2-Segal objects and algebras in spans

Abstract

We define a category parameterizing Calabi–Yau algebra objects in an infinity category of spans. Using this category, we prove that there are equivalences of infinity categories relating, firstly: 2-Segal simplicial objects in C to algebra objects in Span(C); and secondly: 2-Segal cyclic objects in C to Calabi–Yau algebra objects in Span(C).

Appendices

The localization: associative case

This appendix is given over to the proof that the functor ${\mathcal {L}}:\varOmega \rightarrow \varDelta ^\star$ constructed in Sect. 2 satisfies the conditions of Lemma 2.15, and thus is an $\infty$-categorical localization at the morphisms of E. By necessity, this involves fairly intricate combinatorial verifications.

1.1 Decomposing morphisms

Construction A.1

Given a morphism

$$\begin{aligned} f:[m]\rightarrow [n] \end{aligned}$$

in $\varDelta$, we can uniquely decompose it as follows: Let $[1]=:[1_i]\subset [m]$ be the interval $\{i-1\le i\}$, and let $[n_i]\subset [n]$ be the interval $\{f(i-1)\le f(i)\}$. Moreover, let $[n_{left}]$ and $[n_{right}]$ be the intervals $\{0\le f(0)\}$ and $\{f(m)\le n\}$ in [n] respectively. Then f is completely determined by the decomposition of [n], since, given such a decomposition, we can reconstruct f by defining $f_i:[1_i]\rightarrow [n_i]$ to be the unique map preserving maximal and minimal elements, so that f is the composition

$$\begin{aligned} f=f_1\star \cdots \star f_m:[1_1]\star \cdots \star [1_m] \rightarrow [n_1]\star \cdots [n_m]\hookrightarrow [n_{left}]\star [n_1]\star \cdots [n_m]\star [n_{right}]. \end{aligned}$$

We can clarify the indexing of the decomposition of [n] by noting that the pairs $(i-1,i)$ considered above are precisely the inner interstices of [m]. Hence, we have decomposed f as a morphism

$$\begin{aligned} \bigstar _{(i-1,i)\in {\mathbb {I}}([m])} \{i-1,i\} \rightarrow \bigstar _{(i-1,i)\in {\mathbb {I}}([m])} [n_i]. \end{aligned}$$

Definition A.2

Given a morphism $\gamma :[n] \rightarrow [m]$ in $\varDelta$, we can uniquely factor $\gamma$ as

$$\begin{aligned}{}[n]\overset{\gamma _1}{\rightarrow } [m_\gamma ]\overset{\gamma _2}{\hookrightarrow } [m] \end{aligned}$$

where $[m]=[k]\oplus [m_\gamma ]\oplus [\ell ].$ Applying O, we get

$$\begin{aligned} O([m])\rightarrow O([m_\gamma ])\rightarrow O)[n]. \end{aligned}$$

Where $O([m])\rightarrow O([m_\gamma ])$ acts as projection onto a sub-interval. We call $O([m_\gamma ])$ the minimal interval of $\gamma$.

Lemma A.3

Given an interval $\{i,j\}\subset [n]$ and a morphism $\eta :(\{i,j\}\subset [n])\rightarrow (\{r,r+k\} \subset [m])$ in $\varDelta ^{\amalg }$, let $[p,\ldots , q]$ be the minimal interval of $\gamma :={\text {res}}(\eta )$. Then $\eta |_{[p+1,\ldots ,q-1]}= O(\gamma )|_{[p+1,\ldots ,q-1]}$.

Proof

If $[p+1,\ldots ,q-1]$ is empty, the statement is vacuously true. Otherwise, note that for $s\in [p+1,\ldots ,q-1]$, the requirement that ${\text {res}}(\eta )=\gamma$ means that $\gamma (\eta (s))\le s<\gamma (\eta (s)+1)$. Such an $\eta (s)$ always exists, and this inequality uniquely determines $\eta (s)$. (Note that, for p or q in $[p,\ldots ,q]$, we only have one-half of the inequality so that uniqueness need not hold.) $\square$

With these lemmata in hand, we can return to the proof of the localization result.

