# The Fundamental Infinity-Groupoid of a Parametrized Family

## TL;DR

This paper develops a homotopy theory for parametrized families of objects in infinity-categories, generalizing classical space homotopy theory and introducing a Grothendieck topology on Fam(C).

## Contribution

It introduces a homotopy-theoretic framework for parametrized families in infinity-categories and establishes a Grothendieck topology generalizing known topologies on infinity-groupoids.

## Key findings

- Fam(C) admits a Grothendieck topology extending the canonical topology.
- Homotopy-theoretic constructions from fundamental infinity-groupoids are developed.
- The framework generalizes classical homotopy theory of spaces.

## Abstract

Given an infinity-category C, one can naturally construct an infinity-category Fam(C) of families of objects in C indexed by infinity-groupoids. An ordinary categorical version of this construction was used by Borceux and Janelidze in the study of generalized covering maps in categorical Galois theory. In this paper, we develop the homotopy theory of such "parametrized families" as generalization of the classical homotopy theory of spaces. In particular, we study homotopy-theoretical constructions that arise from the fundamental infinity-groupoids of families in an infinity-category. In the same spirit, we show that Fam(C) admits a Grothendieck topology which generalizes the canonical/epimorphism topology on the infinity-topos of infinity-groupoids in the sense of Carchedi.