Grothendieck -groupoids as iterated injectives John Bourke - - PowerPoint PPT Presentation

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck ω-groupoids as iterated injectives

John Bourke

Department of Mathematics and Statistics Masaryk University

CT2016, Halifax

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Background story

◮ End of 2015: Read Grothendieck ω-groupoids . . . by

Maltsiniotis (2010). This brought to attention, and formalised, definition of globular weak ω-groupoids due to Grothendieck (1983).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Background story

◮ End of 2015: Read Grothendieck ω-groupoids . . . by

Maltsiniotis (2010). This brought to attention, and formalised, definition of globular weak ω-groupoids due to Grothendieck (1983).

◮ Also read Types are weak ω-groupoids by van den Berg and

Garner (2011).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Background story

◮ End of 2015: Read Grothendieck ω-groupoids . . . by

Maltsiniotis (2010). This brought to attention, and formalised, definition of globular weak ω-groupoids due to Grothendieck (1983).

◮ Also read Types are weak ω-groupoids by van den Berg and

Garner (2011).

◮ They constructed globular weak ω-groupoids from types using

Batanin ω-categories. Complex construction. (Also LeFanu-Lumsdaine 2010).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Background story

◮ End of 2015: Read Grothendieck ω-groupoids . . . by

Maltsiniotis (2010). This brought to attention, and formalised, definition of globular weak ω-groupoids due to Grothendieck (1983).

◮ Also read Types are weak ω-groupoids by van den Berg and

Garner (2011).

◮ They constructed globular weak ω-groupoids from types using

Batanin ω-categories. Complex construction. (Also LeFanu-Lumsdaine 2010).

◮ Noticed: much easier to communicate their construction using

Grothendieck ω-groupoids because of low-tech formalism.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Background story

◮ End of 2015: Read Grothendieck ω-groupoids . . . by

Maltsiniotis (2010). This brought to attention, and formalised, definition of globular weak ω-groupoids due to Grothendieck (1983).

◮ Also read Types are weak ω-groupoids by van den Berg and

Garner (2011).

◮ They constructed globular weak ω-groupoids from types using

Batanin ω-categories. Complex construction. (Also LeFanu-Lumsdaine 2010).

◮ Noticed: much easier to communicate their construction using

Grothendieck ω-groupoids because of low-tech formalism.

◮ Wrote short paper about this (Bourke 2016).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Background story II

◮ Lots of open questions about Grothendieck ω-groupoids: eg.

model structures, homotopy hypothesis?

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Background story II

◮ Lots of open questions about Grothendieck ω-groupoids: eg.

model structures, homotopy hypothesis?

◮ Started to do some research which I’ll talk about.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Background story II

◮ Lots of open questions about Grothendieck ω-groupoids: eg.

model structures, homotopy hypothesis?

◮ Started to do some research which I’ll talk about. ◮ First half of talk – introduce Grothendieck ω-groupoids.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Background story II

◮ Lots of open questions about Grothendieck ω-groupoids: eg.

model structures, homotopy hypothesis?

◮ Started to do some research which I’ll talk about. ◮ First half of talk – introduce Grothendieck ω-groupoids. ◮ Second half of talk – my own research.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular sets 1

◮ The globe category G is generated by the graph τ

• σ

1

τ

• σ . . .

τ

• σ

n

τ σ n + 1 . . .

subject to the relations σ ◦ σ = σ ◦ τ and τ ◦ σ = τ ◦ τ.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular sets 1

◮ The globe category G is generated by the graph τ

• σ

1

τ

• σ . . .

τ

• σ

n

τ σ n + 1 . . .

subject to the relations σ ◦ σ = σ ◦ τ and τ ◦ σ = τ ◦ τ.

◮ For n < m have exactly two morphisms σ, τ : n ⇒ m.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular sets 1

◮ The globe category G is generated by the graph τ

• σ

1

τ

• σ . . .

τ

• σ

n

τ σ n + 1 . . .

subject to the relations σ ◦ σ = σ ◦ τ and τ ◦ σ = τ ◦ τ.

◮ For n < m have exactly two morphisms σ, τ : n ⇒ m. ◮ A functor A : Gop → C is called a globular object in C.

. . . A(n + 1)

s

• t

A(n)

s

• t

. . .

s

• t

A(1)

s

• t

A(0)

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular sets 1

◮ The globe category G is generated by the graph τ

• σ

1

τ

• σ . . .

τ

• σ

n

τ σ n + 1 . . .

subject to the relations σ ◦ σ = σ ◦ τ and τ ◦ σ = τ ◦ τ.

◮ For n < m have exactly two morphisms σ, τ : n ⇒ m. ◮ A functor A : Gop → C is called a globular object in C.

. . . A(n + 1)

s

• t

A(n)

s

• t

. . .

s

• t

A(1)

s

• t

A(0)

◮ [Gop, Set] is the category of globular sets.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Important globular sets 1

◮ Have Y : G → [Gop, Set]. The globular set Y (n) is the free

n-cell.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Important globular sets 1

◮ Have Y : G → [Gop, Set]. The globular set Y (n) is the free

n-cell.

◮

Y (0) : •

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Important globular sets 1

◮ Have Y : G → [Gop, Set]. The globular set Y (n) is the free

n-cell.

◮

Y (0) : • Y (1) : •

• John Bourke

Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Important globular sets 1

◮ Have Y : G → [Gop, Set]. The globular set Y (n) is the free

n-cell.

◮

Y (0) : • Y (1) : •

• Y (2) : •
• John Bourke

Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

From Lawvere theories to globular theories I

◮ Key diagram for Lawvere theories:

1

1

• F

Set

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

From Lawvere theories to globular theories I

◮ Key diagram for Lawvere theories:

1

1

• F

Set

◮ 1 is the shape category for sets.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

From Lawvere theories to globular theories I

◮ Key diagram for Lawvere theories:

1

1

• F

Set

◮ 1 is the shape category for sets. ◮ F contains the arities for operations in universal algebra – it is

a skeleton of finite sets.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

From Lawvere theories to globular theories I

◮ Key diagram for Lawvere theories:

1

1

• F

Set

◮ 1 is the shape category for sets. ◮ F contains the arities for operations in universal algebra – it is

a skeleton of finite sets.

◮ The finite sets are generated as finite copowers of 1 in Set.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

From Lawvere theories to globular theories I

◮ Key diagram for Lawvere theories:

1

1

• F

Set

◮ 1 is the shape category for sets. ◮ F contains the arities for operations in universal algebra – it is

a skeleton of finite sets.

◮ The finite sets are generated as finite copowers of 1 in Set. ◮ A Lawvere theory is an identity on objects functor

J : Fop → T preserving finite powers of 1.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

From Lawvere theories to globular theories II

◮ Corresponding diagram for globular theories:

G

Y

• Θ0

[Gop, Set]

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

From Lawvere theories to globular theories II

◮ Corresponding diagram for globular theories:

G

Y

• Θ0

[Gop, Set]

◮ G is the shape category for globular sets.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

From Lawvere theories to globular theories II

◮ Corresponding diagram for globular theories:

G

Y

• Θ0

[Gop, Set]

◮ G is the shape category for globular sets. ◮ Θ0 contains the arities for operations in ω-categories,

ω-groupoids and so on. These are the globular cardinals (Street) – certain globular sets.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

From Lawvere theories to globular theories II

◮ Corresponding diagram for globular theories:

G

Y

• Θ0

[Gop, Set]

◮ G is the shape category for globular sets. ◮ Θ0 contains the arities for operations in ω-categories,

ω-groupoids and so on. These are the globular cardinals (Street) – certain globular sets.

