Infinity Giraud's Theorem

Home , Category of sets, Groupoid, Topos

Inﬁnity Giraud’s Theorem 18 and 25 March 2019

The notion of a topos in inﬁnity categories depends only of left exact localization unlike the classical case because there does not exist a bijection between left exact localization and Grothendieck topologies in inﬁnity categories. The main reference is Lurie [2].

Definition 1. Let X be an ∞-category. We will say that X is an ∞-topos if there exists a small ∞-category C and accessible left exact localization functor L : P(C) → X. So the Giraud’s theorem for infinity categories is the following. Theorem 1. Let X be an ∞-category. The following conditions are equivalent: 1. The ∞-category X is an ∞-topos. 2. The ∞-category X satisfies the following ∞-categorical analogues of Giraud’s axioms: (a) The ∞-category X is presentable. (b) Colimits in X are universal. (c) Coproducts in X are disjoint. (d) Every groupoid object of X is effective.

1 Background in ∞-topos theory

The idea of working with ∞-topos is analogous to the classical case, i.e. We prove all in the presheaves categories with help of the properties of the category of spaces S and we send all the constructions to the topos of study, and most of the constructions are conserved thanks to the left exact localization condition.

equivalent ∞-Giraud’s axioms notions Thanks to l.e.l. sends Left exact localizations ∞-topos all the construccions Works in looks like First prove in P(C) S prove "pointwise" in

Generalize in equivalent Giraud’s axioms notions Left exact localizations topos Grothendieck topologies Thanks to l.e.l. sends all the construccions looks Works in like

P(C) Sets prove pointwise in

1.1 Presentability The ﬁrst Giraud axiom tells the category X is a λ-presentable ∞-category for some regular cardinal λ. To prove this axiom the following facts are needed. The ﬁrst one is to characterize the λ-compact objects in P(C).

1 Proposition 1. Let C be a small ∞-category, λ a regular cardinal and F an object of P(C). The following conditions are equivalent:

1. The object F is λ-compact.

2. There exist a diagram p : K → C indexed by a λ-small simplicial set, such that j ◦ p has a colimit G in P(C) and F is a retract 1 of G, where j : C → P(C) is the Yoneda embbeding.

The second one gives the tool of see a presheaf as a colimit of λ-compact objects. Lemma 1. Let C be a small simplicial set and let j : C → P(C) denote the Yoneda embedding. Then idP(C) is a left Kan extension of j along itself. To prove the previous staments we need some previous elements so only we uses their and the proofs can see in [2] as Proposition 5.3.4.17 and Lemma 5.1.5.3.

Proposition 2. Let C be a small ∞-category. The ∞-category of presheaves P(C) is presentable.

Proof. (Step 1) The category P(C) has small colimits.

We need to use the fact the category S has small colimits, and the colimits in P(C) are calculated ”pointwise", this mean if p : K → P(C) is a diagram, p : K. → P(C) is a colimit diagram iff . each pc : K → S is a colimit diagram for pc : K → S, where c is a vertex of C and pc = p ◦ ec with ec : P(C) → S the evaluation map.

. Since S is cocomplete then pc : K → S is a colimit diagram and this happens for each vertex c, we can conclude p : K. → P(C) is a colimit diagram. Therefore P(C) is cocomplete.

In fact, we can deduce this category has λ-ﬁltered colimits using the same argument since S has λ-ﬁltred colimits.

(Step 2) The subcategory of λ-compact objects Pλ (C).

The λ-compacts objects of P(C) are characterizad by the proposition 1. To prove Pλ (C) is small it will sufﬁce to show that the collection of isomorphism classes of objects in the homotopy category Pλ (C) is small.

For this, we use the proposition 1, so every λ-compact object F of P(C) is a retract of some G, wich is itself the colimit of some composition.

p j K / C / P(C) where K is λ-small. Since there is a bounded collection of possibilities for K and p and a bounded collection of idempotent maps of Y in Pλ (C), there is an only bounded number of possibilities for F.

1Let C be a category. Let X,Y be objects of C, we will say Y is a retract of X if there is a commutative diagram:

X ? i r Y / Y idY

2 (Step 3) (Sketch) The subcategory Pλ (C) generates P(C) under ﬁltered colimits.

λ Let F a presheaf, we can deﬁned the slice category P (C)/F over F and construct a subcategory over this slice with the λ-compact objects of Pλ (C) as the following pullback:

λ P (C)/F / P(C)/F

Pλ (C) / P(C)

If we deﬁne the following diagram

λ p : P (C)/F → P(C)/F → P(C)

It is possible to prove F is the colimit of p but we need two tecnical arguments an their are very extensive, the details can be ﬁnd as Lemma 5.4.4.2 and Proposition 5.4.4.3 in [2]. The central idea of that arguments is ilustred in the example 1.

