Lecture 1: Categorical reminders

Teacher: Benjamin Hennion

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Benjemain.hennion at u-psud.fr (office 3E22)

References:

  1. P. Shapira: Categories and homological algebra (www.math.jussieu.fr/~shapira/)

  2. C. Weibel: an introduction to homological algebra

Introduction

$A$ commutative ring, $M$ a $A$-module.

Let

\[0 ⟶ N_1 ⟶ N_2 ⟶ N_3 ⟶ 0\]

be an exact sequence

Now consider:

\[M ⊗_A N_1 \overset{φ}{⟶} M ⊗_A N_2 ⟶ M ⊗_A N_3 ⟶ 0\]

Then:

  • if $M$ is flat (plat in fr),
  • OR: if the sequence is split (scindée in fr)

then $φ$ is injective.

But what if you have neither of these properties? How to compute the kernel of $φ$?

Typically:

\[⋯ ⟶ Tor_1^A(M, N_3) \overset{φ_1}{⟶} M ⊗_A N_1 ⟶ M ⊗_A N_2 ⟶ M ⊗_A N_3 ⟶ 0\]

Then what about $φ_3$? ⟶ other torsion groups

Goal of homological algebra: compute these $Tor$ torsion groups.

Other example: that don’t preserve exactness:

\[0 ⟶ \Hom(N_3, M) ⟶ \Hom(N_2, M) \overset{ψ}{⟶} \Hom(N_3, M)\]

Question: is $ψ$ surjective?

Typically:

\[0 ⟶ \Hom(N_3, M) ⟶ \Hom(N_2, M) \overset{ψ}{⟶} \Hom(N_3, M) ⟶ Ext_A^1(N_3, M) ⟶ Ext_A^1(N_2, M) ⟶ ⋯\]

Outline

  • I. Categories, limits
  • II. Abelian categories
  • III. Complexes
  • IV. Resolutions
  • V. Derived functors
  • VI. Computational tools

I. Categories and limits

1. Categories

Category $𝒞$:

is the datum of:

  • a class of objects $Ob(𝒞)$
  • $∀x,y ∈ Obj(𝒞)$, a class $\Hom_𝒞(x,y)$ of arrows/morphisms/maps from $x$ to $y$ in $𝒞$

  • a composition law:

    \[∀x,y,z ∈ Obj(𝒞), \qquad \circ: \Hom_{𝒞}(x,y) × \Hom_{𝒞}(y,z) ⟶ \Hom_{𝒞}(x,z)\]

  • for all $x∈ 𝒞$, an identity map: \(id_x ∈ \Hom_𝒞(x,x)\)

    such that

  • the identities are identities for the composition law: \(id_x \circ g = g = g \circ id_y\)

  • the composition is associative: \((f \circ g) \circ h = f \circ (g \circ h)\)

Examples:

  • Rings, Groups, Fields, Sets, Top, Manifolds, Preorders, Monoid, Vector spaces on a field $𝕂$, $A$-modules, Algebraic varieties (in alg. geometry)

  • Fix $G$ a monoid: $BG$ is the category with one object $\ast$ and whose endomorphisms are $\Hom_{BG}(\ast, \ast) = G$

  • Fix a set $E$, the discrete category $\underline E$

  • given a category $𝒞$, the opposite category $𝒞^{op}$

NB: Beware, there might some set theoretic issues (e.g. the “set of sets”)

Subcategories

$𝒟 ⊆ 𝒞$ is a subcategory:

if

  • $Ob(𝒟) ⊆ Ob(𝒞)$
  • $∀x, y ∈ 𝒟, \quad \Hom_{𝒟}(x,y) ⊆ \Hom_{𝒞}(x,y)$ such that the $\Hom_𝒟$ are stable by composition and contain identities.
A functor $F: 𝒞 ⟶ 𝒟$:

is the data of

  • $F: Ob(𝒞) ⟶ Ob(𝒟)$
  • $∀x,y ∈ 𝒞, \quad \Hom_𝒞(x,y) \overset{F}{⟶} \Hom_𝒟(F(x), F(y))$

such that

  • $F(id_x) = id_{Fx}$
  • $F(f\circ g) = Ff \circ Fg$

Usually: such functors $𝒞 \overset{F}{⟶} 𝒟$ are called covariant. There are contravariant functors: $G: 𝒞 ⟶ 𝒟$ is contravariant:

  • $G: Ob(𝒞) ⟶ Ob(𝒟)$
  • $∀x,y ∈ 𝒞, \quad G: \Hom_𝒞(x,y) ⟶ \Hom_𝒟(F(y), F(x))$

Actually, a contravariant functor is the same as a covariant functor: $𝒞^{op} ⟶ D$

Examples of functors:

  • forgetful functors: e.g. $Grp ⟶ Sets$ (forgetting it’s a group)
  • Free group functor: $Sets ⟶ Grp$ (free group generated by a set)
  • If $A ∈ CAlg_𝕂$ (commutative algebra over $𝕂$):
    • $U: A\text{-}Mod ⟶ 𝕂\text{-}Mod$
    • $F: 𝕂\text{-}Mod ⟶ A\text{-}Mod, M ⟼ A ⊗_𝕂 M$
  • $\Hom_𝒞(\bullet, x): 𝒞^{op} ⟶ Set$

Category of categories

$Cat$:
  • $Ob(Cat) = $ class of small categories
  • $\Hom_{Cat}(𝒞, 𝒟) = $ functors from $𝒞$ to $𝒟$

Let $f ∈ \Hom_𝒞(x,y)$.

$f$ is an isomorphism:

if there exists $f^{-1} ∈ \Hom_𝒞(y,x)$ (called inverse of $f$) such that: $ff^{-1} = id_y$ and $f^{-1}f = id_x$

Example: take

  • $[0, 1]$: discrete topology
  • a set-theoretic $f: [0, 1] ⟶ [0,1]$ may be an isomorphism in $Set$, but not in $Top$

NB: if $f ∈ 𝒞$ is an isomorphism and $F: 𝒞 ⟶ 𝒟$ is a functor, then $Ff$ is an isomorphism.

Let $F: 𝒞 ⟶ 𝒟$ be a functor.

We say that

  • $F$ is faithful (fidèle in fr) if $∀x,y ∈ 𝒞$, $\Hom_𝒞(x,y) \overset{F}{⟶} \Hom_𝒟(F(x), F(y))$ is injective

  • $F$ is full (plein in fr) if $∀x,y ∈ 𝒞$, $\Hom_𝒞(x,y) \overset{F}{⟶} \Hom_𝒟(F(x), F(y))$ is surjective

  • $F$ is fully faithful (pleinement fidèle in fr) if it is full and faithful, i.e. if $∀x,y ∈ 𝒞$, $\Hom_𝒞(x,y) \overset{F}{⟶} \Hom_𝒟(F(x), F(y))$ is bijective

  • $F$ is essentially surjective if $∀y ∈ 𝒟, \; ∃ x ∈ 𝒞$ and an isomorphism in $𝒟$ s.t. $y ≃ F(x)$

Examples:

  • $U: Grp ⟶ Set$ forgetful functor is faithful
  • $𝒞 ⟶ \ast$ is full
  • $Ab \hookrightarrow Grp$ is fully faithful
  • $FinSet ⟶ FinVect_𝕂, I ⟼ 𝕂^I$: essentially surjective
  • $BGL_n(𝕂) ⟶ Vect_𝕂^{\dim n, \text{ iso }}$
A functor $F: 𝒞 ⟶ 𝒟$ is an equivalence:

if it is fully faithful and essentially surjective

Let $F, G: 𝒞 ⟶ 𝒟$ be two functors.

A natural transformation $θ: F ⇒ G$:

is the data of

  • $∀x ∈ 𝒞,$ a map $θ_x: Fx ⟶ Gx ∈ \Hom_𝒟(Fx, Gx)$

st $∀f: x⟶y ∈ 𝒞$, the following diagram commutes

\[\begin{xy} \xymatrix{ Fx \ar[r]^{θ_x} \ar[d]_{Ff} & Gx \ar[d]^{Gf} \\ Fy \ar[r]_{θ_y} & Gy } \end{xy}\]
An isomorphism of functors $θ: F ⇒ G$:

is a natural transformation such that $∀x ∈ 𝒞, \; θ_x$ is an isomorphism

NB: the category of functors from $𝒞$ to $𝒟$: $Fun(𝒞,𝒟)$ has functors as objects and natural transformations as morphisms. A natural isomorphism is precisely a morphism in this category.

