Definition 1 (via groupoid fibrations)

Recall that a category fibred in groupoids (also called a groupoid fibration) consists of a category ${\displaystyle {\mathcal {C}}}$ together with a functor ${\displaystyle \pi$ :{\mathcal {C}}\to \mathrm {Mfd} } to the category of differentiable manifolds such that

1. ${\displaystyle {\mathcal {C}}}$ is a fibred category, i.e. for any object ${\displaystyle u}$ of ${\displaystyle {\mathcal {C}}}$ and any arrow ${\displaystyle V\to U}$ of ${\displaystyle \mathrm {Mfd} }$ there is an arrow ${\displaystyle v\to u}$ lying over ${\displaystyle V\to U}$;
2. for every commutative triangle ${\displaystyle W\to V\to U}$ in ${\displaystyle \mathrm {Mdf} }$ and every arrows ${\displaystyle w\to u}$ over ${\displaystyle W\to U}$ and ${\displaystyle v\to u}$ over ${\displaystyle V\to U}$, there exists a unique arrow ${\displaystyle w\to v}$ over ${\displaystyle W\to V}$ making the triangle ${\displaystyle w\to v\to u}$ commute.

These properties ensure that, for every object ${\displaystyle U}$ in ${\displaystyle \mathrm {Mfd} }$, one can define its fibre, denoted by ${\displaystyle \pi ^{-1}(U)}$ or ${\displaystyle {\mathcal {C}}_{U}}$, as the subcategory of ${\displaystyle {\mathcal {C}}}$ made up by all objects of ${\displaystyle {\mathcal {C}}}$ lying over ${\displaystyle U}$ and all morphisms of ${\displaystyle {\mathcal {C}}}$ lying over ${\displaystyle id_{U}}$. By construction, ${\displaystyle \pi ^{-1}(U)}$ is a groupoid, thus explaining the name. A stack is a groupoid fibration satisfied further glueing properties, expressed in terms of descent.

Any manifold ${\displaystyle X}$ defines its slice category ${\displaystyle F_{X}=\mathrm {Hom} _{\mathrm {Mdf} }(-,X)}$, whose objects are pairs ${\displaystyle (U,f)}$ of a manifold ${\displaystyle U}$ and a smooth map ${\displaystyle f:U\to X}$; then ${\displaystyle F_{X}\to \mathrm {Mdf} ,(U,f)\mapsto U}$ is a groupoid fibration which is actually also a stack. A morphism ${\displaystyle {\mathcal {C}}\to {\mathcal {D}}}$ of groupoid fibrations is called a representable submersion if

• for every manifold ${\displaystyle U}$ and any morphism ${\displaystyle F_{U}\to {\mathcal {D}}}$, the fibred product ${\displaystyle {\mathcal {C}}\times _{\mathcal {D}}F_{U}}$ is representable, i.e. it is isomorphic to ${\displaystyle F_{V}}$ (for some manifold ${\displaystyle Y}$) as groupoid fibrations;
• the induce smooth map ${\displaystyle V\to U}$ is a submersion.

A differentiable stack is a stack ${\displaystyle \pi$ :{\mathcal {C}}\to \mathrm {Mfd} } together with a special kind of representable submersion ${\displaystyle F_{X}\to {\mathcal {C}}}$ (every submersion ${\displaystyle V\to U}$ described above is asked to be surjective), for some manifold ${\displaystyle X}$. The map ${\displaystyle F_{X}\to {\mathcal {C}}}$ is called atlas, presentation or cover of the stack ${\displaystyle X}$.^[5][6]

Definition 2 (via 2-functors)

Recall that a prestack (of groupoids) on a category ${\displaystyle {\mathcal {C}}}$, also known as a 2-presheaf, is a 2-functor ${\displaystyle X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp} }$, where ${\displaystyle \mathrm {Grp} }$ is the 2-category of (set-theoretical) groupoids, their morphisms, and the natural transformations between them. A stack is a prestack satisfying further glueing properties (analogously to the glueing properties satisfied by a sheaf). In order to state such properties precisely, one needs to define (pre)stacks on a site, i.e. a category equipped with a Grothendieck topology.