◮ The globular cardinals are the globular sums of representables.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

From Lawvere theories to globular theories II

◮ Corresponding diagram for globular theories:

G

Y

• Θ0

[Gop, Set]

◮ G is the shape category for globular sets. ◮ Θ0 contains the arities for operations in ω-categories,

ω-groupoids and so on. These are the globular cardinals (Street) – certain globular sets.

◮ The globular cardinals are the globular sums of representables. ◮ A globular theory is an identity on objects functor

J : Θop

0 → T preserving globular products.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

From Lawvere theories to globular theories II

◮ Corresponding diagram for globular theories:

G

Y

• Θ0

[Gop, Set]

◮ G is the shape category for globular sets. ◮ Θ0 contains the arities for operations in ω-categories,

ω-groupoids and so on. These are the globular cardinals (Street) – certain globular sets.

◮ The globular cardinals are the globular sums of representables. ◮ A globular theory is an identity on objects functor

J : Θop

0 → T preserving globular products.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular cardinals and tables of dimensions

◮ Examples of globular cardinals:

• John Bourke

Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular cardinals and tables of dimensions

◮ Examples of globular cardinals:

• ◮ Indexed by (1, 0, 1) and (1, 0, 2, 1, 2) respectively.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular cardinals and tables of dimensions

◮ Examples of globular cardinals:

• ◮ Indexed by (1, 0, 1) and (1, 0, 2, 1, 2) respectively.

◮ A table of dimensions (t.o.d.) is a sequence n = (n1, . . . , nk)

• f natural numbers such that

n1 > n2 < n3 > n4 < n5 > . . . < nk and of odd length.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular cardinals and tables of dimensions

◮ Examples of globular cardinals:

• ◮ Indexed by (1, 0, 1) and (1, 0, 2, 1, 2) respectively.

◮ A table of dimensions (t.o.d.) is a sequence n = (n1, . . . , nk)

• f natural numbers such that

n1 > n2 < n3 > n4 < n5 > . . . < nk and of odd length.

◮ Index the globular cardinals: Glue the n2-target of a n1-cell to

the n2-source of a n3-cell.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular cardinals and tables of dimensions

◮ Examples of globular cardinals:

• ◮ Indexed by (1, 0, 1) and (1, 0, 2, 1, 2) respectively.

◮ A table of dimensions (t.o.d.) is a sequence n = (n1, . . . , nk)

• f natural numbers such that

n1 > n2 < n3 > n4 < n5 > . . . < nk and of odd length.

◮ Index the globular cardinals: Glue the n2-target of a n1-cell to

the n2-source of a n3-cell.Glue the n4-target of that n3-cell to the n4-source of a n5-cell.. . .

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular cardinals and tables of dimensions

◮ Examples of globular cardinals:

• ◮ Indexed by (1, 0, 1) and (1, 0, 2, 1, 2) respectively.

◮ A table of dimensions (t.o.d.) is a sequence n = (n1, . . . , nk)

• f natural numbers such that

n1 > n2 < n3 > n4 < n5 > . . . < nk and of odd length.

◮ Index the globular cardinals: Glue the n2-target of a n1-cell to

the n2-source of a n3-cell.Glue the n4-target of that n3-cell to the n4-source of a n5-cell.. . .

◮ Colimit formulation of this process is via globular sums.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular sums and globular products

◮ The t.o.d. n = (n1, . . . , nk) determines the diagram

n2 n4 . . . nk−1 n1 n3 n5 nk−2 nk

τ

② ② ② ②

σ

• ❊

❊ ❊ ❊

τ

② ② ② ②

σ

• ❊

❊ ❊ ❊

τ

②②②

σ

• ❊

❊ ❊ ❊

in G.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular sums and globular products

◮ The t.o.d. n = (n1, . . . , nk) determines the diagram

n2 n4 . . . nk−1 n1 n3 n5 nk−2 nk

τ

② ② ② ②

σ

• ❊

❊ ❊ ❊

τ

② ② ② ②

σ

• ❊

❊ ❊ ❊

τ

②②②

σ

• ❊

❊ ❊ ❊

in G.

◮ So given D : G → C we get a diagram

D(n2) D(n4) . . . D(nk−1) D(n1) D(n3) D(n5) D(nk−2) D(nk)

Dτ②

②

Dσ

• ❊

❊

Dτ②

②

Dσ

• ❊

❊

Dτ②

②

Dσ

• ❊

❊

whose colimit, denoted D(n), is called a D-globular sum.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular sums and globular products

◮ The t.o.d. n = (n1, . . . , nk) determines the diagram

n2 n4 . . . nk−1 n1 n3 n5 nk−2 nk

τ

② ② ② ②

σ

• ❊

❊ ❊ ❊

τ

② ② ② ②

σ

• ❊

❊ ❊ ❊

τ

②②②

σ

• ❊

❊ ❊ ❊

in G.

◮ A globular object A : Gop → C determines a diagram

A(n2) A(n4) . . . A(nk−1) A(n1) A(n3) A(n5) A(nk−2) A(nk)

t

❊ ❊

s

② ②

t

❊ ❊

s

② ②

t

❊ ❊

s

② ②

whose limit is called a A-globular product.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular cardinals as globular sums

◮ Consider Y : G → [Gop, Set].

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular cardinals as globular sums

◮ Consider Y : G → [Gop, Set]. ◮ Y -globular sums are exactly the globular cardinals.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular cardinals as globular sums

◮ Consider Y : G → [Gop, Set]. ◮ Y -globular sums are exactly the globular cardinals. ◮

Y (0) Y (1) Y (1) Y (1, 0, 1)

Y τ

t t t

Y σ

• ❏

❏ ❏ t t t

• ❏

❏ ❏

So Y (1, 0, 1) is (• → • → •) as expected.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular cardinals as globular sums

◮ Consider Y : G → [Gop, Set]. ◮ Y -globular sums are exactly the globular cardinals. ◮

Y (0) Y (1) Y (1) Y (2) Y (2) Y (1, 0, 2, 1, 2)

Y τ

t t t

Y σ

• ❏

❏ ❏

Y τ

t t t

Y σ

• ❏

❏ ❏

• ❚

❚ ❚ ❚ ❚ ❚ ❥❥❥❥❥❥

So Y (1, 0, 2, 1, 2) = •

• John Bourke

Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

The category Θ0 and globular theories

◮ Reconsider the diagram:

G

Y

• D

Θ0 [Gop, Set]

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

The category Θ0 and globular theories

◮ Reconsider the diagram:

G

Y

• D

Θ0 [Gop, Set]

◮ Θ0 skeletal full subcategory of [Gop, Set] containing the

globular cardinals (aka Y-globular sums).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

The category Θ0 and globular theories

◮ Reconsider the diagram:

G

Y

• D

Θ0 [Gop, Set]

◮ Θ0 skeletal full subcategory of [Gop, Set] containing the

globular cardinals (aka Y-globular sums).

◮ For canonical representatives, set the objects of Θ0 to be the

t.o.d’s and Θ0(n, m) = [Gop, Set](Y (n), Y (m)).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

The category Θ0 and globular theories

◮ Reconsider the diagram:

G

Y

• D

Θ0 [Gop, Set]

◮ Θ0 skeletal full subcategory of [Gop, Set] containing the

globular cardinals (aka Y-globular sums).

◮ For canonical representatives, set the objects of Θ0 to be the

t.o.d’s and Θ0(n, m) = [Gop, Set](Y (n), Y (m)).

