#
nLab

fibrant replacement

### Context

#### Model category theory

**model category**

## Definitions

## Morphisms

## Universal constructions

## Refinements

## Producing new model structures

## Presentation of $(\infty,1)$-categories

## Model structures

### for $\infty$-groupoids

for ∞-groupoids

### for $n$-groupoids

### for $\infty$-groups

### for $\infty$-algebras

#### general

#### specific

### for stable/spectrum objects

### for $(\infty,1)$-categories

### for stable $(\infty,1)$-categories

### for $(\infty,1)$-operads

### for $(n,r)$-categories

### for $(\infty,1)$-sheaves / $\infty$-stacks

# Contents

## Definition

In a model category every morphism may be factored as a weak equivalence followed by a fibration. Specifically if the morphism is that to the terminal object, this process finds a weakly equivalent fibrant object. This is a *fibrant replacement* or resolution of the original object.

The dual concept is called *cofibrant replacement*.