MAT 589 Algebraic Geometry Jason Starr Stony Brook University Spring 2020 MAT 589 Some Notes on Category Theory Contents 1 Introduction 2 2 Algebraic Objects 2 3 Categories 5 4 Functors 11 5 Natural Transformations 14 6 Adjoint Pairs of Functors 17 7 Adjoint Pairs of Partially Ordered Sets 19 8 Adjoint Pair between Monoids and Semigroups 20 9 Adjoint Pairs of Free Objects 20 10 Adjoint Pairs of Limits and Colimits 33 11 Adjoint Pairs and Yoneda Functors 44 12 Preservation of Exactness by Adjoint Additive Functors 45 13 Derived Functors as Adjoint Pairs 45 14 Constructing Injectives via Adjoint Pairs 58 15 The Koszul Complex via Adjoint Pairs 61 16 Adjoint Pairs of Simplicial and Cosimplicial Objects 70 17 Topology Adjoint Pairs 73 1 MAT 589 Algebraic Geometry Jason Starr Stony Brook University Spring 2020 18 The Adjoint Pair of Discontinuous Sections (Godement Resolution) 88 1 Introduction These are notes cut-and-pasted from some previous courses I taught about the most basic algebraic objects (semigroups, monoids, groups, acts and actions, associative rings, commutative rings, and modules), elementary language of category theory, and adjoint pairs of functors. Most of the notes are exercises working through the details of adjoint pairs of functors that are useful in introductory algebraic geometry. 2 Algebraic Objects Definition 2.1. A semigroup is a pair G; m of a set G and a binary relation, (m G) G G; such that m is associative, i.e., the following∶ diagram× → commutes, m Id G G G × G G G IdG××m × ÐÐÐ→ ×m : × × G × G G× × m × Ö Ö The binary operation is equivalent to a set× function,ÐÐÐ→ L● G HomSets G; G ; g Lg; ′ ′ ′ ′ such that for every g; g G, the composition∶ → Lg ( Lg equals) ↦Lm(g;g ), where m g; g is defined to equal Lg g′ . When no confusion is likely, the element m g; g′ is often denoted g g′. ∈ ○ ( ) For semigroups G; m and G′; m′ a semigroup morphism from the first to the second is a set ( ) ( ) ⋅ map ( ) ( ) u G G′; such that the following diagram commutes, ∶ → u u G G × G′ G′ ′ m× ÐÐÐ→ ×m : × × G× G×′ × u × Ö Ö The set of semigroup morphisms is denoted HomÐÐÐ→Semigroups G; m ; G′; m′ . 2 (( ) ( )) MAT 589 Algebraic Geometry Jason Starr Stony Brook University Spring 2020 Definition 2.2. For a semigroup G; m , an element e of G of is a left identity element, resp. right identity element, if for every g G, g equals m e; g , resp. g equals m e; g . An iden- tity element is an element that( is both) a left identity element and a right identity element. A monoid is a triple G; m; e where G;∈ m is a semigroup( ) and e is an identity( element.) For monoids G; m; e and G′; m′; e′ a monoid morphism from the first monoid to the second is a semigroup morphism that( preserves) identity( ) elements. The set of monoid morphisms is denoted HomMonoids( G; m;) e ; (G′; m′; e′ ) . opp opp Example 2.3.(( For every) ( semigroup)) G; m , the opposite semigroup is G; m , where m g; g′ is defined to equal m g′; g for every g; g′ G G. A left identity element of a semigroup is equiv- alent to a right identity element of the( opposite) semigroup. In particular,( the opposite) semigroup( ) of a monoid is again( a monoid.) ( ) ∈ × Example 2.4. For every set I and for every collection Gα; mα α∈I of semigroups, for the Cartesian product set G α∈I Gα with its projections, ( ) ∶= ∏ prα G Gα; there exists a unique semigroup operation m ∶on →G such that every projection is a morphism of semigroups. Indeed, for every α, the composition prα m G G Gα equals mα prα prα . There exists an identity○ ∶ element× → e of G; m if and only if there exists an identity element eα of Gα; mα for every α, in which case e is the unique element such that prα e equals eα for○ ( every× α )I. ( ) ( ) ( ) Example 2.5. For every∈ set S, the set HomSets S; S of set maps from S to itself has a structure of monoid where the semigroup operation is set composition, f; g f g, and where the identity element of the monoid is the identity function on( S.) For every semigroup G; m , a left act of G; m on S is a semigroup morphism ( ) ↦ ○ ( ) ( ) ρ G; m HomSets S; S ; : For every ordered pair S; ρ ; T; π∶ ( of) sets→ ( with left(G-acts,) ○) a left G-equivariant map from S; ρ to T; π is a set