◮ D : G → Θ0 sends n to the t.o.d.(n) of length 1.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

The category Θ0 and globular theories

◮ Reconsider the diagram:

G

Y

• D

Θ0 [Gop, Set]

◮ Θ0 skeletal full subcategory of [Gop, Set] containing the

globular cardinals (aka Y-globular sums).

◮ For canonical representatives, set the objects of Θ0 to be the

t.o.d’s and Θ0(n, m) = [Gop, Set](Y (n), Y (m)).

◮ D : G → Θ0 sends n to the t.o.d.(n) of length 1. ◮ So Θ0 consists of the D-globular sums

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

The category Θ0 and globular theories

◮ Reconsider the diagram:

G

Y

• D

Θ0 [Gop, Set]

◮ Θ0 skeletal full subcategory of [Gop, Set] containing the

globular cardinals (aka Y-globular sums).

◮ For canonical representatives, set the objects of Θ0 to be the

t.o.d’s and Θ0(n, m) = [Gop, Set](Y (n), Y (m)).

◮ D : G → Θ0 sends n to the t.o.d.(n) of length 1. ◮ So Θ0 consists of the D-globular sums and Θop

consists of the Dop-globular products.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

The category Θ0 and globular theories

◮ Reconsider the diagram:

G

Y

• D

Θ0 [Gop, Set]

◮ Θ0 skeletal full subcategory of [Gop, Set] containing the

globular cardinals (aka Y-globular sums).

◮ For canonical representatives, set the objects of Θ0 to be the

t.o.d’s and Θ0(n, m) = [Gop, Set](Y (n), Y (m)).

◮ D : G → Θ0 sends n to the t.o.d.(n) of length 1. ◮ So Θ0 consists of the D-globular sums and Θop

consists of the Dop-globular products.

◮ A globular theory consists of an identity on objects functor

J : Θop

0 → T preserving globular products.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular theories

◮ A globular theory consists of an identity on objects functor

J : Θop

0 → T preserving globular products.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular theories

◮ A globular theory consists of an identity on objects functor

J : Θop

0 → T preserving globular products. ◮ The category Mod(T, C) of T-models in C is the full

subcategory of [T, C] containing those functors preserving globular products.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular theories

◮ A globular theory consists of an identity on objects functor

J : Θop

0 → T preserving globular products. ◮ The category Mod(T, C) of T-models in C is the full

subcategory of [T, C] containing those functors preserving globular products.

◮ Write Mod(T) for Mod(T, Set).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular theories

◮ A globular theory consists of an identity on objects functor

J : Θop

0 → T preserving globular products. ◮ The category Mod(T, C) of T-models in C is the full

subcategory of [T, C] containing those functors preserving globular products.

◮ Write Mod(T) for Mod(T, Set). ◮ Forgetful functor Mod(T) → [Gop, Set] obtained by restricting

along the inclusion Gop → Θop

0 → T.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Globular theories

◮ A globular theory consists of an identity on objects functor

J : Θop

0 → T preserving globular products. ◮ The category Mod(T, C) of T-models in C is the full

subcategory of [T, C] containing those functors preserving globular products.

◮ Write Mod(T) for Mod(T, Set). ◮ Forgetful functor Mod(T) → [Gop, Set] obtained by restricting

along the inclusion Gop → Θop

0 → T. ◮ Monadic (Ara).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Contractible theories

◮ Let A : Gop → C. A parallel pair of n-cells in A is a pair

f , g : X ⇒ A(n) such that either n = 0 or s ◦ f = s ◦ g and t ◦ f = t ◦ g.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Contractible theories

◮ Let A : Gop → C. A parallel pair of n-cells in A is a pair

f , g : X ⇒ A(n) such that either n = 0 or s ◦ f = s ◦ g and t ◦ f = t ◦ g.

◮ A lifting is an arrow h : X → A(n + 1) such that

A(n + 1)

s

• t
• X

h

• ✇

✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇

f

• g

A(n)

commutes.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Contractible theories

◮ Let A : Gop → C. A parallel pair of n-cells in A is a pair

f , g : X ⇒ A(n) such that either n = 0 or s ◦ f = s ◦ g and t ◦ f = t ◦ g.

◮ A lifting is an arrow h : X → A(n + 1) such that

A(n + 1)

s

• t
• X

h

• ✇

✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇

f

• g

A(n)

commutes.

◮ A is said to be contractible if each parallel pair has a lifting.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Contractible theories

◮ Let A : Gop → C. A parallel pair of n-cells in A is a pair

f , g : X ⇒ A(n) such that either n = 0 or s ◦ f = s ◦ g and t ◦ f = t ◦ g.

◮ A lifting is an arrow h : X → A(n + 1) such that

A(n + 1)

s

• t
• X

h

• ✇

✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇

f

• g

A(n)

commutes.

◮ A is said to be contractible if each parallel pair has a lifting. ◮ J : Θop 0 → T is said to be contractible if its underlying

globular object J ◦ Dop : Gop → T is contractible.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck ω-groupoids

◮ A Grothendieck ω-groupoid is an algebra for some contractible

globular theory T.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck ω-groupoids

◮ A Grothendieck ω-groupoid is an algebra for some contractible

globular theory T.

◮ Where are the operations for a ω-groupoid in such a T?

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck ω-groupoids

◮ A Grothendieck ω-groupoid is an algebra for some contractible

globular theory T.

◮ Where are the operations for a ω-groupoid in such a T? ◮ A function composing 1-cells in a globular set A will be a

suitable function A(m) : A(1, 0, 1) → A(1).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck ω-groupoids

◮ A Grothendieck ω-groupoid is an algebra for some contractible

globular theory T.

◮ Where are the operations for a ω-groupoid in such a T? ◮ A function composing 1-cells in a globular set A will be a

suitable function A(m) : A(1, 0, 1) → A(1). (1, 0, 1)

q

• p
• (1)

s

• (1)

t

(0)

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck ω-groupoids

◮ A Grothendieck ω-groupoid is an algebra for some contractible

globular theory T.

◮ Where are the operations for a ω-groupoid in such a T? ◮ A function composing 1-cells in a globular set A will be a

suitable function A(m) : A(1, 0, 1) → A(1). (1, 0, 1)

q

• p
• (1)

s

• (1)

t

• s
• (1)

t

(0)

(1, 0, 1)

s◦p

• t◦q

(0)

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck ω-groupoids

◮ A Grothendieck ω-groupoid is an algebra for some contractible

globular theory T.

◮ Where are the operations for a ω-groupoid in such a T? ◮ A function composing 1-cells in a globular set A will be a

suitable function A(m) : A(1, 0, 1) → A(1). (1, 0, 1)

q

• p
• (1)

s

• (1)

t

• s
• (1)

t

(0)

(1, 0, 1)

s◦p

• t◦q
• m
• ✇

✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇

(0)

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

◮ What about the associator?

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

◮ What about the associator?

(2)

t

• s
• (1, 0, 1, 0, 1)

m◦(m,1)

• m◦(1,m)

(1)

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

◮ What about the associator?

(2)

t

• s
• (1, 0, 1, 0, 1)

a

• ♠

♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠

m◦(m,1)

• m◦(1,m)

(1)

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

◮ What about the associator?

(2)

t

• s
• (1, 0, 1, 0, 1)

a

• ♠

♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠

m◦(m,1)

• m◦(1,m)

(1)

◮ Equivalence inverses for 1-cells?

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

◮ What about the associator?