1.2 Constructing morphisms

We now prove the first criterion of Lemma 2.15.

Lemma A.4

The object $\{0,k\}\subset [k]\overset{f_M}{\rightarrow } [m]$ is an initial object in $\varOmega _M^{E}$.

Proof

Given another object

$$\begin{aligned} \{i,j\}\subset [n]\overset{f}{\rightarrow } [m^\prime ] \end{aligned}$$

in $\varOmega _M^E$, and a morphism

$\phi$ must be the inclusion of $[i,\ldots , j]$, since any such morphism in E will induce an isomorphism $[k]\rightarrow [i,\ldots , j]$. Moreover, h is clearly uniquely determined by the condition that it maps $[f(i),f(i+1), \ldots , f(j)]$ isomorphically to [m]. $\square$

To show the second criterion of Lemma 2.15, we must show that the inclusion $\varOmega _M^E\hookrightarrow \varOmega _{/M}$ is cofinal. This amounts to showing, for any $g:{\mathcal {L}}(Z)\rightarrow M$ in $\varOmega _{/M}$, the category $(\varOmega _M^E)_{g/}$ is contractible. We will do this by showing that $(\varOmega _M^E)_{g/}$ has an initial object.

To this end, we suppose we are given an object

$$\begin{aligned} Z:=\left\{ \{i,j\}\subset [n]\overset{f}{\rightarrow } [\ell ]\right\} \end{aligned}$$

in $\varOmega$ whose image under ${\mathcal {L}}$ is $([\ell _{i+1}],\ldots ,[\ell _j])$, and a morphism

$$\begin{aligned} g: ([\ell _{i+1}],\ldots ,[\ell _j])\rightarrow ([m_0],\ldots , [m_{k-1}]) \end{aligned}$$

in $\varDelta ^\star$. Write $\gamma :[k-1]\rightarrow [i+1,\ldots , j] \in \varDelta$ and ${\overline{g}}:[m_0]\star \cdots \star [m_{k-1}]\rightarrow [\ell _{i+1}]\star \cdots \star [\ell _j]$ for the morphisms defining g. Denote by $[n_c]:=[p,\ldots ,q]\subset \{i,j\}\subset [n]$ the minimal interval of $\gamma$ and by $\psi : [i,\ldots ,j]\rightarrow [n_c]$ the projection as above, and let $\{0,k\}\subset [k]\overset{f_M}{\rightarrow } [m]:=[m_0]\star \cdots \star [m_{k-1}]$ be the minimal object in $\varOmega$ representing the target.

Note that, by definition, the morphism ${\overline{g}}$ has image contained in $[\ell _{p+1}]\star \cdots \star [\ell _q]=:[\ell _c]$. We introduce some notation for specific decompositions:

$$\begin{aligned}{}[n]= & {} [n_\ell ]\star [n_c]\star [n_r] \\ {}[\ell ]= & {} [{\ell _\ell }]\star [{\ell _c}]\star [{\ell _r}] \end{aligned}$$

Lemma A.5

There is a morphism in $\varOmega$

which extends to a morphism $\mu _{Z,M}$ in $\varOmega$ covering g

Moreover, given any other morphism $Z\rightarrow X$ covering g, there is a unique morphism $Z_M\rightarrow X$ in E such that the diagram

commutes.

