\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$-graph
$\mathbb{Span}_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\natural,\text{{el}}}}(\mathfrak{C})$. At the
level of colours, we just have
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}^{\natural,\text{{el}}}_{/\eta}=\\{\operatorname{id}_{\eta}\\}$.
At the level of operations, writing $e_{1},\dots,e_{a}$ the leaves of the
corolla $\star_{a}$ and $r$ its root, we find that
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}^{\natural,\text{{el}}}_{/\star_{a}}$ is the category

${(\eta\xrightarrow{\ulcorner
e_{1}\urcorner}\star_{a})}$${\cdots}$${(\eta\xrightarrow{\ulcorner
e_{a}\urcorner}\star_{a})}$${\operatorname{id}_{\star_{a}}}$${(\eta\xrightarrow{\ulcorner
r\urcorner}\star_{a})}$ (33)

of $(a+1)$-ary multicospans.

The structure of category objects in multicategories (coloured non-symmetric
operads) was studied in [CGR14, Definition 3.9]. In our case, we get for
$\mathbb{Span}_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}}(\mathfrak{C})$ an operadic
composition of multispans by fibre products along the relevant legs, where
each multispan has a distinguished root as seen in eq. 33.

######  Example 6.5.1.

It is also possible to replace
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$ by the pattern
${{\Xi}^{\mathrm{op}}}^{\natural}$ of [HRY19], whose Segal objects are cyclic
operads. We obtain for
$\mathbb{Span}_{{{\Xi}^{\mathrm{op}}}^{\natural}}(\mathfrak{C})$ the same
structure as above, except that the $(a+1)$-ary spans come without a choice of
root. Note also that in $\Xi$, the nodeless edge $\eta$ is equipped with an
involution, which for Segal objects becomes a “duality” operation on colours.
In our case, it acts as the identity.

Going further, we may also use the pattern
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Upsilon$}
\put(0.5125,0.033){\line(0,1){0.8}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Upsilon$}
\put(0.5125,0.033){\line(0,1){0.8}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Upsilon$}
\put(0.5125,0.033){\line(0,1){0.8}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Upsilon$}
\put(0.5125,0.033){\line(0,1){0.8}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$ of [HRY20] (denoted $\mathbb{U}$
there), whose Segal objects are modular operads. The categorical modular
$\infty$-operad$\mathbb{Span}_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Upsilon$}
\put(0.5125,0.033){\line(0,1){0.8}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Upsilon$}
\put(0.5125,0.033){\line(0,1){0.8}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Upsilon$}
\put(0.5125,0.033){\line(0,1){0.8}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Upsilon$}
\put(0.5125,0.033){\line(0,1){0.8}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}}(\mathfrak{C})$ works much as
$\mathbb{Span}_{{{\Xi}^{\mathrm{op}}}^{\natural}}(\mathfrak{C})$, but with
additional contraction operations that turn the abstract self-duality of
objects into an actual self-duality (in the usual monoidal, or rather
properadic, sense).

The weak Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$-fibrations were identified in
[Ber22] as the “tree-hyperoperads”, which are cumbersome to describe in detail
(_cf._ [GK98, §4.1] or [MSS02, Definition 5.45] for the modular
generalisation, simply called hyperoperads). Nevertheless, we still obtain
from 5.7 that dendroidal monads in the categorical $\infty$-operad of
multispans in $\mathfrak{C}$ are internal operads in $\mathfrak{C}$.

######  Remark 6.5.2.

The definition of operads, and more general multicategorical structures, as
monads in multispans is well-known: it dates to [Bur71], and was independently
rediscovered by both [Her04] and [Lei98], and then further systematised by
[Lei04] and [CS10]. From a cartesian monad $\mathcal{T\\!}\,$ on a category
$\mathfrak{C}$, one constructs a double category of Kleisli
$\mathcal{T\\!}\,$-spans, whose objects are those of $\mathfrak{C}$ and
morphisms from $C$ to $D$ are spans from $\mathcal{T\\!}\,C$ to $D$,
composition of spans using the monad structure.