(2)

t

• s
• (1, 0, 1, 0, 1)

a

• ♠

♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠

m◦(m,1)

• m◦(1,m)

(1)

◮ Equivalence inverses for 1-cells?

(1)

t

• s
• (1)

t

• s

(0)

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

◮ What about the associator?

(2)

t

• s
• (1, 0, 1, 0, 1)

a

• ♠

♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠

m◦(m,1)

• m◦(1,m)

(1)

◮ Equivalence inverses for 1-cells?

(1)

t

• s
• (1)

inv

• q

q q q q q q q q q q q q

t

• s

(0)

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck weak ω-groupoids

◮ I gave slight oversimplification: Grothendieck defined

specifically theories for weak ω-groupoids, but there is a contractible theory ˜ Θ for strict ω-groupoids (Ara).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck weak ω-groupoids

◮ I gave slight oversimplification: Grothendieck defined

specifically theories for weak ω-groupoids, but there is a contractible theory ˜ Θ for strict ω-groupoids (Ara).

◮ Grothendieck: weak ω-groupoid = model of cellular

contractible theory (or coherator).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck weak ω-groupoids

◮ I gave slight oversimplification: Grothendieck defined

specifically theories for weak ω-groupoids, but there is a contractible theory ˜ Θ for strict ω-groupoids (Ara).

◮ Grothendieck: weak ω-groupoid = model of cellular

contractible theory (or coherator).

◮ Note: if T coherator and S contractible there exists T → S

and so Mod(S) → Mod(T).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck weak ω-groupoids

◮ I gave slight oversimplification: Grothendieck defined

specifically theories for weak ω-groupoids, but there is a contractible theory ˜ Θ for strict ω-groupoids (Ara).

◮ Grothendieck: weak ω-groupoid = model of cellular

contractible theory (or coherator).

◮ Note: if T coherator and S contractible there exists T → S

and so Mod(S) → Mod(T). So Grothendieck ω-groupoids are weak ω-groupoids – no contradiction with earlier definition.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck weak ω-groupoids

◮ I gave slight oversimplification: Grothendieck defined

specifically theories for weak ω-groupoids, but there is a contractible theory ˜ Θ for strict ω-groupoids (Ara).

◮ Grothendieck: weak ω-groupoid = model of cellular

contractible theory (or coherator).

◮ Note: if T coherator and S contractible there exists T → S

and so Mod(S) → Mod(T). So Grothendieck ω-groupoids are weak ω-groupoids – no contradiction with earlier definition.

◮ Before definition need to understand the category GlobTh of

globular theories – this is just full subcategory of Θ0/Cat.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck weak ω-groupoids

◮ I gave slight oversimplification: Grothendieck defined

specifically theories for weak ω-groupoids, but there is a contractible theory ˜ Θ for strict ω-groupoids (Ara).

◮ Grothendieck: weak ω-groupoid = model of cellular

contractible theory (or coherator).

◮ Note: if T coherator and S contractible there exists T → S

and so Mod(S) → Mod(T). So Grothendieck ω-groupoids are weak ω-groupoids – no contradiction with earlier definition.

◮ Before definition need to understand the category GlobTh of

globular theories – this is just full subcategory of Θ0/Cat.

◮ Note: GlobTh is locally finitely presentable.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Grothendieck weak ω-groupoids

◮ I gave slight oversimplification: Grothendieck defined

specifically theories for weak ω-groupoids, but there is a contractible theory ˜ Θ for strict ω-groupoids (Ara).

◮ Grothendieck: weak ω-groupoid = model of cellular

contractible theory (or coherator).

◮ Note: if T coherator and S contractible there exists T → S

and so Mod(S) → Mod(T). So Grothendieck ω-groupoids are weak ω-groupoids – no contradiction with earlier definition.

◮ Before definition need to understand the category GlobTh of

globular theories – this is just full subcategory of Θ0/Cat.

◮ Note: GlobTh is locally finitely presentable.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Cellularity and weak ω-groupoids

◮ A cellular presentation of a globular theory T is a chain of

globular theories Θ0 = T0 → T1 → . . . → Tn . . . → T with colimit T,

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Cellularity and weak ω-groupoids

◮ A cellular presentation of a globular theory T is a chain of

globular theories Θ0 = T0 → T1 → . . . → Tn . . . → T with colimit T, together with sets Jn of parallel pairs in Tn,

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Cellularity and weak ω-groupoids

◮ A cellular presentation of a globular theory T is a chain of

globular theories Θ0 = T0 → T1 → . . . → Tn . . . → T with colimit T, together with sets Jn of parallel pairs in Tn, such that Tn+1 is obtained from Tn by freely adjoining a lifting for each parallel pair in Jn.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Cellularity and weak ω-groupoids

◮ A cellular presentation of a globular theory T is a chain of

globular theories Θ0 = T0 → T1 → . . . → Tn . . . → T with colimit T, together with sets Jn of parallel pairs in Tn, such that Tn+1 is obtained from Tn by freely adjoining a lifting for each parallel pair in Jn.

◮ Call T cellular if it admits a cellular presentation.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Cellularity and weak ω-groupoids

◮ A cellular presentation of a globular theory T is a chain of

globular theories Θ0 = T0 → T1 → . . . → Tn . . . → T with colimit T, together with sets Jn of parallel pairs in Tn, such that Tn+1 is obtained from Tn by freely adjoining a lifting for each parallel pair in Jn.

◮ Call T cellular if it admits a cellular presentation. ◮ A Grothendieck weak ω-groupoid is a model for a cellular

contractible theory (a coherator).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Cellularity and weak ω-groupoids

◮ A cellular presentation of a globular theory T is a chain of

globular theories Θ0 = T0 → T1 → . . . → Tn . . . → T with colimit T, together with sets Jn of parallel pairs in Tn, such that Tn+1 is obtained from Tn by freely adjoining a lifting for each parallel pair in Jn.

◮ Call T cellular if it admits a cellular presentation. ◮ A Grothendieck weak ω-groupoid is a model for a cellular

contractible theory (a coherator).

◮ Canonical coherator: Jn = all parallel pairs in Tn.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Cellularity and weak ω-groupoids

◮ A cellular presentation of a globular theory T is a chain of

globular theories Θ0 = T0 → T1 → . . . → Tn . . . → T with colimit T, together with sets Jn of parallel pairs in Tn, such that Tn+1 is obtained from Tn by freely adjoining a lifting for each parallel pair in Jn.

◮ Call T cellular if it admits a cellular presentation. ◮ A Grothendieck weak ω-groupoid is a model for a cellular

contractible theory (a coherator).

◮ Canonical coherator: Jn = all parallel pairs in Tn. In T0 we

have the parallel pair s ◦ p, t ◦ q : (1, 0, 1) ⇒ (0).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Cellularity and weak ω-groupoids

◮ A cellular presentation of a globular theory T is a chain of

globular theories Θ0 = T0 → T1 → . . . → Tn . . . → T with colimit T, together with sets Jn of parallel pairs in Tn, such that Tn+1 is obtained from Tn by freely adjoining a lifting for each parallel pair in Jn.

◮ Call T cellular if it admits a cellular presentation. ◮ A Grothendieck weak ω-groupoid is a model for a cellular

contractible theory (a coherator).

◮ Canonical coherator: Jn = all parallel pairs in Tn. In T0 we

have the parallel pair s ◦ p, t ◦ q : (1, 0, 1) ⇒ (0). So composition m : (1, 0, 1) → (1) appears in T1.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Cellularity and weak ω-groupoids

◮ A cellular presentation of a globular theory T is a chain of

globular theories Θ0 = T0 → T1 → . . . → Tn . . . → T with colimit T, together with sets Jn of parallel pairs in Tn, such that Tn+1 is obtained from Tn by freely adjoining a lifting for each parallel pair in Jn.

