Tanning

The term Gerbe is used in algebraic topology for a specific type of stack over a topological space. Sometimes, however, special cases of such tanning are also referred to for simplicity as tanning, for example bundle tanning or Hitchin tanning . An important characterization of tanning is its bond.

Definition: A Gerbe over a topological space is a stack over in the 2 category of Gruppoids , which satisfies the following two Gerben axioms: ${\ displaystyle X}$ ${\ displaystyle {\ mathcal {G}}}$ ${\ displaystyle X}$ ${\ displaystyle {\ mathfrak {Grpd}}}$

There's an object in with . ${\ displaystyle j \ colon U \ to X}$ ${\ displaystyle {\ mathfrak {Cov}} (X)}$ ${\ displaystyle {\ mathcal {G}} (U) \ neq \ emptyset}$
For every object in and two objects , in there is a morphism in and a morphism in . ${\ displaystyle j \ colon U \ to X}$ ${\ displaystyle {\ mathfrak {Cov}} (X)}$ ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle {\ mathcal {G}} (U)}$ ${\ displaystyle i \ colon V \ to U}$ ${\ displaystyle {\ mathfrak {Cov}} (X)}$ ${\ displaystyle {\ mathcal {G}} (i) (A) \ to {\ mathcal {G}} (i) (B)}$ ${\ displaystyle {\ mathcal {G}} (V)}$

The first axiom thus requires that it is non-trivial, while the second axiom represents a certain transitivity condition. ${\ displaystyle {\ mathcal {G}}}$