For example, taking $\mathcal{T\\!}\,$ to be the monad
$\mathcal{F\\!}\,_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}}^{\flat}}$
for free monoids, a Kleisli $\mathcal{T\\!}\,$-span is a multispan of
arbitrary arity, and monads (in the double-categorical sense) are coloured
operads. Generally speaking, if $\mathcal{T\\!}\,$ is the monad
$\mathcal{F\\!}\,_{\mathfrak{P}}$ for free Segal $\mathfrak{P}$-objects on
$\mathfrak{P}$-graphs for some appropriate algebraic pattern $\mathfrak{P}$,
we expect that monads in Kleisli $\mathcal{F\\!}\,_{\mathfrak{P}}$-spans
should be weak Segal $\mathfrak{P}$-fibrations, obtained as the Segal objects
for a plus construction $\mathfrak{P}^{+}$ of $\mathfrak{P}$ (as in [Ker23,
Proposition 3.2.10]), so as $\mathfrak{P}^{+}$-monads in
$\mathfrak{P}^{+}$-spans.

However, Kleisli
$\mathcal{F\\!}\,_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\flat}}$-spans and
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$-spans, while both admitting a
natural interpretation as multispans, form markedly different structures. On
the one hand,
$\mathbb{Span}_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}}(\mathfrak{C})$ is a categorical
$\infty$-operad, whose operadic composition is given (leg by leg) by simple
pullbacks. On the other hand, Keisli
$\mathcal{F\\!}\,_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\flat}}$-spans only form a double category,
but its composition is more complex and makes full use of the monad structure
on
$\mathcal{F\\!}\,_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}}^{\flat}}$.
For a general algebraic pattern $\mathfrak{P}$, the difference will be
similar: we think of it as moving the structure from the microcosm (on the
Keisli $\mathcal{F\\!}\,_{\mathfrak{P}}$-spans side) to the macrocosm (on the
$\mathfrak{P}^{+}$-spans side).

It is nonetheless unclear what the precise relation between the two
constructions is, if there even is one: the Keisli-type construction can be
abstracted away from a span setting by using general monads acting on virtual
double categories, but it is unlikely to be able to handle non-directed
structures such as the cyclic and modular $\infty$-operads of 6.5.1.

##  7 Conclusion: A fibrational perspective

######  Notation 7.1.

In this section, we will identify $(\infty,1)$-categories with internal
categories in $\mathfrak{\infty\textnormal{-}Grpd}$, where internal categories
are by definition Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$-objects satisfying Rezk’s
univalence-completeness condition. It will also be convenient to see Segal
$\mathfrak{P}$-objects in
$\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}$ — such as, in
particular, the $\mathfrak{P}$-$(\infty,1)$-categories of $\mathfrak{P}$-spans
— as internal categories in
$\mathfrak{Seg}_{\mathfrak{P}}(\mathfrak{\infty\textnormal{-}Grpd})$.

Recall that, for any regular cardinal $\kappa$, the $(\infty,1)$-category
$\mathfrak{\infty\textnormal{-}Grpd}^{(\kappa)}\subset\mathfrak{\infty\textnormal{-}Grpd}$
is the base of the universal discrete cocartesian fibration with
$\kappa$-small fibres
$\mathfrak{\infty\textnormal{-}Grpd}^{(\kappa)}_{\bullet}\to\mathfrak{\infty\textnormal{-}Grpd}^{(\kappa)}$
in the $(\infty,2)$-topos
$\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}$, just as
$\mathfrak{Set}^{(\kappa)}\subset\mathfrak{Set}$ is the universal
$\kappa$-small discrete cocartesian fibration in the $(2,2)$-topos
$\mathfrak{Cat}$. In [Web07, Examples 4.7 and 4.8], it is explained that, for
an algebraic pattern $\mathfrak{P}^{\text{{el}}}$ in which all objects are
elementary and all morphisms inert, the construction
$\mathbb{Span}_{\mathfrak{P}^{\text{{el}}}}(-)$ preserves classifying discrete
fibrations, so that the $2$-topos
$\mathfrak{Cat}\bigl{(}\bigl{\\{}\mathfrak{\mathfrak{P}^{\text{{el}}}},\mathfrak{Set}\bigr{\\}}\bigr{)}$
has a sufficient family of classifying discrete cocartesian fibrations given
by
$\mathbb{Span}_{\mathfrak{P}^{\text{{el}}}}(\mathfrak{Set}_{\bullet}^{(\kappa)})\to\mathbb{Span}_{\mathfrak{P}^{\text{{el}}}}(\mathfrak{Set}^{(\kappa)})$
(where “sufficient” means that every discrete cocartesian fibration is
classified by one in the family).

