ´ etale: � A p ⋆ ( A , α ) trivial E × B A α � B E p ⇐ ⇒ : ( A , α ) split over ( E , p ) Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 11 / 29
� � � � � ✤ The machinery of adjunctions I � X , C with pullbacks, B ∈ C C ⊥ H I B � ( X ↓ IB ) � HX ( C ↓ B ) B × HIB HX ⊥ H B π 1 H ϕ � HIB � ( IA , I α ) B ( A , α ) ✤ η B ( B × HIB HX , π 1 ) ( X , ϕ ) Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 12 / 29
� � � Split objects η A � HIA ( A , α ) trivial : ⇐ ⇒ pullback A α HI α � HIB B η B ⇒ p ∗ ( A , α ) trivial ( A , α ) split over ( E , p ) : ⇐ Example 1 � AbGrp Grp ⊥ α , p surjective, E free Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 13 / 29
� � � ✤ Split objects, continued Example 2 � FinSet ( CR op ↓ k ) fin ⊥ � { minimal non-zero idempotents } A ✤ k × . . . × k X � �� � X times p � � fields E B Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 14 / 29
� � ✤ Split objects, continued Example 3 � Set LCTop ⊥ � π 0 B B ✤ (discrete) X X p : E − → B surjective, ´ etale Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 15 / 29
� George’s Galois Theorem I � X C F ⊆ mor C , Φ ⊆ X : “ fibrations ” ⊥ H Hypothesis – pullbacks of fibrations exist and are fibrations – isomorphisms are fibrations, closed under composition – I and H preserve fibrations – (“ Admissibility ”) φ : X − → IB fibration ⇒ ( I ( B × HIB HX ) − → IHX − → X ) isomorphism Theorem → F ( E ) monadic ⇒ Spl ( E , p ) ≃ X Gal ( E , p ) � Φ p ∗ : F ( B ) − Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 16 / 29
� � � � � � � � � � George’s Galois Theorem (continued) Spl ( E , p ) TrivCov ( E ) Φ( IE ) ≃ (admissible) H E pullback � Φ( E ) Φ( B ) Φ( E ) Id Gal ( E , p ) = I ( Eq ( p )) = ( I ( E × B E × B E ) ��� I ( E × B E ) � I ( E ) ) Ic X Gal ( E , p ) ∋ ( A 0 , π, ξ ) ξ � A 0 I ( E × B E ) × ( Id ,π ) A 0 π � IE I ( E × B E ) Ic First proof generalizing Magid’s Theorem: 1984. In full generality: 1991 Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 17 / 29
� � � � � � � � � � Descent Theory � C p : E − → B effective (for) descent E × B C ⇒ p ! ⊣ p ∗ : F ( B ) − γ ⇐ → F ( E ) monadic � E E × B E ⇐ ⇒ rebuild F ( B ) from F ( E ) as p { ( C , γ ; ξ ) : ξ : E × B C − → C , 2 equations } � B E p Equivalent presentation of ξ : ξ C p · γ π 2 ξ � E × B C E × B C B π 1 p E γ · π 2 Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 18 / 29
� � � Descent Theory (continued) F ( B ) = ( Top ↓ B ) ξ x , y � γ − 1 y γ − 1 x ( x , y ) ∈ E × B E c ❴ j y , x j x , y ξ ( x , c ) � E × B C E × B C ξ x , x = id, ξ x , z = ξ y , z · ξ x , y ( p ( x ) = p ( y ) = p ( z )) , Glueing Condition Example p E = � � B = � U i ( U i ⊆ B open) U i i ∈ I i ∈ I � U j ) � U j ) satisfying the Cocycle Condition ξ i , j : γ − 1 � γ − 1 ( U i ( U i i j Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 19 / 29
� � � � � � � � � � � � Descent Theory (continued) Categorical Algebra Internal Category Topology (commutators) Theory (crossed modules) Sheaf & Galois Theory Descent Theory Cohomology Theory Monad Theory Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 20 / 29
Descent Theory (continued) Two of George’s “simple” observations: descent � = effective descent, even in algebra: { A ∈ AbGrp | n 2 x = 0 ⇒ nx = 0 } , p : Z − → Z / n Z C � � � D closed under pullbacks, p : E − → B in C , effective descent in D . Then: p effective descent in C ⇐ ⇒ ∀ ( A , α ) ∈ ( D ↓ B ) : p ∗ ( A , α ) ∈ ( C ↓ E ) ⇒ ( A , α ) ∈ ( C ↓ B ) � Reiterman-T characterization of effective descent morphisms in Top � Clementino-Hofmann characterization of effective descent mor- phisms in Top Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 21 / 29
� � � � � � � Descent Theory (continued) PreOrd ∼ FinPreOrd ∼ = Alexandroff , = FinTop � y x E universal quotient: p (=descent) � v u B � = � y � z x effective descent: � v � w u � = � x n − 1 � . . . � x 1 � x 0 x n triquotient: � u n − 1 � . . . � u 1 � u 0 u n Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 22 / 29
� � � � � � � � Semi-Abelian Categories Mac Lane, Duality for groups, Bull. AMS 1950 “Abelian bicategory” � “exact category” (Buchsbaum 1955) = abelian category � abelian category AbGrp Grp � ? Old-style generalizations in the realm of pointed/additive categories Walter Tholen (York University, Toronto) George’s Category Theory Coimbra, 2012 23 / 29
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