Proof

In the first diagram, we define the map $\nu$ on $[p+1,\ldots , q-1]$ to be the unique map from Lemma A.3 dual to $\gamma$ under ${\text {res}}$, and send the endpoints to the endpoints of $[1]\star [k]\star [1]$. Then we write

$$\begin{aligned}{}[\ell _c]=[\ell ^1]\star [\ell _c^m]\star [\ell ^2], \end{aligned}$$

where $[\ell _c^m]$ is the minimal interval containing the image of ${\overline{g}}:[m]\rightarrow [\ell ]$. Note that ${\overline{g}}:[m]\rightarrow [\ell _c^m]$ hits both endpoints. We then define

$$\begin{aligned} {\overline{g}}^\prime :={{\text {id}}}_{[\ell ^1]}\star {\overline{g}}\star {{\text {id}}}_{[\ell ^2]}:[\ell ^1]\star [m]\star [\ell ^2]\rightarrow [\ell _c] \end{aligned}$$

(which then, by definition, hits both endpoints), and

$$\begin{aligned} f_M^\prime : [1]\star [k]\star [1]\rightarrow [\ell ^1]\star [m]\star [\ell ^2] \end{aligned}$$

to be $f_M$ on [k], and to send endpoints to endpoints. Then we can decompose the diagram as

by decomposing the morphisms $\nu$, $f|_{\{p,q\}}$, and $f_M^\prime \circ \nu$. The condition that the diagram commute is then equivalent to the conditions that, (1) for each $r\in \{p+2,\ldots , q-1\}$, the endpoints of $[m_r]$ are sent to the endpoints of $[\ell _r]$ by ${\overline{g}}$, and (2) that ${\overline{g}}$ sends the endpoints of $[\ell ^1]\star [m_{p+1}]$ and $[m_q]\star [\ell ^2]$ to the endpoints of $[\ell _{p+1}]$ and $[\ell _q]$, respectively. Since

$$\begin{aligned}{}[m_r]=\bigstar _{a\in {\mathbb {I}}([k_r])} [f_M^\prime (a-1),f_M^\prime (a-1)+1,\ldots , f_M^\prime (a)] \end{aligned}$$

we see that case (1) is true by the definition of $\varDelta ^\star$. Case (2) is true by construction.

This diagram is defined so that the maps $\nu$, $f_M^\prime$, ${\overline{g}}^\prime$, and $f|_{\{p,q\}}$ preserve endpoints. Therefore, we can take the appropriate star products with the morphisms ${{\text {id}}}_{[n_\ell ]}$, ${{\text {id}}}_{[n_r]}$, ${{\text {id}}}_{[\ell _\ell ]}$, ${{\text {id}}}_{[\ell _r]}$, $f|_{[n_\ell ]}:[n_\ell ]\rightarrow [\ell _\ell ]$, and $f|_{[n_r]}:[n_r]\rightarrow [\ell _r]$ to get a commutative diagram

By construction, the morphism ${\text {res}}(\nu ):[k-1] \rightarrow \langle i+1,\ldots , j\rangle$ is $\gamma$, and the morphism ${\overline{g}}^\prime$ restricts to ${\overline{g}}$ on [m], so this diagram determines a morphism in $\varOmega$ covering g. Call this morphism $\mu _{Z,M}:Z\rightarrow Z_M$.

Now suppose we are given a morphism

covering g. We can decompose this into

where $\{0,k\}\subset [a_c]$. By Lemma A.3, we know that $\rho$ is uniquely determined on all of $[n_c]$ except the endpoints. This allows us to further decompose the diagram

as a diagram where the bottom map is a star product with $f_M$.

If there is morphism $Z_M\rightarrow X$ in E commuting with the morphisms $Z\rightarrow X$ and $\mu _{Z,M}$, it must, in particular, restrict to a commutative diagram

Moreover, since the morphism is in E, the bottom square must restrict to the commutative diagram

As a result, the component morphism $[1]\star [k]\star [1]\rightarrow [a_c^1]\star [k]\star [a_c^2]$ is uniquely determined by the commutativity of the left-hand triangle. Additionally, since $w:[b]\rightarrow [\ell ]$ must restrict to ${\overline{g}}$ on [m], we can decompose w as a star product

$$\begin{aligned} w=w^1 \star {\overline{g}}\star w^2: [b_c^1]\star [m]\star [b_c^2]\rightarrow [\ell ^1]\star [\ell _c^m]\star [\ell ^2] \end{aligned}$$

Therefore, the component morphism

$$\begin{aligned}{}[b_c^1]\star [m]\star [b_c^2]\rightarrow [\ell ^1]\star [m]\star [\ell ^2] \end{aligned}$$

is uniquely determined, and must be $w^1\star {{\text {id}}}_{[m]}\star w^2$.