Example 1. Let C be a small category and let F a object of P(C). The Grothendieck construc- R tion C F is the category with objects the pairs (C,x) where C is a object of C and x ∈ F(C) and morphisms the arrows u : (C,x) → (C0,y) such that u : C → C0 is a morphism in C and u(y) = x.

Fu F(C) F(C0)

x y

R λ We can see C F as a slice category, in fact like P (C)/F , the proof of this generalize the next R one: C y(C) = C/c where y is the Yoneda embedding and y(C) = HomC(−,C).

R Let u : (X, f ) → (Y,g) be a morphism in C y(C), by deﬁnition f ,g can seen as objects in the slice category C/C since f ∈ HomC(X,C) and g ∈ HomC(Y,C); also u : f → g is a morphism of C/C because u(g) = f .

λ R P (C)/F has a terminal object and therefore C F need be a ω-ﬁltred category. Additionaly,

R O y F = colim( C F / C / P(C)) where O is a forgetful functor, O(C,x) = C.

1.2 Local Cartesian Closed Categories A category X is called local cartesian closed when the pullback functor f ∗ has a right adjoint for each morphims f . In case X be a presentable category the notion of l.c.c. is equivalent to the following deﬁnition. This follows from adjoint functor theorem.

Deﬁnition 2. Let X be a presentable ∞-category. We will say the colimits in C are universal if ∗ for any morphism f : X → Y in X the associated pullback functor f : X/Y → X/X preserves colimits.

3 To prove the colimits in an ∞-topos are universal we need the following lemma. Lemma 2. Let X be a presentable ∞-category and L : X → Y be an accessible left exact localization. If colimits in X are universal, the colimits in Y are universal. Therefore everything lies, just like the classical case, in proving that P(C) has the property. Proposition 3. Let C be an ∞-category. The colimits in P(C) are universal. Proof. We need to use the following facts: 1. The colimits in S are universal. 2. The colimits in P(C) are calculated "pointwise".

So let τ : F → G a morphism in P(C) and a colimit colimWi in P(C)/G, we can calculate the following pullback. colimWi ×G F / colimWi τ∗( f ) f F τ / G ∗ ∼ To say that the functor τ preserves colimits means that (colimWi)×G F = colim(Wi ×G F), but this holds iff for each vertex c of C the following equation holds ∼ (colimWi(c)) ×G(c) F(c) = colim(Wi(c) ×G(c) F(c)) Its holds because this equation lives in S and this category has the property, so the colimits are universal in P(C). Proposition 4. Let X be an ∞-topos. The colimits in X are universal. Proof. Applying Lemma 2 and proposition 3. Other way to see universal colimits is the following: Let two diagrams in X with α a cartesian natural transformation between them: p K α X q

Suppose both diagrams extend to

p KB α X q

Then the colimits are universal can be reformulated as: If α is cartesian and q is a colimit diagram, then also p is a colimit diagram.

To prove this formulation, let α : p → q be a cartesian transformacion and let q a colimit cone in X, exist a correspondence between the colimits of X and colimits in slices categories of X, this means q can see as a colimit cone in X/y with x := p(∞) and y := q(∞), where ∞ is the terminal . object of K and f := α∞.

∗ ∗ ∗ We can construct the pullback functor f : X/y → X/x so p = f q. Using the fact f preserves colimits and since q is a colimit diagram then p is a colimit diagram.

4 1.3 Descent Argument Deﬁnition 3. Let X be an ∞-category is said to satisfy descent if let p,q two diagrams in X with α a cartesian natural transformation between them: p K α X q

Suppose both diagrams extend to

p KB α X q

If p and q are colimit diagrams, then α is a cartesian transformation. The category of spaces S satisﬁes descent, the proof of this requires model-category arguments and can ﬁnd as part of the proof of Lemma 6.1.3.14 in [2]. Proposition 5. Let P(C) be a presheaf ∞-category. Then for every pushout diagram.

α f / g

β β 0 f 0 / g0 α0

1 0 0 in OP(C) = Fun(∆ ,P(C)). If α and β are cartesian transformations, then α and β are also cartesian. The previous pushout diagram in OP(C) can be seen in P(C) has the following cube.

