PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The fundamental group is a functor $π_{1} : {T o p}_{*} \to G r p$

Statement

Let ${T o p}_{*}$ have pointed spaces $(X, x_{0})$ as objects and basepoint-preserving continuous maps as morphisms. The assignment

$π_{1} : {T o p}_{*} ⟶ G r p, (X, x_{0}) ⟼ π_{1} (X, x_{0})$

and $f \mapsto f_{*}$ is a functor.

Facts & Assumptions

Given: Pointed spaces and basepoint-preserving continuous maps.

[L1]

Spaces and continuous maps form $T o p$ (Topological spaces and continuous maps form the large locally small category $T o p$ ), and groups and homomorphisms form $G r p$ (Groups and group homomorphisms form the large locally small category $G r p$ ).

[L2]

The induced map $f_{*}$ on fundamental groups is the homomorphism of The homomorphism on fundamental groups induced by a pointed continuous map, and induced maps satisfy $(g \circ f)_{*} = g_{*} \circ f_{*}$ and $(1_{X})_{*} = 1_{π_{1} (X, x_{0})}$ (Induced fundamental-group maps are well defined, functorial and invariant under based homotopy).

[L3]

A functor preserves identities and composition (Covariant functor, identity functor, composite functor, and contravariant functor).

Proof

technique · direct

1.1

Basepoint-preserving continuous maps contain identities and are closed under composition, so they form the stated pointed category.

givenL1

2.1

By [L2], every morphism $f$ is sent to a group homomorphism $f_{*}$ , and the identity and composite equations required in [L3] hold.

step 1.1L2L3

3.1

Therefore the object and morphism assignments define a functor $π_{1} : {T o p}_{*} \to G r p$ .

step 2.1L1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)

The fundamental group is a functor π1:Top∗→Grpπ1​:Top∗​→Grp