Double Categories

and Pseudo Algebras

Tom Fiore

www.math.uchicago.edu/∼fiore/

Overview

2-categories

[figure]

[figure]

double categories pseudo algebras over the 2-theory of categories

[figure] [figure]

crossed modules

[figure] [figure]

Chronology

1942-1945 Eilenberg-Mac Lane: category theory

1946, 1950 Whitehead-Mac Lane: crossed modules, homotopy 2-types

1963 Ehresmann: double categories Lawvere: Theories

1970’s R. Brown: 2-groups, crossed modules, Van Kampen Theorems

1988 Segal Bourbaki talk: a CFT “is” a cocycle for elliptic cohomology

1991 Mac Lane: coherence in CFT

2002-2005 Fiore, Hu, Kriz: pseudo algebras over theories and 2-theories as a rigorous foundation of CFT

2-Categories

Definition 1 A 2-category C is a category enriched in categories, i.e.

• a set of objects Obj C

• for each object A and B a category Mor_{C}(A, B)

• composition functors

\xymatrix{Mor_{C}(B, C) × Mor_{C}(A, B) \ar[r]|{∘} & Mor_{C}(A, C)}

• identities 1_{A} ∈ Mor_{C}(A, A) which satisfy the usual axioms for a category.

Examples

[figure: A; f_{1}; f_{2}; B; α]

Example 1 Any category is a 2-category with discrete morphism categories.

Example 2 Topological spaces, continuous maps, homotopy classes of homotopies.

Example 3 Categories, functors, and natural transformations form the 2-category Cat.

Example 4 Rings, bimodules, bimodule maps form a bicategory.

Double Categories

Definition 2 (Ehresmann 1963) A double category 𝔻 is an internal category in Cat.

Definition 3 A double category 𝔻 consists of a set of objects, a set of horizontal morphisms, a set of vertical morphisms, and a class of squares with source and target as follows

[figure: A; f; B; A; j; A; f; j; α; B; k; C; C; g; D]

and compositions and units that satisfy axioms.

Compositions and Units for Morphisms in a Double Category

Horizontal:

\xymatrix{A \ar[r]^{f_{1}} & B \ar[r]^{f_{2}} & C} = [f_{1} f_{2}] = f_{2} ∘ f_{1}

\xymatrix{A \ar[r]^{1_{A}^{h}} & A \ar[r]^{f_{1}} & B} = f_{1} = \xymatrix{A \ar[r]^{f_{1}} & B \ar[r]^{1_{B}^{h}} & B}

Vertical:

\xymatrix{A \ar[d]^{j_{1}} \\ B \ar[d]^{j_{2}} \\ C} = [\begin{matrix}j_{1} \\ j_{2} \end{matrix}] = j_{2} ∘ j_{1}

\xymatrix{A \ar[d]^{1_{A}^{v}} \\ A \ar[d]^{j_{1}} \\ B} = \xymatrix{A \ar[d]^{j_{1}} \\ B} = \xymatrix{A \ar[d]^{j_{1}} \\ B \ar[d]^{1_{B}^{v}} \\ B}

Compositions for Squares in a Double Category

Horizontal:

[figure: A; f_{1}; j; α; B; f_{2}; k; β; C; ℓ; D; g_{1}; E; g_{2}; F] = [figure: A; [f_{1}f_{2}]; j; [αβ]; C; ℓ; D; [g_{1}g_{2}]; F]

Vertical:

[figure: A; j_{1}; f; α; B; k_{1}; C; g; β; j_{2}; D; k_{2}; E; h; F] = \xymatrix{A \ar[rr]^{f} & & B \\ _{\left[{j1 \atop j2}\right]} \ar[d] & _{\left[{α \atop β}\right]} & _{\left[{k1 \atop k2}\right]} \ar[d] \\ E \ar[rr]_{h} & & F}

Units for Squares in a Double Category

Horizontal:

\begin{aligned}[figure: A; 1_{A}^{h}; j; i_{j}^{h}; A; f; j; α; B; k; C; 1_{C}^{h}; C; g; D] & = [figure: A; f; j; α; B; k; C; g; D] \\ & = [figure: A; f; j; α; B; k; 1_{B}^{h}; i_{k}^{h}; B; k; C; g; D; 1_{D}^{h}; D] \end{aligned}

Vertical:

[figure: A; f; 1_{A}^{v}; i_{f}^{v}; B; 1_{B}^{v}; A; f; j; α; B; k; C; g; D] = [figure: A; f; j; α; B; k; C; g; D] = [figure: A; f; j; α; B; k; C; g; 1_{C}^{v}; i_{g}^{v}; D; 1_{D}^{v}; C; g; D]

Axioms for a Double Category

All compositions are associative and unital (as above) and

[\begin{matrix}i_{j_{1}}^{h} \\ i_{j_{2}}^{h} \end{matrix} ] = i\left[{j1 \atop j2}\right]^{h}

[i_{f_{1}}^{v} i_{f_{2}}^{v}] = i_{[f_{1}f_{2}]}^{v}.