◮ Call T cellular if it admits a cellular presentation. ◮ A Grothendieck weak ω-groupoid is a model for a cellular

contractible theory (a coherator).

◮ Canonical coherator: Jn = all parallel pairs in Tn. In T0 we

have the parallel pair s ◦ p, t ◦ q : (1, 0, 1) ⇒ (0). So composition m : (1, 0, 1) → (1) appears in T1. Associator appears in T2 . . .

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

Cellularity and weak ω-groupoids

◮ A cellular presentation of a globular theory T is a chain of

globular theories Θ0 = T0 → T1 → . . . → Tn . . . → T with colimit T, together with sets Jn of parallel pairs in Tn, such that Tn+1 is obtained from Tn by freely adjoining a lifting for each parallel pair in Jn.

◮ Call T cellular if it admits a cellular presentation. ◮ A Grothendieck weak ω-groupoid is a model for a cellular

contractible theory (a coherator).

◮ Canonical coherator: Jn = all parallel pairs in Tn. In T0 we

have the parallel pair s ◦ p, t ◦ q : (1, 0, 1) ⇒ (0). So composition m : (1, 0, 1) → (1) appears in T1. Associator appears in T2 . . .

◮ Only add liftings – no equalities. Cellularity = weakness.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

What has been done?

◮ Consider D : G → Top sending n to n-disk.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

What has been done?

◮ Consider D : G → Top sending n to n-disk. Induces globular

theory TD with TD(n, m) = Top(D(m), D(n)).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

What has been done?

◮ Consider D : G → Top sending n to n-disk. Induces globular

theory TD with TD(n, m) = Top(D(m), D(n)). TD is contractible since each globular sum D(n) is a contractible space.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

What has been done?

◮ Consider D : G → Top sending n to n-disk. Induces globular

theory TD with TD(n, m) = Top(D(m), D(n)). TD is contractible since each globular sum D(n) is a contractible

• space. Kan construction gives fundamental ω-groupoid

functor ˜ D : Top → Mod(TD) where ˜ DX(n) = Top(D(n), X). (Grothendieck 1983, Maltsiniotis 2010).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

What has been done?

◮ Consider D : G → Top sending n to n-disk. Induces globular

theory TD with TD(n, m) = Top(D(m), D(n)). TD is contractible since each globular sum D(n) is a contractible

• space. Kan construction gives fundamental ω-groupoid

functor ˜ D : Top → Mod(TD) where ˜ DX(n) = Top(D(n), X). (Grothendieck 1983, Maltsiniotis 2010).Very simple!

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

What has been done?

◮ Consider D : G → Top sending n to n-disk. Induces globular

theory TD with TD(n, m) = Top(D(m), D(n)). TD is contractible since each globular sum D(n) is a contractible

• space. Kan construction gives fundamental ω-groupoid

functor ˜ D : Top → Mod(TD) where ˜ DX(n) = Top(D(n), X). (Grothendieck 1983, Maltsiniotis 2010).Very simple!

◮ Each type in dependent type theory with identity types gives

rise to a Batanin ω-groupoid (van den Berg–Garner, LeFanu-Lumsdaine around 2009/2010). More complex than

• above. Easily done using Grothendieck ω-groupoids (Bourke

2016).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

What has been done?

◮ Consider D : G → Top sending n to n-disk. Induces globular

theory TD with TD(n, m) = Top(D(m), D(n)). TD is contractible since each globular sum D(n) is a contractible

• space. Kan construction gives fundamental ω-groupoid

functor ˜ D : Top → Mod(TD) where ˜ DX(n) = Top(D(n), X). (Grothendieck 1983, Maltsiniotis 2010).Very simple!

◮ Each type in dependent type theory with identity types gives

rise to a Batanin ω-groupoid (van den Berg–Garner, LeFanu-Lumsdaine around 2009/2010). More complex than

• above. Easily done using Grothendieck ω-groupoids (Bourke

2016).

◮ Dimitri Ara’s thesis (2010) contains many things. It showed

that Batanin’s globular operads are the same as augmented homogenous globular theories.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

What has been done II?

◮ Guillaume Brunerie (2013) constructed a dependent type

theory with identity types capturing ω-groupoids. Natural counterpart of Grothendieck ω-groupoids. Precise relationship expected to be written up soon.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity

What has been done II?

◮ Guillaume Brunerie (2013) constructed a dependent type

theory with identity types capturing ω-groupoids. Natural counterpart of Grothendieck ω-groupoids. Precise relationship expected to be written up soon.

◮ Recent thesis of Remy Tuyeras relating to model structures

and homotopy hypothesis. Last talk today!

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Algebraic injectivity

◮ Goal now: reformulate Grothendieck weak ω-groupoids as

iterated algebraic injectives.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Algebraic injectivity

◮ Goal now: reformulate Grothendieck weak ω-groupoids as

iterated algebraic injectives.

◮ Algebraic injectivity first arose from Garner’s study of

algebraic weak factorisation systems.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Algebraic injectivity

◮ Goal now: reformulate Grothendieck weak ω-groupoids as

iterated algebraic injectives.

◮ Algebraic injectivity first arose from Garner’s study of

algebraic weak factorisation systems.

◮ We work in context of locally finitely presentable categories: C

l.f.p. and J a set of morphisms between f.p. objects.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Algebraic injectivity

◮ Goal now: reformulate Grothendieck weak ω-groupoids as

iterated algebraic injectives.

◮ Algebraic injectivity first arose from Garner’s study of

algebraic weak factorisation systems.

◮ We work in context of locally finitely presentable categories: C

l.f.p. and J a set of morphisms between f.p. objects.

◮ Objects of Inj(J) are pairs (X, x) where X ∈ C and x is a

lifting function i

α∈J

• f

X

j

x(α,f )

• ✁

✁ ✁ ✁ ✁ ✁ ✁ ✁

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Algebraic injectivity

◮ Goal now: reformulate Grothendieck weak ω-groupoids as

iterated algebraic injectives.

◮ Algebraic injectivity first arose from Garner’s study of

algebraic weak factorisation systems.

◮ We work in context of locally finitely presentable categories: C

l.f.p. and J a set of morphisms between f.p. objects.

◮ Objects of Inj(J) are pairs (X, x) where X ∈ C and x is a

lifting function i

α∈J

• f

X

j

x(α,f )

• ✁

✁ ✁ ✁ ✁ ✁ ✁ ✁

◮ Morphisms of Inj(J) preserve the given liftings.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Some examples of algebraic injectives

◮ Inclusion j : 2 → 3 of 2-element set to 3-element set.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Some examples of algebraic injectives

◮ Inclusion j : 2 → 3 of 2-element set to 3-element set.

2

• (a1,a2) X

3

(a1,a2,x(a1,a2))

• ⑧

⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Some examples of algebraic injectives

◮ Inclusion j : 2 → 3 of 2-element set to 3-element set.

2

• (a1,a2) X

3

(a1,a2,x(a1,a2))

• ⑧

⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Some examples of algebraic injectives

◮ Inclusion j : 2 → 3 of 2-element set to 3-element set.

2

• (a1,a2) X

3

(a1,a2,x(a1,a2))

• ⑧

⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧

◮ Inj(J) is the category Mag of sets equipped with a binary

• peration.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Some examples of algebraic injectives

◮ Inclusion j : 2 → 3 of 2-element set to 3-element set.