Characterisation of equivalences

$F$ is an equivalence of categories iff there exists $G: 𝒟 ⟶ 𝒞$ and isomorphisms of functors \(F \circ G ≃ id_𝒟 \qquad G \circ F ≃ id_𝒞\)

Warning: the inverse $G$ is only unique up to (unique) isomorphism. In particular: $F$ iso in $Cat$ ⟹ $F$ equivalence BUT the converse is not true.

Proof:

$⟸$: easy, beause $F \circ G ≃ id_𝒟 ⟹ F \text{ essentially surj.}$ and $G \circ F ≃ id_𝒞 ⟹ F \text{ fully faithful}$

$⟹$: Fix $z ∈ 𝒟$. Choose from essential surjectivity of $F$:

\[x ∈ 𝒞 \text{ and } β_z: F(x) ≃ z\]

And set $G(z) ≝ x$.

\[∀z_1, z_2 ∈ 𝒟, \\ \underbrace{\Hom_{𝒟}(z_1, z_2)}_{≃ \, \Hom_{𝒟}(F(x_1), F(x_2)) \, ≃ \, \Hom_{𝒟}(x_1, x_2)} ⟶ \Hom_{𝒞}(Gz_1, Gz_2)\]

and by def: $\Hom_{𝒟}(x_1, x_2) = \Hom_{𝒞}(Gz_1, Gz_2)$ ($x_i ≝ G z_i$)

Exercise:

  • Check it works…

  • What does it mean to have a left/right quasi-inverse?

Example: the category $ℕ$

  • objects: natural numbers

  • \[\Hom_ℕ(n,m) = \begin{cases} \lbrace\ast\rbrace &&\text{ if } n ≤ m \\ ∅ &&\text{ else} \end{cases}\]

and the category $Surj$ of finite sets and surjections.

Consider the functors:

\[\begin{cases} ℕ^{op} &⟶ Surj \\ n &⟼ ⟦1, n⟧ ≝ \lbrace 1, …, n\rbrace\\ n ≤ m &⟼ \begin{cases} ⟦1, m⟧ &⟶ ⟦1, n⟧ \\ i &⟼ i &&\text{if } i ≤ n\\ j &⟼ n &&\text{ else if } i > n \end{cases} \end{cases}\]

and

\[\begin{cases} Surj &⟶ ℕ^{op} \\ I &⟼ \vert I \vert \end{cases}\]

Can it help answering the previous question?

Yoneda Lemma

Let $𝒞, 𝒟$ be categories. Recall that $Fun(𝒞, 𝒟)$ is the category of functors from $𝒞$ to $𝒟$ and natural transformations between them.

Category of presheaves (préfaisceaux in fr) on $𝒞$:

category $Fun(𝒞^{op}, Set)$

The Yoneda functor:
\[h: \begin{cases} 𝒞 &⟶ Fun(𝒞^{op}, Set) \\ x &⟼ h_x ≝ \Hom(\bullet, x)\\ f: x→y &⟼ \Hom_𝒞(\bullet, x) ⇒ \Hom_𝒞(\bullet, y) \end{cases}\]

Yoneda Lemma: the functor $h$ is fully faithful

Proof: Let $x,y∈𝒞$.

Let’s show that

\[\Hom_𝒞(x,y) \overset{h}{⟶} \Hom_{Fun(𝒞^{op}, Set)}(h_x,h_y)\]

is bijective.

Take a natural transformation:

\[∀z, \quad θ_z: \Hom(z,x) ⟶ \Hom(z,y)\]

Take $z=x$ (a lot of the data in this natural transformation is redundant):

\[θ_x: \begin{cases} \Hom(x,x) &⟶ \Hom(x,y) \\ id_x &⟼ h^{-1}(θ) \end{cases}\]

Check that $h^{-1} \circ h = id$ (by def) and $h \circ h^{-1} = id$.

