nLab
functor of descent type
Functors of descent type
Context
Category theory
category theory

Concepts Universal constructions Theorems Extensions Applications
Locality and descent
Functors of descent type
Idea
A functor is of descent type if it satisfies “half” of the condition to be monadic .

Definition
A functor $U:B\to A$ is of descent type if

it has a left adjoint $F:A\to B$ , and
for all $b\in B$ the canonical fork $F U F U b \underoverset{F U \epsilon}{\epsilon F U}{\rightrightarrows} F U b \xrightarrow{\epsilon} b$ is a coequalizer .
The second condition is equivalent to asking that the comparison functor $B \to A^{U F}$ from $B$ to the Eilenberg-Moore category of the monad $U F$ is fully faithful .

Examples
By monadic descent , a morphism $f$ in the base of a fibration is a descent morphism if and only if $f^*$ is of descent type. This is presumably the origin of the terminology; $f$ is an effective descent morphism when $f^*$ is monadic.

Related pages
Last revised on June 26, 2017 at 12:43:17.
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