2

• (a1,a2) X

3

(a1,a2,x(a1,a2))

• ⑧

⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧

◮ Inj(J) is the category Mag of sets equipped with a binary

• peration.

◮ Algebraic injectivity against sets of inclusions of the form

{jn : n → n + 1} for finite sets n gives exactly the categories

• f Ω-algebras for Ω a signature.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Some examples of algebraic injectives

◮ Inclusion j : 2 → 3 of 2-element set to 3-element set.

2

• (a1,a2) X

3

(a1,a2,x(a1,a2))

• ⑧

⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧

◮ Inj(J) is the category Mag of sets equipped with a binary

• peration.

◮ Algebraic injectivity against sets of inclusions of the form

{jn : n → n + 1} for finite sets n gives exactly the categories

• f Ω-algebras for Ω a signature.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Some examples of algebraic injectives 2

◮ Algebraic injectives in Mag?

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Some examples of algebraic injectives 2

◮ Algebraic injectives in Mag? ◮ Consider terms s = (xy)z and t = x(yz) in free magma F3

and form quotient p : F3 → F3/(s = t).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Some examples of algebraic injectives 2

◮ Algebraic injectives in Mag? ◮ Consider terms s = (xy)z and t = x(yz) in free magma F3

and form quotient p : F3 → F3/(s = t). F3

p

• (a,b,c)

X

F3/(s = t)

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Some examples of algebraic injectives 2

◮ Algebraic injectives in Mag? ◮ Consider terms s = (xy)z and t = x(yz) in free magma F3

and form quotient p : F3 → F3/(s = t). F3

p

• (a,b,c)

X

F3/(s = t)

(ab)c=a(bc)

• t

t t t t t t t t t

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Some examples of algebraic injectives 2

◮ Algebraic injectives in Mag? ◮ Consider terms s = (xy)z and t = x(yz) in free magma F3

and form quotient p : F3 → F3/(s = t). F3

p

• (a,b,c)

X

F3/(s = t)

(ab)c=a(bc)

• t

t t t t t t t t t

◮ Inj({p}) is the category of associative magmas.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Some examples of algebraic injectives 2

◮ Algebraic injectives in Mag? ◮ Consider terms s = (xy)z and t = x(yz) in free magma F3

and form quotient p : F3 → F3/(s = t). F3

p

• (a,b,c)

X

F3/(s = t)

(ab)c=a(bc)

• t

t t t t t t t t t

◮ Inj({p}) is the category of associative magmas. ◮ Algebraic injectivity in Ω-Alg against quotient maps

Fn → Fn/(t = s) gives (Ω, E)-algebras.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Iterated injectivity

◮ Forgetful functor U : Inj(J) → C is finitary monadic so Inj(J)

is l.f.p.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Iterated injectivity

◮ Forgetful functor U : Inj(J) → C is finitary monadic so Inj(J)

is l.f.p.

◮ So can iterate.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Iterated injectivity

◮ Forgetful functor U : Inj(J) → C is finitary monadic so Inj(J)

is l.f.p.

◮ So can iterate. J1 ⊆ Arr(C),

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Iterated injectivity

◮ Forgetful functor U : Inj(J) → C is finitary monadic so Inj(J)

is l.f.p.

◮ So can iterate. J1 ⊆ Arr(C), Jn+1 ⊆ Arr(Inj(Jn)) between f.p.

• bjects for each n ∈ N.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Iterated injectivity

◮ Forgetful functor U : Inj(J) → C is finitary monadic so Inj(J)

is l.f.p.

◮ So can iterate. J1 ⊆ Arr(C), Jn+1 ⊆ Arr(Inj(Jn)) between f.p.

• bjects for each n ∈ N.

◮ Consider chain of forgetful functors on bottom row:

Injω(I)

• ▼

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

U

• ❨

❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨

. . . Inj(In+1) Un

n+1

Inj(In) . . . Inj(I1)

U1

C

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Iterated injectivity

◮ Forgetful functor U : Inj(J) → C is finitary monadic so Inj(J)

is l.f.p.

◮ So can iterate. J1 ⊆ Arr(C), Jn+1 ⊆ Arr(Inj(Jn)) between f.p.

• bjects for each n ∈ N.

◮ Consider chain of forgetful functors on bottom row:

Injω(I)

• ▼

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

U

• ❨

❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨

. . . Inj(In+1) Un

n+1

Inj(In) . . . Inj(I1)

U1

C

◮ We call the cofiltered limit Injω(I) an iterated injectivity

category.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Iterated injectivity

◮ Forgetful functor U : Inj(J) → C is finitary monadic so Inj(J)

is l.f.p.

◮ So can iterate. J1 ⊆ Arr(C), Jn+1 ⊆ Arr(Inj(Jn)) between f.p.

• bjects for each n ∈ N.

◮ Consider chain of forgetful functors on bottom row:

Injω(I)

• ▼

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

U

• ❨

❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨

. . . Inj(In+1) Un

n+1

Inj(In) . . . Inj(I1)

U1

C

◮ We call the cofiltered limit Injω(I) an iterated injectivity

category.

◮ Injω(I) is also l.f.p. and each Un finitary right adjoint.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Iterated injectivity II

◮ Write Un : Inj(In) → C for composite and Fn ⊣ Un.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Iterated injectivity II

◮ Write Un : Inj(In) → C for composite and Fn ⊣ Un. ◮ Let Θ ⊆ Ob(Cf .p.) and J ⊆ Mor(Cf .p.). ◮ Θ=shapes of operations, J =how we can glue.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Iterated injectivity II

◮ Write Un : Inj(In) → C for composite and Fn ⊣ Un. ◮ Let Θ ⊆ Ob(Cf .p.) and J ⊆ Mor(Cf .p.). ◮ Θ=shapes of operations, J =how we can glue. ◮ Form a pushout in Inj(In)

Fni

Fnα

• u

FnX

pu,α

• Fnj

qu,α X(u, α)

for each pair (α, u) where α ∈ J and X ∈ Θ.

◮ If In+1 ⊆ {pu,α : FnX → X(u, α); X ∈ Θ, α ∈ J} then we call

Injω(I) a (Θ, J)-iterated injectivity category.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Iterated injectivity II

◮ Write Un : Inj(In) → C for composite and Fn ⊣ Un. ◮ Let Θ ⊆ Ob(Cf .p.) and J ⊆ Mor(Cf .p.). ◮ Θ=shapes of operations, J =how we can glue. ◮ Form a pushout in Inj(In)

Fni

Fnα

• u

FnX

pu,α

• Fnj

qu,α X(u, α)

for each pair (α, u) where α ∈ J and X ∈ Θ.

◮ If In+1 ⊆ {pu,α : FnX → X(u, α); X ∈ Θ, α ∈ J} then we call

Injω(I) a (Θ, J)-iterated injectivity category.

◮ These are what we are interested in.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Examples of (Θ, J)-iterated injectives

◮ Consider C = Set, Θ = FinSet and J = {∅ → 1, 2 → 1}.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Examples of (Θ, J)-iterated injectives

◮ Consider C = Set, Θ = FinSet and J = {∅ → 1, 2 → 1}. ◮ The squares

∅

!

• 2
• F2

F!

• s,t

F3

p

• 1

3

F1

F3/(s = t)

present associative magmas as a 2-step (Θ, J)-iterated injectivity category.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Examples of (Θ, J)-iterated injectives

◮ Consider C = Set, Θ = FinSet and J = {∅ → 1, 2 → 1}. ◮ The squares

∅

!

• 2
• F2

F!