F f F α g G G β β 0 F0 F0 f 0 α0 g0 G0 G0

Proof. We need use the following facts: • The proposition is true in S. • The colimits and cartesian transformation are calcluted ”pointwise". So α0 and β 0 are cartesian transformations iff α0(c) and β 0(c) are cartesian, where c is a vertex of C. Evaluating c in each functor of the previous cube the following pullback diagram can be built in OS αc fc / gc

0 βc βc 0 0 fc 0 / gc αc

5 0 0 Since αc and βc are cartesian squares and the property holds in S then αc and βc are a cartesian transformations, so this happens for each vertex of C. Therefore α0 and β 0 are cartesian transformations.

Additionally, the property it is preserved for left exact localizations. Proposition 6. Let X be an ∞-topos. Then for every pushout diagram

α f / g

β β 0 f 0 / g0 α0

1 0 0 in OX = Fun(∆ ,X), if α and β are cartesian transformations, then α and β are also cartesian.

o Proof. Let L : P(C) / X : R be a left exact localization and the following diagram in X:

α f / g

β f 0

Applying the functor R and consider the fact that P(C) is cocomplete, then the following is a pushout diagram in P(C). Rα R f / Rg

0 Rβ β R f 0 / g0 α0 0 Since R is left exact, Rα and Rβ are cartesian transformations. Since α0 and β are cartesian transformations in P(C) (Proposition 5), applying L to the pushout diagram and the left exact 0 condition of L, the following diagram is a pushout in X and Lα0 and Lβ are cartesian transformations. LRα LR f / LRg

0 LRβ Lβ LR f 0 / Lg0 Lα0 But the left exact conditions of L allows us to rewrite the diagram as:

α f / g

β β 0 f 0 / g0 α0 where α0 and β 0 are cartesian transformations. Example 2. Descent argument is satisﬁes by S on the contrary of the category of sets, for example take the functions f ,g : {4} → {2} and form the cube diagram in Sets, it’s easy check α0 is not a cartesian transformation

6 4 2 f g α • 2 1 • • • g • • • • 2 4 2 2 1 f g α0 1 1

Note that α is a pullback but α0 doesn’t because the pullback of the diagram 1 → 1 ← 1 is 1 ← 1 → 1.

The lemma 3 can be see as a equivalent notion of the proposition 5 if the colimits are universal, the proof need a few of equivalent terminology, we only use the result and the proof can be ﬁnd as part of the theorem 6.1.3.9 in [2].

Lemma 3. Let X be an presentable ∞-category where the colimits are universal and satisﬁes the descent argument. Let K be a small simplicial set and α : p → q a natural transformation of . diagrams p,q : K → X. Suppose that q is a colimit diagram, and that α = α|K is a cartesian transformation. Then p is a colimit diagram iff α is a cartesian transformation.

p p

K α X KB α X q q

Example 3. Let X be an presentable ∞-category where the colimits are universal and satifying general descent argument. Let K be the following diagram shape • ← • → •, p,q : K → X two diagrams and α : p → q a natural transformation.

V 0 V01 V1 U01 U1 α

U 0 U01 U1 U0 U q

We would like construct an object V over U by gluing it together from a local date. This means, we can calculate the colimit diagram of p and this ﬁts correctly (be Cartesian) with the natural transformation that is created thanks to the universal property of pushouts. The following diagram ilustre that.

V01 V1

U1 V0 V w

U0 U

So we can see the above diagram as following:

7 p

V0 V V1

w α

U0 U U1 q

1.4 Disjoint coproducts condition The third Giraud’s axiom is the following. Deﬁnition 4. Let X be an ∞-category with coproducts and initial object. The coproducts in X are disjoint, if the following diagram is a pullback in X:

/0 / Y

X / X qY To prove the coproducts are disjoint in an ∞-topos, we need the following results. Lemma 4. Let X be a presentable category in which the colimits are universal and f : X → /0 be a morphism, where /0 is an initial object of X. Then X is also initial. The previous lemma tell us that for an accessible inﬁnity category the initial object is strict.

The following cube can be deﬁne in an ∞-topos. /0 Y id /0 α idY /0 Y idY β 0 β g f /0 Y f 0 g α ` X X Y

Where α is cartesian because /0is initial, also β is cartesian thanks to the Lemma 4. Proposition 7. Let X be an ∞-topos. The coproducts in X are disjoint. Proof. (Option 1) Let α0 be the following pushout diagram in X.

/0 / Y

X / X qY

0 The following diagram is constructed from α in OX.

α id/0 / idY

β β 0 f / g α0

8 We know α and β are cartesian and applying proposition 6, we have α0 is cartesian.

(Option 2) We need use the following facts: 1. The property holds in S. 2. The coproducts are disjoint in P(C) because the following diagram is a pullback diagram.

