nLab locally cartesian (infinity,1)-category

Redirected from "locally cartesian (∞,1)-category".

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Definition

An ( , 1 ) (\infty, 1) -category CC is locally cartesian if each of its slice ( , 1 ) (\infty, 1) -categories C/xC/x is a cartesian monoidal ( , 1 ) (\infty, 1) -category, meaning that C/xC/x has all finite products. Another way to say this is that CC has all (,1)(\infty, 1)-pullbacks.

Properties

In any locally cartesian (,1)(\infty, 1)-category, the base change ( , 1 ) (\infty, 1) -functor has a left adjoint ( , 1 ) (\infty, 1) -functor called the dependent sum.

The internal logic of a locally cartesian (,1)(\infty, 1)-category is expected to be a dependent type theory with identity types and dependent sum types.

 Examples

Created on March 8, 2024 at 05:14:23. See the history of this page for a list of all contributions to it.