• s,t

F3

p

• 1

3

F1

F3/(s = t)

present associative magmas as a 2-step (Θ, J)-iterated injectivity category.

◮ More generally each category of the form (Ω, E)-Alg is

naturally a 2-step (Θ, J)-iterated injectivity category.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Examples of (Θ, J)-iterated injectives

◮ Consider C = Set, Θ = FinSet and J = {∅ → 1, 2 → 1}. ◮ The squares

∅

!

• 2
• F2

F!

• s,t

F3

p

• 1

3

F1

F3/(s = t)

present associative magmas as a 2-step (Θ, J)-iterated injectivity category.

◮ More generally each category of the form (Ω, E)-Alg is

naturally a 2-step (Θ, J)-iterated injectivity category. And indeed all (Θ, J)-iterated injectivity category are of this form.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Cellularity and iterated injectivity

◮ C = [Gop, Set], Θ = Θ0 ⊆ [Gop, Set] the skeletal category of

globular cardinals, B = {δn → Yn : n ∈ N} where δn is the boundary of Yn.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Cellularity and iterated injectivity

◮ C = [Gop, Set], Θ = Θ0 ⊆ [Gop, Set] the skeletal category of

globular cardinals, B = {δn → Yn : n ∈ N} where δn is the boundary of Yn.

◮ For example δ1 = {•

• }.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Cellularity and iterated injectivity

◮ C = [Gop, Set], Θ = Θ0 ⊆ [Gop, Set] the skeletal category of

globular cardinals, B = {δn → Yn : n ∈ N} where δn is the boundary of Yn.

◮ For example δ1 = {•

• }.

Theorem

Categories of models of cellular theories and of (Θ0, B)-iterated injectives coincide (up to equivalence over [Gop, Set]).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Cellularity and iterated injectivity

◮ C = [Gop, Set], Θ = Θ0 ⊆ [Gop, Set] the skeletal category of

globular cardinals, B = {δn → Yn : n ∈ N} where δn is the boundary of Yn.

◮ For example δ1 = {•

• }.

Theorem

Categories of models of cellular theories and of (Θ0, B)-iterated injectives coincide (up to equivalence over [Gop, Set]).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Cellularity and iterated injectivity

◮ C = [Gop, Set], Θ = Θ0 ⊆ [Gop, Set] the skeletal category of

globular cardinals, B = {δn → Yn : n ∈ N} where δn is the boundary of Yn.

◮ For example δ1 = {•

• }.

Theorem

Categories of models of cellular theories and of (Θ0, B)-iterated injectives coincide (up to equivalence over [Gop, Set]).

Proof.

Direct and straightforward.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Grothendieck weak ω-groupoids as iterated injectives

Theorem

• 1. Categories of models of cellular theories and of

(Θ0, B)-iterated injectives coincide (up to equivalence over [Gop, Set]).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Grothendieck weak ω-groupoids as iterated injectives

Theorem

• 1. Categories of models of cellular theories and of

(Θ0, B)-iterated injectives coincide (up to equivalence over [Gop, Set]).

• 2. Categories of models of cellular contractible theories coincide

with those categories of (Θ0, B)-iterated injectives U : Injω(I) → [Gop, Set] with the property that the free iterated injective on a globular cardinal is contractible.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Grothendieck weak ω-groupoids as iterated injectives

Theorem

• 1. Categories of models of cellular theories and of

(Θ0, B)-iterated injectives coincide (up to equivalence over [Gop, Set]).

• 2. Categories of models of cellular contractible theories coincide

with those categories of (Θ0, B)-iterated injectives U : Injω(I) → [Gop, Set] with the property that the free iterated injective on a globular cardinal is contractible.

◮ Upshot: study categories of weak ω-groupoids using iterated

injectivity without reference to globular theories.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Nikolaus on algebraic injectives I

Concerned algebraic Kan complexes: Inj(J) for C the category of simplicial sets and J the horn inclusions.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Nikolaus on algebraic injectives I

Concerned algebraic Kan complexes: Inj(J) for C the category of simplicial sets and J the horn inclusions. Nikolaus’ goals:

• 1. Transfer model structure from C along U : Inj(J) → C.
• 2. Show that U a Quillen equivalence.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Nikolaus on algebraic injectives I

Concerned algebraic Kan complexes: Inj(J) for C the category of simplicial sets and J the horn inclusions. Nikolaus’ goals:

• 1. Transfer model structure from C along U : Inj(J) → C.
• 2. Show that U a Quillen equivalence.

Hard parts:

• 1. Show that pushouts of FJ-maps (pushouts of free maps) are

weak equivalences.

• 2. Show that the unit X → UFX a weak equivalence.

Nikolaus’ achieved (1) and (2) by giving elegant constructions of pushouts of free maps and of free injectives.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Nikolaus on algebraic injectives II

◮ Cell(J): transfinite composites of pushouts of maps in J.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Nikolaus on algebraic injectives II

◮ Cell(J): transfinite composites of pushouts of maps in J. ◮ Let us say that U : (A, J) → (B, K) is cellular if U : A → B

takes J-cellular maps to K-cellular maps.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Nikolaus on algebraic injectives II

◮ Cell(J): transfinite composites of pushouts of maps in J. ◮ Let us say that U : (A, J) → (B, K) is cellular if U : A → B

takes J-cellular maps to K-cellular maps.

◮ Following is slight abstraction of Nikolaus’ result.

Theorem (Nikolaus)

Let C be l.f.p and J a set of monos between f.p. objects with Cell(J) ⊆ Mono.

• 1. Then U : (Inj(J), FJ) → (C, J) is cellular.
• 2. The unit ηX : X → UFX is J-cellular.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Nikolaus on algebraic injectives II

◮ Cell(J): transfinite composites of pushouts of maps in J. ◮ Let us say that U : (A, J) → (B, K) is cellular if U : A → B

takes J-cellular maps to K-cellular maps.

◮ Following is slight abstraction of Nikolaus’ result.

Theorem (Nikolaus)

Let C be l.f.p and J a set of monos between f.p. objects with Cell(J) ⊆ Mono.

• 1. Then U : (Inj(J), FJ) → (C, J) is cellular.
• 2. The unit ηX : X → UFX is J-cellular.

As U reflects monos each FJ-cellular map is mono.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Nikolaus on algebraic injectives II

◮ Cell(J): transfinite composites of pushouts of maps in J. ◮ Let us say that U : (A, J) → (B, K) is cellular if U : A → B

takes J-cellular maps to K-cellular maps.

◮ Following is slight abstraction of Nikolaus’ result.

Theorem (Nikolaus)

Let C be l.f.p and J a set of monos between f.p. objects with Cell(J) ⊆ Mono.

• 1. Then U : (Inj(J), FJ) → (C, J) is cellular.
• 2. The unit ηX : X → UFX is J-cellular.

As U reflects monos each FJ-cellular map is mono. So can iterate.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Nikolaus’ theorem iterated

Theorem

Let C be l.f.p and J a set of morphisms between f.p. objects with Cell(J) ⊆ Mono. Consider (Θ, J)-iterated category of injectives.

• 1. Then each forgetful functor in the chain

. . . (Inj(In+1), Fn+1J) Un

n+1

(Inj(In), FnJ) . . . (C, J)

is cellular.

• 2. The unit ηX : X → Un

n+1F n n+1X is FnJ-cellular.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Nikolaus’ theorem iterated

Theorem

Let C be l.f.p and J a set of morphisms between f.p. objects with Cell(J) ⊆ Mono. Consider (Θ, J)-iterated category of injectives.

