Tau-Exponential

The tau-exponential E_tau: HolEnd(tau) -> HolEnd(tau) sends A to [A, A]. It is sigma-equivariant: E_tau(sigma(A)) = sigma(E_tau(A)) for the involution sigma.

Tau-Exponential

Summary

The tau-exponential E_tau: HolEnd(tau) -> HolEnd(tau) sends A to [A, A]. It is sigma-equivariant: E_tau(sigma(A)) = sigma(E_tau(A)) for the involution sigma.

Statement

%
\label{def:tau-exponential}
The \textbf{tau-exponential}
is the endofunctor
\[
    \boxed{%
    E_\tau : \mathrm{Obj}(\tau) \;\longrightarrow\; \mathrm{Obj}(\tau),
    \qquad
    A \;\longmapsto\; [A, A],}
\]
where $[A, A]$ is the Hom object
(Definition~\ref{def:hom-object}, II.D54)
--- the internal endomorphism space of~$A$.
On morphisms, $E_\tau$ acts by conjugation:
for $\varphi \colon A \to B$,
\[
    E_\tau(\varphi) \colon [A, A] \to [B, B],
    \qquad
    f \mapsto \varphi \circ f \circ \varphi^{-1},
\]
whenever $\varphi$ is invertible.
For non-invertible $\varphi$,
$E_\tau(\varphi)$ is defined via postcomposition:
$f \mapsto \varphi \circ f$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-02.jsonl line 218
• Manuscript source: 2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch43-yoneda-theorem.tex lines 987-1011

Lean / Formalization Notes

• Formalization: planned
• Module: None
• Name: None

Dependencies

• Canonical: II.P21, II.D53

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.