We now extend back to the full diagram

and note that, since the vertical components of the back square restrict to identities on $[n_\ell ]$, $[n_r]$, $[\ell _\ell ]$, and $[\ell _r]$, the bottom square is uniquely determined by the morphisms $[n_{\ell }]\rightarrow [a_{\ell }]$, $[n_r]\rightarrow [a_r]$, $b_\ell ]\rightarrow [\ell _\ell ]$, and $[b_{r}]\rightarrow [\ell _{r}]$. So there is a unique morphism $Z_M\rightarrow X$ in $\varOmega$ with the desired properties. $\square$

The localization: Calabi–Yau case

We can immediately verify that each $\varOmega _M^E$ has an initial object — showing the first criterion of Lemma 2.15.

Proposition B.1

For every M in $\varLambda ^\star$, there is an initial element in $\varOmega _M^E$.

Proof

We will complete the proof in two cases:

Suppose first that $M=\{[m_i]\}_{i\in P}$. Then the weak fiber only involves morphisms in $\mathcal {A}\!{\text {ss}}\subset \mathcal {A}\!{\text {ss}}_{{\text {CY}}}$. We define a set

$$\begin{aligned} T:= \coprod _{i\in P} {\mathbb {I}}([m_i]) \end{aligned}$$

and a morphism $f_M: T\rightarrow P$ by setting $f_M({\mathbb {I}}([m_i]))=i$. The canonical isomorphisms

$$\begin{aligned} \eta _i:O({\mathbb {I}}([m_i]))\cong [m_i] \end{aligned}$$

equip $P\subset P \overset{f_M}{\longleftarrow } T$ with the structure of an object of $\varOmega _M^E$. Given an element

$$\begin{aligned} P\subset U\overset{f}{\leftarrow } V \end{aligned}$$

and an isomorphism $\phi _i: O(f^{-1}(i))\cong [m_i]$, we define a unique morphism $\mu$ in $\varOmega _M^E$ given by

as follows. Since this must be a morphism in E, we see that g must map P identically to P, and send $U^\circ {\setminus } P$ to the basepoint. On fibers, we consider the isomorphisms

$$\begin{aligned} \eta _i^{-1}\circ \phi _i: O(f^{-1}(i))\rightarrow O(I[m_i]) \end{aligned}$$

Since O is fully faithful, this lifts to a unique isomorphism $I(\phi _i):I([m_i])\cong f^{-1}(i)$. We therefore see that ${\overline{g}}$ must be the coproduct of these morphisms if $\mu$ is to be a morphism in the weak fiber. It is immediate that this does, indeed, define a morphism in $\varOmega _M^E$.

Now suppose instead $M=\langle m\rangle$. We define $f_M: D(\langle m\rangle ) \rightarrow \diamond$ to be the morphism with $f_M^{-1}(\diamond )=D(\langle m\rangle )$. Since D is an equivalence, we choose the isomorphism

$$\begin{aligned} \eta : D^2(\langle m \rangle )\cong \langle m\rangle \end{aligned}$$