NB:

  • $k: 𝒞^{op} ⟶ Fun(𝒞, Set)$ is also fully faithful (deduce it from $(𝒞^{op})^{op} = 𝒞$)
  • $k$ and $h$ are “never” equivalences

Representable functors

A functor $F: 𝒞^{op} ⟶ Set$ (resp. $G: 𝒞 ⟶ Set$) is said to be representable if it is in the essential image of $h$ (resp. $k$): i.e. there exits $x ∈ 𝒞$ and an iso $h_x ≃ F$ (resp. $k_x ≃ G$).

NB: the object $x$ is then unique up to a unique isomorphism: assume that $x’$ then $h_x ≃ F ≃ h_{x’}$, so

\[h_x(x') ≃ h_{x'}(x')\\ \underbrace{\Hom(x', x)}_{\text{ in there: the unique isomorphism}} ≃ \Hom(x',x')\]

If $F ∈ Fun(𝒞^{op}, Set)$ is isomorphic to $h_x$, we say that $x$ represents $F$. (likewise: if $G ∈ Fun(𝒞, Set)$ is isomorphic to $k_x$, we say that $x$ represents $G$).

We also say that $x$ satisfies the universal property given by $F$.

Examples: take

\[\begin{cases} Grp &⟶ Set \\ G &⟼ G^n \end{cases}\]

is it representable? That is, is there a group st:

\[∀G, \quad G^n ≃ \Hom_{Grp}(H, G)\]

Yes, $H ≝$ free group with n generators.

For commutative algebras: algebra of polynomials with $n$ variables

Another one: let $k$ be a field, $V_1, V_2 ∈ Vect_k$

\[\begin{cases} Vect_k &⟶ Set \\ W &⟼ Bilinear(V_1, V_2; W) \end{cases}\]

is represented by $V_1 ⊗_k V_2$

These examples are special cases of left universal properties: representation of contravariant functors

\[\Hom_k(V_1 ⊗ V_2, W) ≃ Bilinear(V_1, V_2; W)\]

But there are also right universal properties (representation of covariant functors):

Ex: $A$ a ring, $f: M_1 ⟶ M_2$ a map of left modules.

\[\begin{cases} A \text{-}Mod^{op} &⟶ Set \\ N &⟼ \lbrace g: N ⟶ M \; \mid \; fg = 0\rbrace \end{cases}\]

looking for a module $P$ st

\[\Hom(N, P) ≃ \lbrace g: N ⟶ M \; \mid \; fg = 0\rbrace\]

$P$ is the kernel of $f$.

Ex: $G$ group $H ⊆ G$ subgroup.

\[\begin{cases} Grp &⟶ Set \\ G' &⟼ \lbrace f:G →G' \; \mid \; f_{| H} = 1\rbrace \end{cases}\]

We would like to say the representation of this functor is $G/H$, but $H$ is not necessarily normal. Actually, it is $G/H^{norm}$, where $H^{norm}$ is the smallest group such that

\[H ⊆ H^{norm} \overset{\text{normal}}{⊆} G\]

Non-representable functors are pretty far-fetched: even the stupidest functor:

\[\begin{cases} Grp &⟶ Set \\ G &⟼ \ast \end{cases}\]

is represented by $1$.

Limits, colimits

Example: Let $𝒞$ be a category, and $(x_i)_{i ∈ I} ∈ 𝒞^I$ a family of objects in $𝒞$.

Look at

\[\begin{cases} 𝒞^{op} &⟶ Set \\ y &⟼ \prod_{i ∈ I} \Hom(y, x_i) \end{cases}\]

If it exists, we denote by a $\prod_{i ∈ I} x_i$ a representative of this functor, and we call it the product.

This generalises the notion of cartesian product. Indeed, assume $\prod_{i ∈ I} x_i$ exists, then:

\[\Hom\left(\prod_{i ∈ I} x_i, \prod_{i ∈ I} x_i\right) ≃ \prod_{j ∈ I} \Hom\left(\prod_{i ∈ I} x_i, x_j\right)\]

then you have projections:

\[p_j: \prod_{i ∈ I} x_i ⟶ x_j\]

Similarly, if the functor:

\[𝒞 \ni y ⟼ \prod_{i ∈ I} \Hom(x_i, y) ∈ Set\]

is represented, we denoted by $\coprod_{i∈ I} x_i$ and call it the coproduct a representative of that functor.