Any object ${\displaystyle M\in \mathrm {Obj} ({\mathcal {C}})}$ defines a stack ${\displaystyle {\underline {M}}:=\mathrm {Hom} _{\mathcal {C}}(-,M)}$, which associated to another object ${\displaystyle N\in \mathrm {Obj} ({\mathcal {C}})}$ the groupoid ${\displaystyle \mathrm {Hom} _{\mathcal {C}}(N,M)}$ of morphisms from ${\displaystyle N}$ to ${\displaystyle M}$. A stack ${\displaystyle X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp} }$ is called geometric if there is an object ${\displaystyle M\in \mathrm {Obj} ({\mathcal {C}})}$ and a morphism of stacks ${\displaystyle {\underline {M}}\to X}$ (often called atlas, presentation or cover of the stack ${\displaystyle X}$) such that

• the morphism ${\displaystyle {\underline {M}}\to X}$ is representable, i.e. for every object ${\displaystyle Y}$ in ${\displaystyle {\mathcal {C}}}$ and any morphism ${\displaystyle Y\to X}$ the fibred product ${\displaystyle {\underline {M}}\times _{X}{\underline {Y}}}$ is isomorphic to ${\displaystyle {\underline {Z}}}$ (for some object ${\displaystyle Z}$) as stacks;
• the induces morphism ${\displaystyle Z\to Y}$ satisfies a further property depending on the category ${\displaystyle {\mathcal {C}}}$ (e.g., for manifold it is asked to be a submersion).

A differentiable stack is a stack on ${\displaystyle {\mathcal {C}}=\mathrm {Mfd} }$, the category of differentiable manifolds (viewed as a site with the usual open covering topology), i.e. a 2-functor ${\displaystyle X:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }$, which is also geometric, i.e. admits an atlas ${\displaystyle {\underline {M}}\to X}$ as described above.^[7][8]

Note that, replacing ${\displaystyle \mathrm {Mfd} }$ with the category of affine schemes, one recovers the standard notion of algebraic stack. Similarly, replacing ${\displaystyle \mathrm {Mfd} }$ with the category of topological spaces, one obtains the definition of topological stack.

Definition 3 (via Morita equivalences)

Recall that a Lie groupoid consists of two differentiable manifolds ${\displaystyle G}$ and ${\displaystyle M}$, together with two surjective submersions ${\displaystyle s,t:G\to M}$, as well as a partial multiplication map ${\displaystyle m:G\times _{M}G\to G}$, a unit map ${\displaystyle u:M\to G}$, and an inverse map ${\displaystyle i:G\to G}$, satisfying group-like compatibilities.

Two Lie groupoids ${\displaystyle G\rightrightarrows M}$ and ${\displaystyle H\rightrightarrows N}$ are Morita equivalent if there is a principal bi-bundle ${\displaystyle P}$ between them, i.e. a principal right ${\displaystyle H}$-bundle ${\displaystyle P\to M}$, a principal left ${\displaystyle G}$-bundle ${\displaystyle P\to N}$, such that the two actions on ${\displaystyle P}$ commutes. Morita equivalence is an equivalence relation between Lie groupoids, weaker than isomorphism but strong enough to preserve many geometric properties.