Suppose given another element

$$\begin{aligned} \{1\}\subset \diamond \overset{f}{\longleftarrow } T \end{aligned}$$

with $\phi : D(f^{-1}(\diamond ))\cong \langle n\rangle$ in the weak fiber. We define a unique morphism $\mu \in \varOmega _M^E$ given by

as follows. The morphism g must be the identity, so we need only define ${\overline{g}}$. The condition that $\mu$ be in the weak fiber implies that $\eta _i\circ D({\overline{g}}|_{D(\langle m\rangle )})=\phi _i$, i.e. $D({\overline{g}}|_{D(\langle m\rangle )})=\eta _i^{-1}\circ \phi _i$. However, since D is fully faithful, this condition defines a unique isomorphism $D(\langle m\rangle )\cong f^{-1}(\diamond )$, determining ${\overline{g}}$, and thus $\mu$, uniquely. $\square$

1.1 Cofinality

We now prove the second criterion of Lemma 2.15: that the inclusion $\varOmega _M^M\rightarrow \varOmega _{/M}$ is cofinal. As before, we do this by finding an initial object in each slice of the inclusion. Unlike the associative case, however, we must do this in two steps, depending on whether $M=\{[m_i]\}_{i\in P}$ or $M=\langle m\rangle$.

Proposition B.2

Suppose given an object $M=\{[m_i]\}_{i\in P}$ in $\varLambda ^\star$, an object

$$\begin{aligned} Z:= \left\{ Q\subset S\overset{f_Z}{\leftarrow } T \right\} \end{aligned}$$

in $\varOmega$, and a morphism

$$\begin{aligned} (\phi , \{\gamma _i\}_{i\in Q}):{\mathcal {L}}(Z)\rightarrow M \end{aligned}$$

in $\varLambda ^\star$. Then there is an element $X_{M,Z}$ in $\varOmega _M^E$ and a morphism $\varPhi :Z\rightarrow X_{M,Z}$ in $\varOmega$ covering $(\phi ,\{\gamma _i\}_{i\in Q})$ such that, for any other morphism $\varPsi :Z\rightarrow X$ covering $(\phi ,\{\gamma _i\}_{i\in Q})$, there is a unique morphism $\tau : X_{M,Z}\rightarrow X$ which makes the diagram

commute.

Proof

There are two cases to consider, corresponding to whether or not $S=\diamond$.

Case 1: First suppose $S\in \mathcal {A}\!{\text {ss}}$. In this case, we construct $X_{M,Z}$ as follows. Let

$$\begin{aligned} P\subset P \overset{f_M}{\longleftarrow } U \end{aligned}$$

be the object constructed in Proposition B.2. Then, in particular, $\phi :P\rightarrow Q\subset S$.

For each $i\in Q$, we have a morphism

$$\begin{aligned} \gamma _i: \bigoplus _{j\in \phi ^{-1}(i)}[m_i]\rightarrow O(f_Z^{-1}(i)) \end{aligned}$$

For each $j\in \phi ^{-1}(i)$ denote by $\gamma _i([m_j])$ the smallest subinterval of $O(f_Z^{-1}(i))$ containing the image of $[m_j]$ under $\gamma _i$. Then $\gamma _i|_{[m_j]}\rightarrow \gamma _i([m_j])$ preserves boundary, and thus corresponds to a map ${\overline{g}}_{j}:I(\gamma _i([m_j]))\rightarrow I([m_j])$ of linearly ordered sets. Moreover, ${\overline{g}}_j$ fits into a commutative diagram

in $\mathcal {A}\!{\text {ss}}$. We here use the identification of $I(\gamma _i([m_j]))$ with a subset of T.

Since, by definition, $U=\coprod _{j\in P} I([m_j])$, we can then write down a commutative diagram

(7)

in $\mathcal {A}\!{\text {ss}}$.

For each $i\in Q$, this restricts to a diagram of ordered sets

We denote $L_i:=f_Z^{-1}(i){\setminus }\coprod _{j\in \phi ^{-1}(i)}I(\gamma _i([m_j])$, and proceed as follows.

