In mathematics, a pseudofunctor ''F'' is a mapping between
2-categories, or from a
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
to a
2-category
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of catego ...
, that is just like a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
except that
and
do not hold as exact equalities but only up to ''
coherent isomorphism
In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold " up to homotopy" or "up to isomorphism".
The adjectives such as "pseudo-" and "lax- ...
s''.
The
Grothendieck construction The Grothendieck construction (named after Alexander Grothendieck) is a construction used in the mathematical field of category theory.
Definition
Let F\colon \mathcal \rightarrow \mathbf be a functor from any small category to the category of sma ...
associates to a pseudofunctor a
fibered category
Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which ''inverse images'' (or ''pull-backs'') of ...
.
See also
*
Lax functor
In category theory, a discipline within mathematics, the notion of lax functor between bicategories generalizes that of functors between categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, ca ...
*
Prestack
In algebraic geometry, a prestack ''F'' over a category ''C'' equipped with some Grothendieck topology is a category together with a functor ''p'': ''F'' → ''C'' satisfying a certain lifting condition and such that (when the fibers are groupoids ...
(an example of pseudofunctor)
*
Fibered category
Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which ''inverse images'' (or ''pull-backs'') of ...
References
*C. Sorger
Lectures on moduli of principal G-bundles over algebraic curves
External links
*http://ncatlab.org/nlab/show/pseudofunctor
Functors
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