Refocusing

Formal definition

A Refocusing is a morphism $\mathfrak{R} : (H, t^*) \to (H', t')$ in the category of pointed presheaves:

$$\mathfrak{R} = (\rho : H \to H',\; \phi : t^* \leadsto t')$$

where:

• $(H, t^*)$ is the source MacroFocus — the presheaf $H : T^{\mathrm{op}} \to \mathbf{HA}_{\mathrm{nucl}}$ with designated history $t^*$
• $(H', t')$ is the target MacroFocus — the presheaf $H'$ (possibly equal to $H$) with the new designated history $t'$
• $\rho : H \to H'$ is a presheaf morphism — a natural transformation between the ambient presheaves; when $H' = H$, this is $\mathrm{id}_H$
• $\phi : t^* \leadsto t'$ is the history reassignment — the record of which history is now focal; in the forward extension case, $\phi$ is the step morphism $s \star t^* = t'$; in the backward and lateral cases, $\phi$ is a morphism in $T$ or a choice from $T$

Three invariants. $\mathfrak{R}$ is a refocusing iff:

1. Presheaf coherence: if $\rho \neq \mathrm{id}_H$, then $\rho$ is a natural transformation — every component $\rho_t : H_t \to H'_t$ commutes with the restriction maps; $\rho_{t} \circ H'(f) = H(f) \circ \rho_{t'}$ for all $f : t \to t'$. The nuclei are automatically preserved (Natural Transformation).

2. Fiber availability: $H'_{t'} \in \mathbf{HA}_{\mathrm{nucl}}$ — the target fiber at $t'$ is a Heyting algebra with commuting nuclear pair; the designated position $t'$ is a valid focal position in $H'$.

3. Field reconstitution: the Gurwitsch tripartition around $t'$ in $H'$ is fully determined by the refocusing — a refocusing is not just a pointer change but a full reconstitution of Theme ($H'_{t'}$), Thematic Field ($\{H'_t : t < t'\}$), and Margin ($\{H'_t : t \not\leq t'\}$). The new field is determined by the target position, not inherited from the source.

The three cases

Forward extension: $(H, t^*) \to (G_s H, s \star t^*)$

• $H' = G_s H$ where $G_s$ is the directed comonad (Passage): $(G_s H)(t) = H(s \star t)$
• $t' = s \star t^*$ — the history extended by generator $s$
• $\rho = \gamma$ — the carrier stepping map $\gamma_{t^*} : H_{t^*} \to H_{s \star t^*}$ (the covariant forward map of the Carrier)
• Effect: the old theme $H_{t^*}$ moves into the new thematic field (since $t^* < s \star t^*$); the new theme $H_{s \star t^*}$ is a fresh fiber above the old one

This is the RelationalMachine’s step operation. It is the only refocusing case that changes $H$ — the other two cases preserve $H$ and change only the pointer.

Backward restriction: $(H, t^*) \to (H, t)$ where $t < t^*$

• $H' = H$ — same presheaf; $\rho = \mathrm{id}_H$
• $t' = t$ — a history strictly below $t^*$ in $T$; currently in the thematic field of the source MacroFocus
• $\phi$ is the inclusion $t \hookrightarrow t^*$ (the restriction map direction)
• Effect: the old theme $H_{t^*}$ moves to the margin of the new focus (since $t^* \not\leq t$ in a non-trivial partial order); the thematic field shrinks to $\downarrow t$; the old thematic field elements above $t$ become marginal

This is context inspection — retreating to an accessible fiber. It is lossless: the full presheaf $H$ is retained; only the pointer changes. The old theme is not discarded; it moves to the margin and becomes re-accessible via lateral shift.

Lateral shift: $(H, t^*) \to (H, t')$ where $t' \not\leq t^*$ and $t^* \not\leq t'$

• $H' = H$ — same presheaf; $\rho = \mathrm{id}_H$
• $t'$ is a history currently in the margin $\mathcal{M}$ of the source MacroFocus
• $\phi$ is not a morphism in $T$ between $t^*$ and $t'$ — the two histories are incomparable; the shift is a pointer reassignment, not a path-following
• Effect: the old theme $H_{t^*}$ and its thematic field $\downarrow t^*$ may be entirely disjoint from the new thematic field $\downarrow t'$; this is a hard context switch

Lateral shift is the only refocusing that does not preserve context continuity. The new thematic field $\{H_t : t < t'\}$ may have no overlap with the old thematic field $\{H_t : t < t^*\}$.

Math grounding

The three cases correspond to the two poles of the Passage and lateral choice:

Case	Math structure	Direction
Forward extension	Carrier stepping map $\gamma_t : X(t) \to X(s \star t)$	Covariant — forward along histories
Backward restriction	Restriction map $H(f) : H_{t'} \to H_t$	Contravariant — backward along histories
Lateral shift	Pointer reassignment in $T$	Neither — incomparable position

The Reindexing structure gives the formal account of what the presheaf $H$ provides at each position: refocusing moves through the index set $T$ while the presheaf $H$ provides the fiber contents. The Natural Transformation structure ensures that in the forward extension case, the stepping map $\gamma$ is coherent with the nuclear pair.

Composition of refocusings

Refocusings compose in the category of pointed presheaves. Two significant compositions:

Forward then backward $(H, t^*) \to (G_s H, s \star t^*) \to (G_s H, t^*)$: extend by $s$ then retreat to the original position. This is not the identity: the presheaf has changed from $H$ to $G_s H$; the theme returns to position $t^*$ but the fiber $G_s H_{t^*} = H_{s \star t^*}$ is different from the original $H_{t^*}$.

Two backward restrictions $(H, t^*) \to (H, t) \to (H, t'')$ where $t'' < t < t^*$: a double retreat. This composes by transitivity in $T$: the path $t'' < t < t^*$ gives a path of inclusions. The new thematic field is $\downarrow t'' \subset \downarrow t \subset \downarrow t^*$.

Open questions

• Whether there is a canonical refocusing distance — a metric on $T$ that measures how many refocusing steps are needed to reach $t'$ from $t^*$, and whether this metric has the same structure as the edit distance in the history monoid.
• Whether the category of MacroFoci and refocusings has a terminal object — a canonical “most focused” position — and whether that terminal object corresponds to the Grundnorm anchor of the fiber’s normative structure.
• Whether the three refocusing types (forward, backward, lateral) generate all morphisms in the pointed-presheaf category, or whether there are morphisms between MacroFoci that are not decomposable into these three.