• For $p, p+1$ in $\phi ^{-1}(i)$, if there is at least one $k\in L_i$ such that

  $$\begin{aligned} I(\gamma _i([m_{p}]))<k< I(\gamma _i([m_{p+1}])) \end{aligned}$$

  we define a new element $r_p$ and append it to $\phi ^{-1}(i)$ between p and $p+1$.

• If there exists $k\in L_i$ such that

  $$\begin{aligned} k< I(\gamma _i([m_{p}])) \end{aligned}$$

  for all $p\in \phi ^{-1}(i)$, then we append a new minimal element $r_{min}$ to $\phi ^{-1}(i)$.

• If there exists $k\in L_i$ such that

  $$\begin{aligned} I(\gamma _i([m_{p}]))< k \end{aligned}$$

  for all $p\in \phi ^{-1}(i)$, then we append a new maximal element to $\phi ^{-1}(i)$.

Call the resulting set $W_i\supset \phi ^{-1}(i)$. We then set

$$\begin{aligned} R_i:= U\amalg L_i \end{aligned}$$

and define $f_{i}: R_i \rightarrow W_i$ to act as $f_{M}$ on U and on $L_i$ to send

• $k\mapsto r_p$ if

  $$\begin{aligned} I(\gamma _i([m_{p}]))<k< I(\gamma _i([m_{p+1}])) \end{aligned}$$

• $k\mapsto r_{min}$ if

  $$\begin{aligned} k< I(\gamma _i([m_{p}])) \end{aligned}$$

  for all $p\in \phi ^{-1}(i)$

• $k\mapsto r_{max}$ if

  $$\begin{aligned} I(\gamma _i([m_{p}]))< k \end{aligned}$$

  for all $p\in \phi ^{-1}(i)$

We make $f_i$ into a morphism in $\mathcal {A}\!{\text {ss}}$ by taking the linear order induced by $L_i$ on the fibers over the $r_{p}$, $r_{min}$ and $r_{max}$. We then define

$$\begin{aligned} {\overline{g}}^i: f_Z^{-1}(i)\rightarrow R_i \end{aligned}$$

to act as $\coprod _{j\in \phi ^{-1}(i)} {\overline{g}}_j$ on $\coprod _{j\in \phi ^{-1}(i)} I(\gamma _i([m_j]))$ and as the identity on $L_i$. We further define $\phi _i: W_i\rightarrow \{i\}$ to send every element to i. We thus have a commutative diagram

in $\mathcal {A}\!{\text {ss}}$, which covers the morphism $\gamma _i:\bigoplus _{j\in \phi ^{-1}(i)} [m_j] \rightarrow O(f_Z^{-1}(i))$. Taking the coproduct over $i\in {\text {Im}}(\phi )$ gives us a morphism

Finally, we set

$$\begin{aligned} W=\left( \coprod _{i\in {\text {Im}}(\phi )}W_i\right) \amalg (S{\setminus } {\text {Im}}(\phi )) \end{aligned}$$

and

$$\begin{aligned} R=\left( \coprod _{i\in {\text {Im}}(\phi )}R_i\right) \amalg (T{\setminus } f_{Z}^{-1}({\text {Im}}(\phi ))) \end{aligned}$$

We then define morphisms:

• $g:W\rightarrow S$ to act as $\coprod _{i\in {\text {Im}}(\phi )} \phi _i$ on $\coprod _{i\in {\text {Im}}(\phi )} W_i$ and as the identity otherwise.

• $f_{M,Z}: R\rightarrow W$ to act as $f_i$ on $R_i$ and as $f_Z$ on $T{\setminus } f_Z^{-1}({\text {Im}}(\phi ))$.

• ${\overline{g}}:T\rightarrow R$ to act as $\coprod _{i\in {\text {Im}}(\phi )} {\overline{g}}^{i}$ on $f_Z^{-1}({\text {Im}}(\phi ))$ and the identity elsewhere.