Interchange Law:

If [figure: α; β; γ; δ], then [\begin{matrix}[ & α β & ] \\ [ & γ δ & ] \end{matrix}] = [[\begin{matrix}α \\ γ \end{matrix}] [\begin{matrix}β \\ δ \end{matrix}]] and

we write [\begin{matrix}α & β \\ γ & δ \end{matrix}] .

Examples of Double Categories

Let I be a 1-category.

□I := double category of commutative squares in I

Obj □I := Obj I Hor □I := Mor I V er □I := Mor I Sq □I := commutative squares in I

\xymatrix{A \ar[d]_{j_{1}} \ar[r]^{f} & B \ar[d]^{k_{1}} \\ C \ar[r]|{g} \ar[d]_{j_{2}} & D \ar[d]^{k_{2}} \\ E \ar[r]_{h} & F} = \xymatrix{A \ar[r]^{f} \ar[d]_{j_{2}∘j_{1}} & B \ar[d]^{k_{2}∘k_{1}} \\ E \ar[r]_{h} & F}

⊡I := double category of not necessarily commutative squares in I Every 2-category C is a double category with trivial vertical morphisms.

[figure: A; f; 1_{A}^{v}; α; B; 1_{B}^{v}; A; g; B] := [figure: A; f; g; B; α]

Definition 4 The horizontal 2-category H𝔻 of a double category 𝔻 has objects Obj 𝔻, morphisms Hor 𝔻, and 2-cells

[figure: A; f; g; B; α] := [figure: A; f; 1_{A}^{v}; α; B; 1_{B}^{v}; A; g; B] .

Let C be a 2-category.

ℚC := Ehresmann’s double category of quintets in C (1963)

Obj ℚC := Obj C Hor ℚC := Mor C V er ℚC := Mor C

Sq ℚC := [figure: {; A; f; j; α; B; k; C; g; D; |; A; k∘f; g∘j; D; α; }]

Theorem 1 (Grandis-Paré 2004) The functor \xymatrix{ℚ : 2-Cat \ar[r] & Dbl} admits a right adjoint.

ℝng:= pseudo double category or rings, bimodules, and equivariant maps

Obj ℝng := rings with identity Hor ℝng := bimodules V er ℝng := homomorphisms of rings Sq ℝng :=

[figure: R; M; j; α; S; k; T; N; U] |

\xymatrix{α : M \ar[r] & N} group homomorphism
α(smr) = k(s)α(m)j(r)

Examples of Double Categories

Let C be a topological category, i.e. Obj C and Mor C are topological spaces. ℙ′C:= double category of Moore paths on C. Obj ℙ′C := Obj C Hor ℙ′C := Mor C V er ℙ′C := P′(Obj C) = Moore paths in Obj C Sq ℙ′C := P′(Mor C) = Moore paths in Mor C

P′X := \{(w, s) : s ≥ 0, \xymatrix{w : [0, s] \ar[r] & X}\}

A worldsheet is a real, compact, not necessarily connected, two dimensional, smooth manifold with complex structure and real analytically parametrized boundary components.

𝕎:= pseudo double category of worldsheets

Obj 𝕎 := finite sets Hor 𝕎(A, B) := worldsheets with inbound components labelled by A and outbound components by B V er 𝕎 := bijections of finite sets Sq 𝕎 :=

[figure: A; x; j; α; B; k; C; y; D]

|

\xymatrix{α : x \ar[r] & y} holomorphic diffeo.

α compatible with j and k α preserves boundary params.

Folding Structures

We introduce folding structures to compare algebras over the 2-theory of categories with double categories.

Definition 5 A holonomy on a double category 𝔻 is a 2-functor

\xymatrix{(V𝔻)_{0} \ar[r] & H𝔻}
\xymatrix{A \ar@{|->}[r] & \overline{A} = A}

[figure: A; j; B; A; \overline{j}; B]

Example 5 For a topological category C, a holonomy

[figure: (Vℙ′C)_{0}; Hℙ′C]

assigns to a path of objects a morphism from the initial point to the terminal point, like in differential geometry.

Folding Structures

Definition 6 A folding structure on a double category 𝔻 consists of a holonomy \xymatrix{j \ar@{|->}[r] & \overline{j}} and bijections

[figure: A; f; j; α; B; k; C; g; D; Λ; A; [f\overline{k}]; 1_{A}^{v}; Λ(α); D; 1_{D}^{v}; A; [\overline{j}g]; D]

compatible with compositions and units.

A folding structure horizontalizes a double category.

Examples of Folding Structures

Let I be a 1-category. □I=double category of commutative squares in I ⊡I=double category of not necessarily commutative squares in I

Then □I and ⊡I each admit a unique folding structure.