We have natural “inclusions”:

\[x_j ⟶ \coprod_{i ∈ I} x_i\]

Examples:

  • in $Set$: usual ones
  • in $Top$: product topology, union
  • in commutative algebras $CAlg_k$: the usual set product with algebra structure, and the coproduct is the tensor product $⊗_k$ ($k[x] \coprod k[y] = k[x,y] ≠ k[x] ⊕ k[y]$)
  • in $Grp$: the product is the one on sets, the coproduct is the free product $G \ast H$
  • In $Cat$, the product is the usual one, and what about the coproduct?

A category where there’s no such thing as products:

  • $Surj$, where

    • objects: finite non-empty sets
    • $\Hom(I, J)$: surjections from $I$ to $J$

Indeed: fix $J$ and $J’$.

\[\begin{cases} Surj^{op} &⟶ Set \\ I &⟼ \Hom(I,J) × Hom(I, J') \end{cases}\]

is it represented by $J × J’$? No, because the map from $I$ to $J × J’$ is not surjective! There is no product in general of $J$ and $J’$ in $Surj$.

Notation: $𝒞$ a category.

If it exists, we denote by $\ast$ the product in $𝒞$ of the empty family. It’s called the final object.

If it exists, we denote by $∅$ the coproduct in $𝒞$ of the empty family. It’s called the initial object.

Examples:

  • in $Set$: $∅ ≝ ∅, \ast = $ the singleton
  • in $Vect_k$: $∅ = \ast = 0$
  • in $CAlg_k$, $∅ = k, \ast = 0$

NB: products in $𝒞^{op}$ are coproducts in $𝒞$ and vice-versa.

Limits: definition

Fix $I$ a category and $F: I ⟶ Set$ a functor.

A limit of $F$, denoted by $\lim F ≝ \; \lim\limits_{←} F = \lim_{i ∈ I} F(i)$:

is defined as \(\lim F ≝ \; \Big\lbrace (x_i) ∈ \prod_{i ∈ Ob(I)} F(i) \; \mid \; ∀f:i ⟶j ∈ I, F(f)(x_i) = x_j\Big\rbrace\)

This is called the projective limit of $F$.

Fix $F: I ⟶ 𝒞$ a functor. The (projective) limit of $F$, if it exists, is a representative of the functor

\[\begin{cases} 𝒞^{op} &⟶ Set \\ y &⟼ \lim_{i ∈ I} \Hom_𝒞(y, F(i)) \end{cases}\]

We denote it by $\lim F ≝ \; \lim\limits_{←} F = \lim_{i ∈ I} F(i)$.

Fix $G: I^{op} ⟶ 𝒞$ a functor. The colimit (inductive limit), if it exists, is a representative of the functor

\[\begin{cases} 𝒞 &⟶ Set \\ y &⟼ \lim_{i ∈ I} \Hom_𝒞(F(i), y) \end{cases}\]

We denote it by $\colim F, \, \colim_{i ∈ I} F(i), \, \lim\limits_{→} F$.

Sum up:

\[\Hom(\bullet, \lim(\bullet)) = \lim \Hom(\bullet, \bullet)\\ \Hom(\colim \bullet, \bullet) = \lim \Hom(\bullet, \bullet)\\\]

Examples:

  • Limits: kernel/equalizers.
  • $ℤ_p = \lim_n ℤ/p^n ℤ$
\[ℤ_p = \lim\limits_{←} ℤ/p^n ℤ ⋯ ⟶ ℤ/p^2 ℤ ⟶ ℤ/p ℤ ⟶ ℤ/p^0 ℤ = 0\]

Other example:

\[\underbrace{k⟦t⟧ = \lim\limits_{←} k[t]/t^n}_{\text{formal series}} ⋯ ⟶ k[t]/t^n ⟶ k[t]/t^{n-1} ⟶ ⋯ ⟶ k\]

Other example: let $Incl$ be the category of sets, maps are inclusions.

