#
nLab

fibrant replacement

Contents
### 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 rational $\infty$-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**.

If the factorization is functorial, then it yields a **fibrant replacement functor**. See at *functorial factorization*.

Last revised on June 4, 2020 at 02:02:13.
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