Stack (category theory)

In algebraic topology is meant by a stack ( English for "stack") one (a certain way) kategorifizierte Garbe . The categorization consists of two steps: the categorization of a coarse and that of the axiom of descent, the fulfillment of which turns a coarse into a sheaf.

For a topological space, let the category whose objects are surjective continuous mappings and whose morphisms are surjective continuous mappings , so that applies. ${\displaystyle X}$ ${\displaystyle {\mathfrak {Cov}}(X)}$ ${\displaystyle j\colon U\to X}$ ${\displaystyle i\colon U_{1}\to U_{2}}$ ${\displaystyle j_{1}=j_{2}\circ i}$

• A preamble over in a category is a contravariant functor . ${\displaystyle X}$ ${\displaystyle {\mathfrak {C}}}$ ${\displaystyle {\mathcal {F}}\colon {\mathfrak {Cov}}(X)\to {\mathfrak {C}}}$

For every morphism and a pullback ${\displaystyle i\colon U\to V}$

${\displaystyle U^{2}}$	${\displaystyle \to }$	${\displaystyle U}$
${\displaystyle \downarrow }$		${\displaystyle \downarrow }$
${\displaystyle U}$	${\displaystyle \to }$	${\displaystyle V}$

you get an induced commuting diagram

${\displaystyle {\mathcal {F}}(V)}$	${\displaystyle \to }$	${\displaystyle {\mathcal {F}}(U)}$
${\displaystyle \downarrow }$		${\displaystyle \downarrow }$
${\displaystyle {\mathcal {F}}(U)}$	${\displaystyle \to }$	${\displaystyle {\mathcal {F}}(U^{2})}$

(with upside down arrows). According to the universal property of a pullback

${\displaystyle D}$	${\displaystyle \to }$	${\displaystyle {\mathcal {F}}(U)}$
${\displaystyle \downarrow }$		${\displaystyle \downarrow }$
${\displaystyle {\mathcal {F}}(U)}$	${\displaystyle \to }$	${\displaystyle {\mathcal {F}}(U^{2})}$

there is a definite morphism in the category . ${\displaystyle \mathrm {Des} (i,{\mathcal {F}}):{\mathcal {F}}(V)\to D}$ ${\displaystyle {\mathfrak {C}}}$

• The axiom of descent for the prawn is: For everyone the morphism is an isomorphism. ${\displaystyle {\mathcal {F}}}$ ${\displaystyle i\colon U\to V}$ ${\displaystyle \mathrm {Des} (i,{\mathcal {F}})}$

One can now consider that these definitions coincide with the more common ones from the article on sheaves . In any case, they allow categorization in a natural way: categories become 2-categories, functors become 2-functors, objects become categories, morphisms become functors, and equations of morphisms become natural equivalences. The category becomes a 2-category in that only identities as 2-morphisms are allowed. ${\displaystyle {\mathfrak {Cov}}(X)}$

This results in the following definitions:

• A fibered category over in a 2-category is a contravariant 2-functor . ${\displaystyle X}$ ${\displaystyle {\mathfrak {C}}}$ ${\displaystyle {\mathcal {F}}:{\mathfrak {Cov}}(X)\to {\mathfrak {C}}}$
• The axiom of descent for a fibered category is: For every 1-morphism the functor is an equivalence of categories. ${\displaystyle {\mathcal {F}}}$ ${\displaystyle i\colon U\to V}$ ${\displaystyle \mathrm {Des} (i,{\mathcal {F}})}$
• A stack is a fibered category that fulfills the axiom of relegation.

Note: A fibered category should actually be called “pre-stack”, but this term is already covered by a slightly different, non-equivalent definition.

literature

• Ieke Moerdijk: Introduction to the language of stacks and gerbes . University of Utrecht, 2002 (English) arxiv : math.AT/0212266