The marked boundary-relative open-closed normal word

The normal word is first a formal composite of the ten typed surface generators, with $C$ the circle symbol and $A_{ab}$ the interval symbol $I_{ab}$. Tensor, associators, unitors and flips are those of their symmetric monoidal presentation with the elementary sewing relations. The formulas below also specify its evaluation in ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathrm{f}\mathrm{d}}(k)$ after choosing knowledgeable data. Equality of such vector-space evaluations is not used to infer equality of formal words or surfaces; that distinction is essential before the normal-form theorem.

Let $M$ be a connected morphism in ${\mathsf{C}\mathsf{o}\mathsf{b}}_2^{\mathrm{o}\mathrm{c}}(S)$, with the marked incoming and outgoing black boundary parameterizations of The oriented open-closed surface category with colored free boundary. The marked data of $M$ are:

1. the partition of its marked black boundary components into connected components of $M$;
2. the genus $g\ge 0$ of each connected component;
3. for each color $s\in S$, the number $w_s\ge 0$ of free-boundary circles (windows) of color $s$ on that component;
4. the cyclically ordered open-boundary cycles, each recorded as the ordered list of its marked interval slots together with the full endpoint-color sequence along those intervals and free edges.

An open-boundary cycle is obtained by following the oriented boundary of $M$ through alternating black interval collars and the intervening free-boundary interval faces. Its successive black slots are distinct within the cycle and each slot appears in exactly one cycle. For the multiplication word after bending all open slots to inputs, traverse this cycle in the direction opposite to the induced surface-boundary orientation. If its slots in this reverse traversal are $j_1,\dots ,j_q$, write their endpoint labels in their incoming object orientations as

$$I_{a_0{a}_1},I_{a_1{a}_2},\dots ,I_{a_{q-1}{a}_q},a_q=a_0\mathrm{.}$$

Each incoming interval is oppositely oriented to its surface-boundary segment, so this reverse traversal follows its object orientation. The color at each join agrees because the free face has a single label. In particular, an induced-boundary cycle through incoming $I_{ab},I_{ca},I_{bc}$ has multiplication order $I_{ab},I_{bc},I_{ca}$ after starting at the first slot; no factor is replaced by its opposite-colored space. Distinct cycles may have different endpoint-color sequences even when their lengths agree.

Use the snakes of Typed algebraic derivations of the Frobenius sewing moves to bend every outgoing interval to an incoming interval of reversed endpoint order and every incoming circle to an outgoing circle. Thus a connected component is reduced to a typed word with $m$ incoming intervals, $n$ outgoing circles, and the same marked open-boundary cycles. Let $c_1,\dots ,c_r$ be those cycles, ordered by their least marked interval slot; within each cycle start at its least slot and use the reverse-boundary multiplication order just specified. For cycle $c_j$, let its slots be $j_{j,1},\dots ,j_{j,q_j}$ and its cyclic endpoint colors be $a_{j,0},\dots ,a_{j,q_j}=a_{j,0}$. Define

$$T_j:=i_{a_{j,0}}^{\ast }\circ {\mu }_{a_{j,0},a_{j,1},\dots ,a_{j,q_j}}^{(q_j)}:\underset{\mathrm{\ell }=1}{\overset{q_j}{⨂}}{A}_{a_{j,\mathrm{\ell }-1}{a}_{j,\mathrm{\ell }}}⟶C\mathrm{.}$$

Let $P$ be the symmetry isomorphism that permutes the $m$ interval factors into this listed cycle order, preserving each factor and its endpoint colors. Set

$$T_M:=(T_1\otimes \cdots \otimes T_r)\circ P\mathrm{.}$$

For $r=0$, set $T_M=1_k:k\to k$, the identity on the empty tensor. Let ${\mu }_C^{[r]}:C^{\otimes r}\to C$ be the iterated closed multiplication for $r\ge 2$, with ${\mu }_C^{[1]}=1_C$ and ${\mu }_C^{[0]}={\eta }_C$. Let ${\mathrm{\Delta }}_C^{[n]}:C\to C^{\otimes n}$ be the iterated closed comultiplication for $n\ge 2$, with ${\mathrm{\Delta }}_C^{[1]}=1_C$ and ${\mathrm{\Delta }}_C^{[0]}={ϵ}_C$. The ordered product of window operators is

where the product is a finite composite: compactness gives only finitely many windows on $M$. Its order is immaterial by the $C$-linearity of the $W_s$; if there are no windows, $W_M=1_C$. Let $H={\mu }_C{\mathrm{\Delta }}_C$ and define the candidate normal word

$$N(M):={\mathrm{\Delta }}_C^{[n]}\circ H^g\circ W_M\circ {\mu }_C^{[r]}\circ T_M,$$

with the canonical associators, unitors, and symmetry maps inserted. This formula is the word for the bent surface; for the original marked source and target, apply the inverse bends and restore their slot permutations to define $N(M)$. The identity on each marked closed output slot is retained by the final output permutation. If $n=0$, the final map is ${ϵ}_C$; if $r=0$, the initial closed input is ${\eta }_C$. In particular a component with no black boundary has the scalar word ${ϵ}_C{H}^g{W}_M{\eta }_C$ and keeps every colored window count.

For a disconnected morphism, define its candidate normal word as the ordered tensor product of its connected candidate words, preceded and followed by the symmetry maps that restore the original marked source and target slot orders. No assertion that $N(M)$ is independent of a surface decomposition is part of this definition; that assertion is the claim of The complete colored boundary-relative surface normal-form theorem ↗.

For $r,n\ge 1$ the candidate map factors through the open boundary cycles, their cozipper images in $C$, and the closed multiplication/window/handle/comultiplication word. The displayed arrows have exactly the domains specified above; when $r=0$ or $n=0$, replace the corresponding map using the stated unit or counit convention.

\bigotimes_{\ell=1}^{m} A_{a_\ell b_\ell}
  \arrow[r, "P"]
& \bigotimes_{j=1}^{r}\bigotimes_{\ell=1}^{q_j} A_{a_{j,\ell-1}a_{j,\ell}}
  \arrow[r, "\bigotimes_j(i_{a_{j,0}}^*\mu^{(q_j)}_{a_{j,0},\ldots,a_{j,q_j}})"]
& C^{\otimes r} \arrow[r, "\mu_C^{[r]}"]
& C \arrow[r, "W_M"]
& C \arrow[r, "H^g"]
& C \arrow[r, "\Delta_C^{[n]}"]
& C^{\otimes n}