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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Vector spaces over a fixed field and linear maps form the large locally small category ${V e c t}_{F}$

Statement

For a fixed field $F$ , vector spaces over $F$ and linear maps form a large locally small category ${V e c t}_{F}$ .

Facts & Assumptions

Given: $F$ -vector spaces $U, V, W$ and linear maps $S : U \to V$ , $T : V \to W$ .

[L1]

The vector-space axioms are those of Vector space over a field, and a linear map preserves vector addition and scalar multiplication (Linear map between vector spaces over the same field).

[L2]

The functions $A \to B$ between fixed sets form the set $B^{A}$ (The set $B^{A}$ of all functions $A \to B$ ); category size is governed by Category, object, morphism, domain, codomain, identity, composition, and hom-collection and Small, locally small, and large categories, and the ordinals form a proper class (Burali-Forti: there is no set of all ordinals).

Proof

technique · direct

1.1

Identity maps preserve addition and scalar multiplication, and $T \circ S$ does so by applying linearity first to $S$ and then to $T$ ; composition is associative and unital as function composition.

givenL1

2.1

Hence $F$ -vector spaces and linear maps form a category, and every hom-collection is a set of functions.

step 1.1L2

3.1

Each singleton ${α}$ , for an ordinal $α$ , carries a transported zero-dimensional $F$ -vector-space structure; these distinct objects and [L2] show that ${V e c t}_{F}$ is large and locally small.

step 2.1L1L2∎

Depends on

Used by

Dependency tree · two levels

19 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)

Vector spaces over a fixed field and linear maps form the large locally small category VectFVectF​