By construction, this defines a commutative diagram

(8)

in $\mathcal {A}\!{\text {ss}}$, covering $(\phi ,\{\gamma _i\})$, and the bottom row is in $\varOmega _M$. We therefore define $X_{M,Z}$ to be the bottom row, and $\varPhi$ to be the morphism defined by the diagram (8).

To check the remaining universal property, we let

$$\begin{aligned} P\subset A\overset{f_X}{\leftarrow } B \end{aligned}$$

and $\beta _i: O(f_X^{-1}(i))\cong [m_i]$ be another element in $\varOmega _M^E$, and let $\nu$ be a morphism

covering $(\phi , \{\gamma _i\}_{i\in Q})$.

For each $i\in {\text {Im}}(\phi )$, the identity on P and the condition nothing be sent to the basepoint uniquely determines a map of ordered sets

$$\begin{aligned} \zeta _i:\rho ^{-1}(i)\rightarrow W_i. \end{aligned}$$

Moreover, the $\zeta _i$ together with the restriction of $\rho$ to $A{\setminus } \rho ^{-1}({\text {Im}}(\phi ))$ uniquely determines a map

$$\begin{aligned} \zeta : A\rightarrow W \end{aligned}$$

such that the diagram

commutes. Note that $\zeta |_P$ induces the identity $P\rightarrow P$.

Moreover, for each $i\in {\text {Im}}(\phi )$ the isomorphisms $I(\beta _i)$ on $I([m_j])$ and restriction ${\overline{\rho }}_i:L_i\rightarrow f_X^{-1}(\rho ^{-1}(i))$ uniquely determine a map

$$\begin{aligned} {\overline{\zeta }}_i:R_i \rightarrow f_X^{-1}(\rho ^{-1}(i)). \end{aligned}$$

These, together with the restriction of ${\overline{\rho }}$ to $T{\setminus } f_Z^{-1}({\text {Im}}(\phi ))$ uniquely determine a morphism

$$\begin{aligned} {\overline{\zeta }} R\rightarrow B \end{aligned}$$

such that the diagram

commutes, and the restriction of ${\overline{\zeta }}$ to $f_X^{-1}(P)$ is the isomorphism $\coprod _{i}I(\beta _i)$.

We therefore have constructed a unique morphism

$$\begin{aligned} (\zeta ,{\overline{\zeta }}):X_{Z,M} \rightarrow X \end{aligned}$$

in $\varOmega _M^E$ such that the diagram

commutes.

Case 2: Now suppose that $S=\diamond$. Then $\phi$ is completely determined by a cyclic order on P, and $\gamma$ is a morphism

$$\begin{aligned} \gamma : \bigcup \nolimits ^S [m_i] \rightarrow D(f_Z^{-1}(\diamond )). \end{aligned}$$

We note that, given any morphism

a choice of linear order on $g^{-1}(\diamond )$ compatible with the cyclic order uniquely determines a factorization

Similarly, given a morphism $(\psi , \eta ): \langle n\rangle \rightarrow \{[n_i]\}_{i\in S}$, a choice of linear order on S compatible with the cyclic order uniquely determines a factorization

$$\begin{aligned} \langle n\rangle \rightarrow \{[n]\} \rightarrow \{[n_i]\}. \end{aligned}$$

We can therefore choose a linear order on P and define Y to be the object

$$\begin{aligned} \{1\} \subset \{1\} \overset{f_Z}{\leftarrow } T. \end{aligned}$$

Then take $(\phi _Y,\gamma _Y)$ to be the unique morphism yielding a factorization

$$\begin{aligned} {\mathcal {L}}(\phi ,\gamma ): D(f^{-1}_{Z}(\diamond ))\rightarrow K(f^{-1}_Z(\diamond ))\overset{(\phi _Y,\gamma _Y)}{\longrightarrow } \{[m_i]\}_{i\in P} \end{aligned}$$

We can then construct $X_{M,Y}$ as in case 1. It is immediate that

defines a morphism $\varPhi$ in $\varOmega$ covering $(\phi ,\gamma )$.

