Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Euler Characteristics of Categories and Homotopy Colimits Thomas M. Fiore joint work with Wolfgang L¨ uck and Roman Sauer http://www-personal.umd.umich.edu/~tmfiore/
Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Outline 1 Introduction 2 Finiteness Obstructions and Euler Characteristics for Categories 3 Classifying I -Spaces 4 Homotopy Colimit Formula and the Inclusion-Exclusion Principle 5 Comparison with Leinster’s Notions 6 Applications and Summary
Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary I. Introduction.
Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Introduction The most basic invariant of a finite CW -complex is the Euler characteristic. χ : finite CW -complexes � R Remarkable connections to geometry: χ (compact connected orientable surface) = 2 − 2 · genus, Theorem of Gauss-Bonnet � χ ( M ) = 1 curvature dA 2 π M for M any compact 2-dimensional Riemannian manifold.
� � Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Introduction Problem: meaningfully define χ purely in terms of the combinatorial models finite skeletal categories without loops ? |−| � R finite CW -complexes χ
� � � � � � Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Introduction More generally: categories ? finite skeletal categories without loops � � χ ( |−| ) � � � |−| � � � � � � � � � � � � R finite CW -complexes χ
Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Trivial Example Presents Challenges Γ = � Z 2 , that is, Γ has one object ∗ and mor Γ ( ∗ , ∗ ) = Z 2 . | � Z 2 | = geometric realization of nerve of � Z 2 0-cells of | � ob( � Z 2 | Z 2 ) = {∗} = 1-cells of | � Z 2 | = non-identity maps = {∗ → ∗} 2-cells of | � Z 2 | = paths of 2 non-id maps = {∗ → ∗ → ∗} etc. = etc. � χ ( | � ( − 1) n card( n -cells of | � Z 2 | ) = Z 2 | ) n ≥ 0 � 1 − ( − 1) = 1 1 ( − 1) n Leinster − Berger = ===== 2 . n ≥ 0
Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Desiderata for Invariants Desiderata for χ, χ (2) : categories → R 1. Geometric relevance 2. Compatibility with equivalence of categories coverings of groupoids: if p : E → B , then χ (2) ( E ) = n · χ (2) ( B ) isofibrations: if f : E → B , then χ (2) ( E ) = χ (2) ( f − 1 ( b 0 )) · χ (2) ( B ) finite products finite coproducts “pushouts” (Inclusion-Exclusion Principle) χ ( A ∪ B ) = χ ( A ) + χ ( B ) − χ ( A ∩ B ) homotopy colimits. Our work achieves this.
Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Desiderata for Invariants Desiderata for χ, χ (2) : categories → R 1. Geometric relevance 2. Compatibility with equivalence of categories coverings of groupoids: if p : E → B , then χ (2) ( E ) = n · χ (2) ( B ) isofibrations: if f : E → B , then χ (2) ( E ) = χ (2) ( f − 1 ( b 0 )) · χ (2) ( B ) finite products finite coproducts “pushouts” (Inclusion-Exclusion Principle) χ ( A ∪ B ) = χ ( A ) + χ ( B ) − χ ( A ∩ B ) homotopy colimits. Our work achieves this.
� � � � � � Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Pushouts in Cat and χ { 0 , 1 } { 0 → 1 } χ ( | � N | ) = χ ( S 1 ) = 0 = 1 + 1 − 2 ✧ pushout � � {∗} N {∗ ′ } { 0 , 1 } χ ( {∗} ) =1 � = 1 + 1 − 2 ✪ pushout � {∗} {∗} Colimits are not homotopy invariant, cannot expect compatibility of χ with pushouts.
� � � � � � � � � � Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Homotopy Pushouts in Cat and χ � • • { 0 , 1 } { 0 → 1 } � � � � � � � � Homotopy p.o. is • • � � � � {∗} • {∗ ′ } • { 0 , 1 } � � � � � � � � � Homotopy p.o. is • • � � � � � � � {∗} � � • In both cases, χ = 0 = 1 + 1 − 2. ✧
Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Main Theorem of this Talk Theorem (Fiore-L¨ uck-Sauer) Let C : I → Cat be a pseudo functor such that I is directly finite: ab = id ⇒ ba = id ; I admits a finite I -CW -model, Λ n := the finite set of n-cells λ = mor(? , i λ ) × D n ; each C ( i ) is of type (FP R ). � � ( − 1) n · Then: χ (hocolim I C ; R ) = χ ( C ( i λ ); R ) . n ≥ 0 λ ∈ Λ n Similar formulas hold for the L 2 -Euler characteristic, the functorial characteristics, and the finiteness obstruction.
Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary II. Finiteness Obstructions and Euler Characteristics for Categories.
Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Modules and the Projective Class Group R = an associative commutative ring with 1 Γ= a small category An R Γ -module is a functor M : Γ op → R - MOD . K 0 ( R Γ) := projective class group = Z { iso classes of finitely generated projective R Γ-modules } modulo the relation [ P 0 ] − [ P 1 ] + [ P 2 ] = 0 for every exact sequence 0 → P 0 → P 1 → P 2 → 0 of finitely generated projective R Γ-modules.
Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Type (FP) and Finiteness Obstruction Γ is of type (FP R ) if there is a finite projective R Γ-resolution P ∗ → R . In this case, the finiteness obstruction is � ( − 1) n · [ P n ] ∈ K 0 ( R Γ) . o (Γ; R ) := n ≥ 0 Remark Suppose G is a finitely presented group of type (FP Z ). Then o ( � G ; Z ) = o Wall ( BG ; Z ) .
Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Examples of Type (FP) Example Suppose Γ is a finite category in which every endo is an iso, that is, Γ is an EI-category. If | aut Γ ( x ) | ∈ R × for all x ∈ ob(Γ) , then Γ is of type (FP R ). Thus finite groupoids, finite posets, finite transport groupoids, and orbit categories of finite groups are all of type (FP Q ).
� Introduction Finiteness Obstructions and Euler Characteristics for Categories Classifying I -Spaces Homotopy Colimit Formula and the Inclusion-Exclusion Principle Comparison with Leinster’s Notions Applications and Summary Splitting Theorem of L¨ uck Theorem If Γ is an EI-category, then � S K 0 ( R Γ) K 0 ( R aut Γ ( x )) x ∈ iso(Γ) is an isomorphism, where S x ( M ) is the quotient of the R-module M ( x ) by the R-submodule generated by all images of M ( u ) for all non-invertible morphisms u : x → y in Γ .
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