Intersection: $E ∩ E’$ is the pullback

\[\begin{xy} \xymatrix{ E \ar[r]^{ f } \ar@{<-}[d]_{ ι } & F \ar@{<-}[d]^{ ι' } \\ E ∩ E' \ar[r]_{ g } & E' } \end{xy}\]

In any $𝒞$, if $X ⟶ Y ⟵ Z$ has a limit, it is denoted by $X ×_Y Z$. It’s called the pullback/fiber product.

Examples of colimits: dually: quotients/coequalizers, unions, pushouts $X \coprod_Y Z$.

Pushouts:

  • in commutative algebra (in $CAlg$): $B \coprod_A C = B ⊗_A C$.

  • in $Incl$: the pushout is the union.

  • in $Grp$ the amalgated sum: $H_1 \ast_G H_2 = H_1 \ast_G H_2$

  • in $A$-$Mod$: the pushout of $N \overset{g}{⟵} P \overset{f}{⟶} M$ is $(M ⊕ N)/Im(f-g)$

Exercise 1: in $Top$:

\[X ≝ (0, 1) ∪ (2,3) ⟶ (0, 3) ≝ Y\]

Exercise 2: Define equalizers in any category, and show that if all limits exist: $X ×_Y Z$ is the equalizer of

\[\begin{xy} \xymatrix{ X × Z \ar@<+3pt>[r] \ar@<-3pt>[r] & Y } \end{xy}\]

Adjunctions

Assume $F: 𝒞 ⟶ 𝒟$ and $G: 𝒟 ⟶ 𝒞$ are functors.

We say that $F$ is left adjoint to $G$, $G$ is right adjoint to $F$ if

\[∀x∈𝒞, y ∈ 𝒟; \quad \Hom_𝒟(F(x), y) ≃ \Hom_𝒞(x, Gy)\]

functorially in $x$ and $y$.

Prop: if $F: 𝒞 ⟶ 𝒟$ admits a right (left) adjoint, then the adjoint is unique up to unique isomorphism.

Proof: for any $y ∈ 𝒟$, $G(y)$ represents the functor $\Hom(F(\bullet), y)$.

Examples:

  • $F: Set ⟶ Grp ⊣ U: Grp ⟶ Set$
  • $A ⟶ B$ map in $CAlg_k$, $\bullet ⊗_A B ⊣ θ$
  • let $I$ be a category, $𝒞$ a category: $cst: 𝒞 ⟶ Fun(I, 𝒞) ⊣ \lim$ if all limits exist

Why do we care about right adjoints?

  • Uniqueness enables us to show that two functors are isomorphic provided that they have the same left/right adjoint

  • Other reason:

Prop: Left/right adjoints preserve colimits/limits.

$I \overset{E}{⟶} 𝒞 \overset{F}{⟶} 𝒟$, $F$ left adjoint.

\(F(\colim_i E(i)) ⟵ \colim_i F \circ E(i)\) is an iso as soon as $\colim E(i)$ exists

Proof: $F ⊣ G$:

\[\Hom(F(\colim E), \bullet) ≃ \Hom(\colim E, G(\bullet))\\ ≃ \lim_i \Hom(E(i), G(\bullet)) ≃ \lim_i \Hom(F(E(i)), \bullet)\\ ≃ Hom(\colim_i F(E(i)), \bullet)\]

NB: so limits commute with limits.

Instance of that:

\[k[x] ⊗ k[y] ≃ k[x,y]\]

Another instance: $A ∈ CAlg$, $M ∈ Mod_A$

\[0 ⟶ N_1 ⟶ N_2 ⟶ N_3 ⟶ 0\]

then

\[M ⊗_A N_1 ⟶ M ⊗_A N_2 ⟶ M ⊗_A N_3 ⟶ 0\]

and

\[0 ⟶ \Hom(M,N_1) ⟶ \Hom(M,N_2) ⟶ \Hom(M,N_3)\]

because $M ⊗_A \bullet ⊣ \Hom(M,\bullet)$ and kernels/cokernels are limits/colimits.

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