function u S T such that u ρ g s equals π g u s for every g G and for every s S. (( ) ( )) ( ) ( ) ∶ → ( ( ) ) ( ) ( ) ∈ For each set S, a right act of G on S is a semigroup morphism ρ from G; m to the opposite ∈ opp semigroup of HomSets S; S . Note, this is equivalent to a left act of the opposite semigroup G on S. For every ordered pair S; ρ ; T; π of sets with a right G-act, a(right) G-equivariant map is a set function(u S) T such that u sρ g equals u s π g for every g G and for every s S. Note, this is equivalent to(( a left) (Gopp))-equivariant map. ∶ → ( ( )) ( ) ( ) ∈ For an ordered pair G; m ; H; n of semigroups, for each set S, a G H-act on S is an ordered ∈ pair ρ, π of a left G-act on S, ρ, and a right H-act on S, π, such that ρ g s π h equals (( ) ( )) − ( ) 3 ( ( ) ) ( ) MAT 589 Algebraic Geometry Jason Starr Stony Brook University Spring 2020 ρ g sπ h for every g G, for every h H, and for every s S. This is equivalent to a left act on S by the product semigroup of G and Hopp.A G H-equivariant map is a map that is left equivariant( )( ( )) for the associated∈ left act by G∈ Hopp. ∈ − For every monoid G; m; e , a left action of G; m; e on S is a monoid morphism from G; m; e × to HomSets S; S . There is a category G Sets whose objects are pairs S; ρ of a set S and a left action of G; m; e ( on S,) whose morphisms are( left G) -equivariant maps, and where composition( ) is usual set( function) composition. A right− action is a monoid morphism( ) from G; m; e to the opposite monoid( of) HomSets S; S . There is a category Sets G whose objects are pairs S; ρ of a set S and a right action of G; m; e on S, whose morphisms are left G-equivariant maps,( and) where composition is usual set function( ) composition. Finally, for every− ordered pair G; m; e ;(H; n;) f of monoids, a G H-action( on S is) a G H act ρ, π such that each of ρ and π is an action. There is a category G H Sets whose objects are sets together with a G H-action,(( whose) morphisms( )) are G H-equivariant− maps, and where− composition( ) is usual set function composition. − − − Definition 2.6. A semigroup G; is called left cancellative, resp. right cancellative, if for − every f; g; h in G, if f g equals f h, resp. if g f equals h f, then g equals h. A semigroup is cancellative if it is both left cancellative( ⋅) and right cancellative. A semigroup is commutative if for every f; g G, f⋅ g equals g⋅ f, i.e., the identity⋅ function⋅ from G to itself is a semigroup morphism from G to the opposite semigroup. For an element f of a monoid, a left inverse, resp. right inverse, is∈ an element⋅ g such⋅ that g f equals the identity, resp. such that f g equals the identity. An inverse of f is an element that is both a left inverse and a right inverse. An element f is invertible if it has an inverse. ⋅ ⋅ Definition 2.7. A group is a monoid such that every element is invertible. If the monoid operation is commutative, the group is Abelian. A monoid morphism between groups is a group homomor- phism, and the set of monoid morphisms between two groups is denoted HomGroups G; m; e ; G′; m′; e′ . ′ ′ ′ If both groups happen to be Abelian, this is also denoted HomZ−mod G; m; e ; G ; m ; e . In this case, this set is itself naturally an Abelian group for the operation that associates(( to a pair) u;( v )) of group homomorphisms the group homomorphism u v defined by((u v g) ( m′ u g )); v g . ( ) Definition 2.8. An associative ring is an ordered pair A; ; 0 ;L of an Abelian group A; ; 0 ⋅ ●( ⋅ )( ) = ( ( ) ( )) and a homomorphism of Abelian groups, (( + ) ) ( + ) L● A HomZ−mod A; A ; a La A A such that for every a; a′ A, the composition L L ′ equals L ′ , where a a′ denotes L a′ . The ∶ → (a )a ↦ ( a⋅∶a → ) a set map L● is equivalent to a biadditive binary operation, ∈ ○ ⋅ ( ) A A A; a; a′ a a′; that is also associative, i.e., for every⋅ ∶ a;× a′;→ a′′ in A(, the) = element⋅ a a′ a′′ equals a a′ a′′ . In particular, A; is a semigroup. For associative rings A; ; 0; and A′; ′; 0′; ′ , a ring homo- morphism from the first to the second is a set function that is( simultaneously⋅ ) ⋅ a morphism⋅ ( ⋅ ) of Abelian groups( ⋅) from A; ; 0 to A′; ′; 0′ and a morphism( + of⋅) semigroups( + from⋅ ) A; to A′; ′ .