Now suppose given any other morphism $\varPsi =(\psi ,{\overline{\psi }}): Z\rightarrow X$ covering $(\phi , \gamma )$. A choice of linear order on $\psi ^{-1}(\diamond )$ compatible with the chosen linear order on P uniquely factors $\varPsi$ through Y. We therefore get a morphism $\tau : X_{M,Y}\rightarrow X$ such that the diagram

commutes.

To see that this morphism is unique, suppose that $(\xi ,{\overline{\xi }}), (\zeta ,{\overline{\zeta }}): X_{M,Y}\rightarrow X$ are two such morphisms. Then, choosing a linear order on $\psi ^{-1}(\diamond )$ compatible with the chosen linear order on P uniquely factors the diagram as

But, by case 1, there is a unique morphism making the bottom triangle commute. Therefore, $(\xi ,{\overline{\xi }})=(\zeta ,{\overline{\zeta }})$, proving the proposition. $\square$

In the second case, that of $M=\langle m\rangle$, the computation is somewhat simpler.

Proposition B.3

Suppose given an object $M=\langle m\rangle$ in $\varLambda ^\star$, an object

$$\begin{aligned} Z:= \left\{ Q\subset S\overset{f_Z}{\leftarrow } T \right\} \end{aligned}$$

in $\varOmega$, and a morphism

$$\begin{aligned} (\phi , \{\gamma _i\}_{i\in Q}):{\mathcal {L}}(Z)\rightarrow M \end{aligned}$$

in $\varLambda ^\star$. Then there is an element $X_{M,Z}$ in $\varOmega _M^E$ and a morphism $\varPhi :Z\rightarrow X_{M,Z}$ in $\varOmega$ covering $(\phi ,\{\gamma _i\}_{i\in Q})$ such that, for any other morphism $\varPsi :Z\rightarrow X$ covering $(\phi ,\{\gamma _i\}_{i\in Q})$, there is a unique morphism $\tau : X_{M,Z}\rightarrow X$ which makes the diagram

commute.

Proof

We first note that $S=\diamond$, since otherwise no such morphism $(\phi ,\{\gamma _i\}_{i\in Q})$ can exist. Consequently, $\phi ={{\text {id}}}_\diamond$, and $\gamma$ is a morphism of cyclically ordered sets $\langle m\rangle \rightarrow D(f_Z^{-1}(\diamond ))$. We can therefore take $X_{Z,M}$ to be the object

$$\begin{aligned} \{1\}\subset \diamond \overset{f_M}{\leftarrow } D(\langle m\rangle ) \end{aligned}$$

constructed in the proof of Proposition B.1. We then get a commutative diagram

where ${\overline{g}}$ acts as $D(\gamma )$ on $f_Z^{-1}(\diamond )$ and the identity on $T{\setminus } f_Z^{-1}(\diamond )$. This morphism in $\varOmega$ clearly covers $({{\text {id}}},\gamma )$.

Given $X\in \varOmega _M^E$ and $\varPsi :Z\rightarrow X$, represented by a diagram

by B.1 that there is a unique morphism

in $\varOmega _M^E$. Via the restriction of ${\overline{\ell }}$ to $T{\setminus } f^{-1}_Z(\diamond )$, this extends to a morphism

in $\varOmega _M^E$.

Since all of the left-hand vertical morphisms are required to be identities, we only need to check that ${\overline{\xi }}\circ {\overline{g}}={\overline{\ell }}$, which is true by construction. The requirement that ${\overline{\xi }}$ define a morphism in $\varOmega _M^E$ uniquely determines ${\overline{\xi }}$ on $D(\langle m\rangle )$ and the requirement that ${\overline{\xi }}\circ {\overline{g}}={\overline{\ell }}$ uniquely determines ${\overline{\xi }}$ on $T{\setminus } f_Z^{-1}(\diamond )$. $\square$