/0 / G

F / X qY iff for each vertex c of C the following diagram is a pullback in S.

/0 / G(c)

F(c) / F(c)qG(c)

but the previous diagram is a pullback because the property holds in S. Therefore the coproducts are disjoint in P(C). So let X,Y objects of X, applying the right adjoint R of the left exact localization L : P(C) → X, and construct the coproduct of RX,RY then we have the following pushout diagram.

/0 / RY

RX / RX qRY So applying the functor L and the conditions of left exacts, we have the following pushout in X.

L/0 / LRY

LRX / L(RX qLRY) but the above diagram for the properties of L can be rewritten as:

/0 / Y

X / X qY

2 Groupoid objects

The fourth Giraud’s axiom in the classical case told us, every equivalence relation R is effective, we can see this property as the following diagram

d0 R X q d1

X q X/R

9 The above diagram can seen as the following simplicial diagram

d q 1 X/R X s0 R d0

By effective condition of R we can garantizate that the previous diagram has all data that we need because after R all the morphims are determined by some composition of morphisms in the above diagram. Additionaly, reﬂective and transitive property tell us R can seen as category and symmetric condition allows to prove the morphisms are invertible. Therefore R is a groupoid.

Since the equivalence relations can seen as groupoids, then the following deﬁnitions come nat- urally to construct the fourth ∞-Giraud’s axiom. Deﬁnition 5. A simplicial object of an ∞-category C is a map of ∞-categories.

op U• : N(∆) → C We need the simplicial objects that meet a good gluing properties. This motivates the following deﬁnition.

Deﬁnition 6. Let C be an ∞-category and U• a simplicial object of C. We will say that U• is a groupoid object if for every n ≥ 0 and every partition n = S ∪ S0 such that S ∩ S0 consists of a single element s, the following diagram is a pullback in C.

U([n]) / U(S0)

U(S) / U({s})

Let ∆+ be the category of ﬁnite (possible empty) linearly ordered sets, we can extend the notion of groupoid object for this category. Deﬁnition 7. An augmented simplicial object of C is a map.

+ op U• : N(∆+) → C For the case of an augmented simplicial object the notion of good gluing properties motivates ≤n the following deﬁniton, where ∆+ denote the full subcategory of ∆+ spanned by the objects {[k]}−1≤k≤n. + Proposition 8. Let C be an ∞-category and let U• be an augmented simplicial object. We will + ˇ 2 say U• is a Cech Nerve if it satisﬁes the equivalent conditions: + 1. The underlying simplicial object of U• is a groupoid object of C and the following dia- + gram U• | ≤1 is a pullback square in C. N(∆+ )

U1 / U0

U0 / U−1 2In case C is a category with pullbacks and given a morphism f : U → X in C, the Cechˇ Nerve of f is the simplicial object

C(X) =( U ×X U ×X U U ×X U U X )

10 + + 2.U • is a right Kan extension of U• | ≤0 . N(∆+ ) Now we have the elements to deﬁne a groupoid object with good gluing properties.

Deﬁnition 8. Let U• be a simplicial object of an ∞-category C. We will say that U• is an effective + + ˇ groupoid if can be extended to a colimit diagram U• such that U• is a Cech Nerve.

U• N(∆)op / C ;

+ U• op N(∆+)

2.1 Dec> Construction To prove the groupoid objects in an ∞-topos are effective we need two special tools, the ﬁrst one is the Dec> construcction of a simplicial object. Let X be a simplicial set (or a simplicial object in an ∞-category, it does matter because the construcction holds in any case), we can deﬁne a new simplicial set removing the top face map d> in all the simplicial diagram of X.

X0 X1 X2 X

d1 d2 d3

d d1 2 X1 X2 X3 Dec>X d 0 d0

Additionally, we can see X0 has a colimit of Dec>X.

d d1 2 X X X2 X 0 d0 1 3 d 0 d0

The last stament holds, if we see the above diagram as the following split augmented diagram

s1 s2 s0 X X X2 X 0 d0 1 3 and using the following lemma.

Lemma 5. Every split augmented diagram of a simplicial set is a colimit diagram

To prove lemma we need some technical arguments, but those generalize the following construction. A fork in X is a diagram of the form

f q A B C g such that, q◦ f = q◦g. A split fork is a fork such that exist t : B → A and s : C → B that satisfying q ◦ s = idC, f ◦ = idB and s ◦ q = g ◦t.