A differentiable stack, denoted as ${\displaystyle [M/G]}$, is the Morita equivalence class of some Lie groupoid ${\displaystyle G\rightrightarrows M}$.^[5][9]

Equivalence between the definitions 1 and 2

Any fibred category ${\displaystyle {\mathcal {C}}\to \mathrm {Mdf} }$ defines the 2-sheaf ${\displaystyle X:\mathrm {Mdf} ^{opp}\to \mathrm {Grp} ,U\mapsto \pi ^{-1}(U)}$. Conversely, any prestack ${\displaystyle X:\mathrm {Mdf} ^{\text{opp}}\to \mathrm {Grp} }$ gives rise to a category ${\displaystyle {\mathcal {C}}}$, whose objects are pairs ${\displaystyle (U,x)}$ of a manifold ${\displaystyle U}$ and an object ${\displaystyle x\in X(U)}$, and whose morphisms are maps ${\displaystyle \phi$ :(U,x)\to (V,y)} such that ${\displaystyle X(\phi )(y)=x}$. Such ${\displaystyle {\mathcal {C}}}$ becomes a fibred category with the functor ${\displaystyle {\mathcal {C}}\to \mathrm {Mdf} ,(U,x)\mapsto U}$.

The glueing properties defining a stack in the first and in the second definition are equivalent; similarly, an atlas in the sense of Definition 1 induces an atlas in the sense of Definition 2 and vice versa.^[5]

Equivalence between the definitions 2 and 3

Every Lie groupoid ${\displaystyle G\rightrightarrows M}$ gives rise to the differentiable stack ${\displaystyle BG:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }$, which sends any manifold ${\displaystyle N}$ to the category of ${\displaystyle G}$-torsors on ${\displaystyle N}$ (i.e. ${\displaystyle G}$-principal bundles). Any other Lie groupoid in the Morita class of ${\displaystyle G\rightrightarrows M}$ induces an isomorphic stack.

Conversely, any differentiable stack ${\displaystyle X:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }$ is of the form ${\displaystyle BG}$, i.e. it can be represented by a Lie groupoid. More precisely, if ${\displaystyle {\underline {M}}\to X}$ is an atlas of the stack ${\displaystyle X}$, then one defines the Lie groupoid ${\displaystyle G_{X}:=M\times _{X}M\rightrightarrows M}$ and checks that ${\displaystyle BG_{X}}$ is isomorphic to ${\displaystyle X}$.

A theorem by Dorette Pronk states an equivalence of bicategories between differentiable stacks according to the first definition and Lie groupoids up to Morita equivalence.^[10]

Quotient differentiable stack

Given a Lie group action ${\displaystyle a:M\times G\to M}$ on ${\displaystyle M}$, its quotient (differentiable) stack is the differential counterpart of the quotient (algebraic) stack in algebraic geometry. It is defined as the stack ${\displaystyle [M/G]}$ associating to any manifold ${\displaystyle X}$ the category of principal ${\displaystyle G}$-bundles ${\displaystyle P\to X}$ and ${\displaystyle G}$-equivariant maps ${\displaystyle \phi :P\to M}$. It is a differentiable stack presented by the stack morphism ${\displaystyle {\underline {M}}\to [M/G]}$ defined for any manifold ${\displaystyle X}$ as

$${\displaystyle {\underline {M}}(X)=\mathrm {Hom} (X,M)\to [M/G](X),\quad f\mapsto (X\times G\to X,\phi _{f})}$$

where ${\displaystyle \phi _{f}:X\times G\to M}$ is the ${\displaystyle G}$-equivariant map ${\displaystyle \phi _{f}=a\circ (f\circ \mathrm {pr} _{1},\mathrm {pr} _{2}):(x,g)\mapsto f(x)\cdot g}$.^[7]

The stack ${\displaystyle [M/G]}$ corresponds to the Morita equivalence class of the action groupoid ${\displaystyle M\times G\rightrightarrows M}$. Accordingly, one recovers the following particular cases:

• if ${\displaystyle M}$ is a point, the differentiable stack ${\displaystyle [M/G]}$ coincides with ${\displaystyle BG}$
• if the action is free and proper (and therefore the quotient ${\displaystyle M/G}$ is a manifold), the differentiable stack ${\displaystyle [M/G]}$ coincides with ${\displaystyle {\underline {M/G}}}$
• if the action is proper (and therefore the quotient ${\displaystyle M/G}$ is an orbifold), the differentiable stack ${\displaystyle [M/G]}$ coincides with the stack defined by the orbifold