[figure: A; f; j; B; k; C; g; D; Λ; A; k∘f; 1_{A}^{v}; D; 1_{D}^{v}; A; g∘j; D]

Examples of Folding Structures

Let C be a 2-category. Then ℚC admits a folding structure by definition. [figure: A; f; j; α; B; k; C; g; D] = [figure: A; k∘f; g∘j; D; α] = [figure: A; k∘f; 1_{A}^{v}; α; D; 1_{D}^{v}; A; g∘j; D]

Examples of Folding Structures

ℝng:= pseudo double category or rings, bimodules, and equivariant maps

Obj ℝng := rings with identity Hor ℝng := bimodules V er ℝng := homomorphisms of rings Sq ℝng :=

[figure: R; M; j; α; S; k; T; N; U] |

\xymatrix{α : M \ar[r] & N} group homomorphism
α(smr) = k(s)α(m)j(r)

Holonomy:

\overline{j} := T_{j} = the (T, R)-module T

Folding:

\xymatrix{Λ(α) : U_{k} ⨂_{S} M \ar@{=>}[r] & N ⨂_{T} T_{j}}
\xymatrix{u ⊗ m \ar@{|->}[r] & (u · α(m)) ⊗ 1_{T}}

Examples of Folding Structures

𝕎:= pseudo double category of worldsheets

Obj 𝕎 := finite sets Hor 𝕎(A, B) := worldsheets with inbound components labelled by A and outbound components by B V er 𝕎 := bijections of finite sets Sq 𝕎 :=

[figure: A; x; j; α; B; k; C; y; D]

|

\xymatrix{α : x \ar[r] & y} holomorphic diffeo.

α compatible with j and k α preserves boundary params.

Holonomy:

bijection

[figure]

labelled union of infinitely thin annuli

Folding:

relabel x and y

Comparison Theorems

Observation 2 (Brown-Mosa 1999, F. 2006) The notions of folding structure and connection pair are equivalent.

Theorem 3 (F. 2006) The 2-category of strict 2-algebras over the 2-theory of categories is 2-equivalent to the 2-category of double categories with folding structures and invertible vertical morphisms.

The pseudo version of the theorem also holds.

Strict 2-Algebras

Definition 7 A strict 2-algebra over the 2-theory of of categories is a groupoid I and a strict 2-functor \xymatrix{X : I^{2} \ar[r] & Cat} with strictly 2-natural functors

\xymatrix{X_{B,C} × X_{A,B} \ar[r]|{∘} & X_{A,C}}
\xymatrix{\{∗\} \ar[r]^{1_{B}} & X_{B,B}}

for all A, B, C ∈ I which satisfy axioms like those of a category.

Towards Strict 2-Algebras and Crossed Modules

Consider one object cases.

groupoids ⊆ 2-groupoids ⊆ double groupoids
groups  ⊆ 2-groups  ⊆ double groups

Theorem 4 (Verdier, Brown-Spencer 1976,...) 2-groups are equivalent to crossed modules.

Theorem 5 (Brown-Spencer 1976) Edge symmetric double groups with folding structure with trivial holonomy are equivalent to crossed modules.

Question: What is a one object strict 2-algebra over the 2-theory of categories with everything iso?

Crossed Modules

Definition 8 A crossed module is a group homomorphism \xymatrix{∂ : H \ar[r] & G} with a left action

\xymatrix{G × H \ar[r] & H}
(g, α) ↦ ^{g}α

by automorphisms such that:

1. ∂(^{g}α) = g∂(α)g^{−1} for all α ∈ H and g ∈ G 2. ^{∂(α)}α_{1} = αα_{1}α^{−1} for all α, α_{1} ∈ H. Example 6 H ◁ G is a crossed module

Crossed Modules

Example 7 Let (X, A, ∗) be a based pair of spaces. Then

\xymatrix{∂ : π_{2}(X, A, ∗) \ar[r] & π_{1}(A, ∗)}

is a crossed module.

Theorem 6 (Mac Lane-Whitehead 1950) Crossed modules model homotopy 2-types via this example.

Crossed Modules with Group Action

Definition 9 Let \xymatrix{∂ : H \ar[r] & G} be a crossed module and I a group. An action of I on the crossed module \xymatrix{∂ : H \ar[r] & G} consists of

• a left action of I on H by automorphisms written (j, α) ↦ ^{j}α

• a left action of I on G as a set written (j, g) ↦ jg

• a right action of I on G as a set written (g, j) ↦ gj

Crossed Modules with Group Action

These actions satisfy the following axioms for all j, k ∈ I, α ∈ H, g, g_{1}, g_{2} ∈ G.

1. (jg)k = j(gk)

2. (jg_{1})g_{2} = j(g_{1}g_{2})

3. (g_{1}g_{2})k = g_{1}(g_{2}k)

4. (g_{1}j)g_{2} = g_{1}(jg_{2})

5. j1_{G} = 1_{G}j

6. ^{(gk)}α = ^{g}(^{k}α)

7. ^{(jg)}α = ^{j}(^{g}α)

8. ∂(^{j}α) = j∂(α)j^{−1}

Strict 2-Algebras and Crossed Modules

Theorem 7 (F. 2006) One object strict 2algebras over the 2-theory of categories with everything iso are equivalent to crossed modules with a group action.

In the case of a trivial group and trivial I, this says 2-groups are equivalent to crossed modules.

Conclusion

2-categories

[figure]

[figure]

double categories pseudo algebras over the 2-theory of categories

[figure] [figure]

crossed modules

[figure] [figure]