11 t s

f A B q C g

Lemma 6. A split fork is a coeaqualizer.

Proof. There exist W and w : B → W that w ◦ f = w ◦ g.

t s

f A B q C g w W

We can deﬁne a morphism w ◦ s from C and W such that satisﬁes the following.

w ◦ (s ◦ q) = w ◦ (g ◦t) = (w ◦ g) ◦t = (w ◦ f ) ◦t = w ◦ ( f ◦t)

= w ◦ idB

Therefore q is the coequalizer of f ,g. To prove the ﬁrst part of lemma 7 is necessary used the split augmented argument uses in the proof of split fork are coequalizer and can be ﬁnd in [2] as Lemma 6.1.3.16 and the second part is deduced from the analogous of the prisma lemma3 for pushout.

op Lemma 7. Let X be an ∞-category and let U• : N(∆) → X be a simplicial object of X. Let Dec>U• be the augmented simplicial object given by composing U• with the functor F. Then:

1. The augmented simplicial object Dec>U• is a colimit diagram.

2. If U• is a groupoid object of X, then α : (Dec>U•)|N(∆)op → U• is cartesian. The groupoid objects over a category, where the colimits are universal and satisﬁes descent argument, are effective.

Theorem 2. Let X be an ∞-category such that:

1. The colimits in X are universal. 3The prisma lemma tells if β is a pullback. Then α is a pullback diagram iff all the diagram is a pullback.

• • •

α β

• • •

12 2. For every pushout diagram: α f / g

β β 0 f 0 / g0 α0 0 0 in OX, if α and β are cartesian transformations then α and β are also cartesian transformations.

Then the groupoid objects of X are effective.

op Proof. Let U• be a groupoid object of X and let U• : N(∆+) → X be a colimit cocone` of U•. op Let Dec>U• : N(∆+) → X be the composition of U• with F : ∆+ → ∆+, F(J) = J {∞}; for the construction of F, we have Dec>Un = Un+1.

And applying lemma 7, Dec>U• is a colimit diagram in X. Additionally for the second impli- cation of the lemma 7, α = α|N(∆)op is cartesian.

U•

U−1 q U0 U1 U2

q d1 d2 d3 α

U0 U1 U2 U3

Dec>U•

Applying the lemma 3, α : Dec>U• → U• is a cartesian transformation because Dec>U• is a colimit diagram. Additionally, we can construct the following pullback diagram in X.

Dec>U0 / Dec>U−1

U0 / U−1

This pullback can be rewritten using Dec>Un = Un+1 and U•|N(∆)op = U•

U1 / U0

U0 / U−1

Then U• is an effective groupoid. Corollary 1. Let X be an ∞-topos. Then the groupoid objects in X are effective.

13 3 Giraud’s theorem

The previous section told us that if the ∞-category is an ∞-topos then satisﬁes the Giraud’s axioms. To prove the converse, we need the following result.

Proposition 9. Let C be a small ∞-category wich admits ﬁnite limits and let X be an ∞-category wich satisﬁes the Giraud’s axioms. Let F : P(C) → X be a colimit preserving functor. Supposse that the composition F ◦ j : C → X is left exact, where j is the Yoneda embedding. Then F is left exact.

Let Indλ C be the subcategory of P(C) formed by those presheaves that classify right ﬁbrations of λ-ﬁltered categories. Simpson’s theorem allow us to see Indλ C as a localization of P(C), in fact, its a left exact localization applying the proposition 9.

Proposition 10. Let X be an ∞-category. Suppose that X satisﬁes the ∞-categorical Giraud’s axioms then there exists a small ∞-category C that admits ﬁnite limits and an accessible left exact localization functor L : P(C) → X. In particular X is an ∞-topos. Proof. The category X is accessible so this is equivalent (Simpson’s theorem) to the exis- tence of a small category C an ∞-category which admits λ-small colimits, and an equivalence Indλ C ≡ X.

So the candidate to construct the left exat localization is the category P(C). We have i : Indλ C ,→ P(C) has a left adjoint L : P(C) → Indλ C. The composition of L with the Yoneda embedding 0 j : C → P(C) can be seen as the Yoneda embedding j : C → Indλ C and therefore preserves all limits that exists in C.

Applying proposition 9, we have that L is left exact, so Indλ C is a left exact localization and therefore X it is also.

References

[1] C. Rezk, Toposes and homotopy toposes.

[2] J. Lurie, Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press, Princeton and Oxford, 2009.

[3] S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic, A ﬁrst introduction to topos theory, Springer-Verlag, New York, 1992.