• 1. Then each forgetful functor in the chain

. . . (Inj(In+1), Fn+1J) Un

n+1

(Inj(In), FnJ) . . . (C, J)

is cellular.

• 2. The unit ηX : X → Un

n+1F n n+1X is FnJ-cellular.

• 3. For all n the FnJ-cellular maps are monic.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Nikolaus’ theorem iterated

Theorem

Let C be l.f.p and J a set of morphisms between f.p. objects with Cell(J) ⊆ Mono. Consider (Θ, J)-iterated category of injectives.

• 1. Then each forgetful functor in the chain

. . . (Inj(In+1), Fn+1J) Un

n+1

(Inj(In), FnJ) . . . (C, J)

is cellular.

• 2. The unit ηX : X → Un

n+1F n n+1X is FnJ-cellular.

• 3. For all n the FnJ-cellular maps are monic.

Moreover, we obtain explicit formulae for free iterated injectives and pushouts of free maps in Injω(I).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Nikolaus’ theorem iterated

Theorem

Let C be l.f.p and J a set of morphisms between f.p. objects with Cell(J) ⊆ Mono. Consider (Θ, J)-iterated category of injectives.

• 1. Then each forgetful functor in the chain

. . . (Inj(In+1), Fn+1J) Un

n+1

(Inj(In), FnJ) . . . (C, J)

is cellular.

• 2. The unit ηX : X → Un

n+1F n n+1X is FnJ-cellular.

• 3. For all n the FnJ-cellular maps are monic.

Moreover, we obtain explicit formulae for free iterated injectives and pushouts of free maps in Injω(I).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

An application

Theorem (Conjecture of Maltsiniotis and Ara)

Let Θ0 = T0 → T1 → . . . → Tn . . . → T be a cellular presentation

• f a globular theory T. Then each functor Tn → Tn+1 is faithful.

Proof.

The corresponding chain of forgetful functors Mod(T)

. . . Mod(Tn+1)

Un

n+1 Mod(Tn) . . .

(Gop, Set)

is a (Θ0, B)-iterated category of injectives.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

An application

Theorem (Conjecture of Maltsiniotis and Ara)

Let Θ0 = T0 → T1 → . . . → Tn . . . → T be a cellular presentation

• f a globular theory T. Then each functor Tn → Tn+1 is faithful.

Proof.

The corresponding chain of forgetful functors Mod(T)

. . . Mod(Tn+1)

Un

n+1 Mod(Tn) . . .

(Gop, Set)

is a (Θ0, B)-iterated category of injectives. Hence unit X → Un

n+1F n n+1X is FnB-cellular (so monic).

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

An application

Theorem (Conjecture of Maltsiniotis and Ara)

Let Θ0 = T0 → T1 → . . . → Tn . . . → T be a cellular presentation

• f a globular theory T. Then each functor Tn → Tn+1 is faithful.

Proof.

The corresponding chain of forgetful functors Mod(T)

. . . Mod(Tn+1)

Un

n+1 Mod(Tn) . . .

(Gop, Set)

is a (Θ0, B)-iterated category of injectives. Hence unit X → Un

n+1F n n+1X is FnB-cellular (so monic). Can identify

Tn(X, Y ) → Tn+1(X, Y ) with Mod(Tn)(FnX, FnY ) → Mod(Tn+1)(F n

n+1FnX, F n n+1FnY )

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

An application

Theorem (Conjecture of Maltsiniotis and Ara)

Let Θ0 = T0 → T1 → . . . → Tn . . . → T be a cellular presentation

• f a globular theory T. Then each functor Tn → Tn+1 is faithful.

Proof.

The corresponding chain of forgetful functors Mod(T)

. . . Mod(Tn+1)

Un

n+1 Mod(Tn) . . .

(Gop, Set)

is a (Θ0, B)-iterated category of injectives. Hence unit X → Un

n+1F n n+1X is FnB-cellular (so monic). Can identify

Tn(X, Y ) → Tn+1(X, Y ) with Mod(Tn)(FnX, FnY ) → Mod(Tn+1)(F n

n+1FnX, F n n+1FnY ) and

Mod(Tn)(FnX, FnY ) → Mod(Tn)(FnX, Un

n+1F n n+1FnY )

postcomposing by the unit.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

An application

Theorem (Conjecture of Maltsiniotis and Ara)

Let Θ0 = T0 → T1 → . . . → Tn . . . → T be a cellular presentation

• f a globular theory T. Then each functor Tn → Tn+1 is faithful.

Proof.

The corresponding chain of forgetful functors Mod(T)

. . . Mod(Tn+1)

Un

n+1 Mod(Tn) . . .

(Gop, Set)

is a (Θ0, B)-iterated category of injectives. Hence unit X → Un

n+1F n n+1X is FnB-cellular (so monic). Can identify

Tn(X, Y ) → Tn+1(X, Y ) with Mod(Tn)(FnX, FnY ) → Mod(Tn+1)(F n

n+1FnX, F n n+1FnY ) and

Mod(Tn)(FnX, FnY ) → Mod(Tn)(FnX, Un

n+1F n n+1FnY )

postcomposing by the unit. Since the unit is monic the last map is injective – as required.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Summary

◮ Grothendieck ω-groupoids: lovely low-tech definition of

ω-groupoid.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Summary

◮ Grothendieck ω-groupoids: lovely low-tech definition of

ω-groupoid.

◮ Can describe using iterated injectivity.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Summary

◮ Grothendieck ω-groupoids: lovely low-tech definition of

ω-groupoid.

◮ Can describe using iterated injectivity. ◮ Building on Nikolaus, this viewpoint has some applications.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Summary

◮ Grothendieck ω-groupoids: lovely low-tech definition of

ω-groupoid.

◮ Can describe using iterated injectivity. ◮ Building on Nikolaus, this viewpoint has some applications. ◮ Gives natural constructions of free iterated injectives and

pushouts of free maps.

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Summary

◮ Grothendieck ω-groupoids: lovely low-tech definition of

ω-groupoid.

◮ Can describe using iterated injectivity. ◮ Building on Nikolaus, this viewpoint has some applications. ◮ Gives natural constructions of free iterated injectives and

pushouts of free maps.

◮ Can it be used to establish Quillen model structure on

ω-groupoids?

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

Summary

◮ Grothendieck ω-groupoids: lovely low-tech definition of

ω-groupoid.

◮ Can describe using iterated injectivity. ◮ Building on Nikolaus, this viewpoint has some applications. ◮ Gives natural constructions of free iterated injectives and

pushouts of free maps.

◮ Can it be used to establish Quillen model structure on

ω-groupoids?

◮ Thanks for listening!

John Bourke Grothendieck ω-groupoids as iterated injectives

Background Grothendieck ω-groupoids Iterated injectivity Grothendieck weak ω-groupoids and iterated injectives Properties of categories of iterated injectives

References

◮ Ara. Sur les ∞-groupoides de Grothendieck et une variante

∞-categorique. (2010).

◮ Bourke. Note on the construction of globular weak

ω-groupoids from types, topological spaces etc. (2016).

◮ Brunerie. Syntactic Grothendieck weak ω-groupoids. (2013). ◮ Grothendieck. Pursuing Stacks. (1983). ◮ LeFanu Lumsdaine. Weak ω-categories from intensional type

• theory. (2010).

◮ Maltsiniotis. Grothendieck ω-groupoids, and still another

definition of ω-categories. (2010).

◮ Nikolaus. Algebraic models for higher categories. (2010). ◮ van den Berg and Garner. Types are weak ω-groupoids.

(2011).

John Bourke Grothendieck ω-groupoids as iterated injectives