List objects – Vikraman’s site

Let $\cat{C}(1, \otimes)$ be a strict monoidal category, and $A$ be an object of $\cat{C}$.

A list algebra on $A$ is an object $L$ with a pair of maps $(1 \xrightarrow{n} L \xleftarrow{c} A \otimes L)$.

A list object on $A$ is a list algebra $(1 \xrightarrow{\nil} L(A) \xleftarrow{\cons} A \otimes L(A))$, such that:

for any $P$-parameterised list algebra $(P \xrightarrow{n} L \xleftarrow{c} A \otimes L)$, there is a unique map $\iter{n}{c} : L(A) \otimes P \to L$ satisfying:

$% https://q.uiver.app/#q=WzAsNixbMCwwLCIxIFxcb3RpbWVzIFAiXSxbMiwwLCJMKEEpIFxcb3RpbWVzIFAiXSxbMCwyLCJQIl0sWzIsMiwiTCJdLFs0LDAsIkEgXFxvdGltZXMgTChBKSBcXG90aW1lcyBQIl0sWzQsMiwiQSBcXG90aW1lcyBMIl0sWzAsMSwiXFxuaWwgXFxvdGltZXMgUCJdLFsyLDMsIm4iLDJdLFs0LDEsIlxcY29ucyBcXG90aW1lcyBQIiwyXSxbNCw1LCJBIFxcb3RpbWVzIFxcaXRlcntufXtjfSJdLFs1LDMsImMiXSxbMSwzLCJcXGl0ZXJ7bn17Y30iLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwyLCIiLDIseyJsZXZlbCI6Miwic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ== \begin{tikzcd}[ampersand replacement=\&,cramped] {1 \otimes P} \&\& {L(A) \otimes P} \&\& {A \otimes L(A) \otimes P} \\ \\ P \&\& L \&\& {A \otimes L} \arrow["{\nil \otimes P}", from=1-1, to=1-3] \arrow[equals, from=1-1, to=3-1] \arrow["{\iter{n}{c}}"', dashed, from=1-3, to=3-3] \arrow["{\cons \otimes P}"', from=1-5, to=1-3] \arrow["{A \otimes \iter{n}{c}}", from=1-5, to=3-5] \arrow["n"', from=3-1, to=3-3] \arrow["c", from=3-5, to=3-3] \end{tikzcd}$

A monoid object $M$ in $\cat{C}$ is a pair of maps $(1 \xrightarrow{\epsilon} M \xleftarrow{\mu} M \otimes M)$ satisfying:

$% https://q.uiver.app/#q=WzAsNCxbMCwwLCIxIFxcb3RpbWVzIE0iXSxbMSwwLCJNIFxcb3RpbWVzIE0iXSxbMSwxLCJNIl0sWzIsMCwiTSBcXG90aW1lcyAxIl0sWzAsMSwiXFxlcHNpbG9uIFxcb3RpbWVzIE0iXSxbMSwyLCJcXG11Il0sWzMsMSwiTSBcXG90aW1lcyBcXGVwc2lsb24iLDJdLFswLDIsIiIsMix7ImxldmVsIjoyLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzIsMywiIiwyLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0= \begin{tikzcd}[ampersand replacement=\&,cramped] {1 \otimes M} \& {M \otimes M} \& {M \otimes 1} \\ \& M \arrow["{\epsilon \otimes M}", from=1-1, to=1-2] \arrow[equals, from=1-1, to=2-2] \arrow["\mu", from=1-2, to=2-2] \arrow["{M \otimes \epsilon}"', from=1-3, to=1-2] \arrow[equals, from=2-2, to=1-3] \end{tikzcd} \qquad % https://q.uiver.app/#q=WzAsNCxbMCwwLCJNIFxcb3RpbWVzIE0gXFxvdGltZXMgTSJdLFsxLDAsIk0gXFxvdGltZXMgTSJdLFswLDEsIk0gXFxvdGltZXMgTSJdLFsxLDEsIk0iXSxbMCwxLCJNIFxcb3RpbWVzIFxcbXUiXSxbMCwyLCJcXG11IFxcb3RpbWVzIE0iLDJdLFsyLDMsIlxcbXUiLDJdLFsxLDMsIlxcbXUiXV0= \begin{tikzcd}[ampersand replacement=\&,cramped] {M \otimes M \otimes M} \& {M \otimes M} \\ {M \otimes M} \& M \arrow["{M \otimes \mu}", from=1-1, to=1-2] \arrow["{\mu \otimes M}"', from=1-1, to=2-1] \arrow["\mu", from=1-2, to=2-2] \arrow["\mu"', from=2-1, to=2-2] \end{tikzcd}$

Lemma 1 $(1 \xrightarrow{\nil} L(A) \xleftarrow{\iter{\id}{\cons}} L(A))$ is a monoid structure on $L(A)$.

Theorem 1 The list object construction gives $L \dashv U : \mathsf{Mon}(\cat{C}) \to \cat{C}$.

Now, we switch to $\cat{C}$ being symmetric strict monoidal, with symmetry $\beta$.

A symmetric list object on $A$ is a list object $(1 \xrightarrow{\nil} L(A) \xleftarrow{\cons} A \otimes L(A))$ satisfying:

$% https://q.uiver.app/#q=WzAsNSxbMSwwLCJBIFxcb3RpbWVzIEEgXFxvdGltZXMgTChBKSJdLFswLDEsIkEgXFxvdGltZXMgQSBcXG90aW1lcyBMKEEpIl0sWzAsMiwiQSBcXG90aW1lcyBMKEEpIl0sWzIsMiwiTChBKSJdLFsyLDEsIkEgXFxvdGltZXMgTChBKSJdLFswLDEsIlxcYmV0YSBcXG90aW1lcyBMKEEpIiwyXSxbMSwyLCJBIFxcb3RpbWVzIFxcY29ucyIsMl0sWzIsMywiXFxjb25zIiwyXSxbMCw0LCJBIFxcb3RpbWVzIFxcY29ucyJdLFs0LDMsIlxcY29ucyJdXQ== \begin{tikzcd}[ampersand replacement=\&,cramped] \& {A \otimes A \otimes L(A)} \& \\ {A \otimes A \otimes L(A)} \&\& {A \otimes L(A)} \\ {A \otimes L(A)} \&\& {L(A)} \arrow["{\beta \otimes L(A)}"', from=1-2, to=2-1] \arrow["{A \otimes \cons}", from=1-2, to=2-3] \arrow["{A \otimes \cons}"', from=2-1, to=3-1] \arrow["\cons", from=2-3, to=3-3] \arrow["\cons"', from=3-1, to=3-3] \end{tikzcd}$

Lemma 2 $(1 \xrightarrow{\nil} L(A) \xleftarrow{\iter{\id}{\cons}} L(A))$ is a commutative monoid structure on $L(A)$.

Lemma 3 If $L(A)$ is a list object on $A$, the following are equivalent:

$L(A)$ is a symmetric list object on $A$.
$(L(A), \nil, \iter{\id}{\cons})$ is a commutative monoid.
$(L(A), \nil, \iter{\id}{\cons}, \eta_A)$ is the free commutative monoid on $A$.

Theorem 2 The symmetric list object construction gives $L \dashv U : \mathsf{CMon}(\cat{C}) \to \cat{C}$.

Lemma 4 If $\cat{C}$ is symmetric monoidal closed and has the relevant colimits, the following are equivalent:

$L(A)$ is a symmetric list object on $A$.
$L(A)$ is the sequential colimit of the symmetric powers of $A$.