In the $\infty$-categorical setting, the properties of universal (or
“classifying”) fibrations are captured by the notion of univalence, which we
restate from [GK16] (see also [Ras21a, Theorem 4.4]) in the internal setting.

######  Construction 7.2.

Let $\mathfrak{C}$ be a finitely complete $(\infty,1)$-category. Recall that a
discrete cocartesian fibration in $\mathfrak{C}$ is an internal functor
$\mathcal{f\\!}\,\colon\mathbb{E}\to\mathbb{B}$ such that
$(d_{1},\mathcal{f\\!}\,_{1})\colon\mathbb{E}_{1}\to\mathbb{E}_{0}\times_{\mathbb{B}_{0}}\mathbb{B}_{1}$
is an equivalence.

Lifting the construction of [GK16, Theorem 2.10] to the cartesian closed
$(\infty,2)$-category $\mathfrak{Cat}(\mathfrak{C})$, one can construct for
any discrete cocartesian fibration
$\mathcal{f\\!}\,\colon\mathbb{E}\to\mathbb{B}$ in $\mathfrak{C}$ an internal
category
$\mathbb{Eq}_{/\mathbb{B}\times\mathbb{B}}(\varpi_{1}^{\ast}\mathbb{E},\varpi_{2}^{\ast}\mathbb{E})$
over $\mathbb{B}\times\mathbb{B}$ (where
$\varpi_{1},\varpi_{2}\colon\mathbb{B}\times\mathbb{B}\to\mathbb{B}$ are the
two projections) characterising equivalences between fibres of
$\mathcal{f\\!}\,$.

######  Definition 7.3 (Univalent fibration).

A discrete cocartesian fibration $\mathbb{E}\to\mathbb{B}$ internal to
$\mathfrak{C}$ is univalent if
$\mathbb{B}\to\mathbb{Eq}_{/\mathbb{B}\times\mathbb{B}}(\varpi_{1}^{\ast}\mathbb{E},\varpi_{2}^{\ast}\mathbb{E})$
is an equivalence.

######  Example 7.4.

It is shown in [Cis19, Proposition 5.3.13] that the universal discrete
cocartesian fibration
$\mathfrak{\infty\textnormal{-}Grpd}_{\bullet}\to\mathfrak{\infty\textnormal{-}Grpd}$
(in the $(\infty,2)$-category
$\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}=\mathfrak{Cat}(\mathfrak{\infty\textnormal{-}Grpd})$)
is univalent.

Using this characterisation, one can show (though we omit the proof here as
this result is only used for motivation) that for any globally saturated
algebraic pattern $\mathfrak{P}^{\text{{inrt}}}$ all of whose morphisms are
inert, the construction
$\mathbb{Span}_{\mathfrak{P}^{\text{{inrt}}}}(-)\colon\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}^{(\text{$\mathfrak{P}$-cplt})}\to\mathfrak{Cat}(\mathfrak{Seg}_{\mathfrak{P}^{\text{{inrt}}}}(\mathfrak{\infty\textnormal{-}Grpd}))$
preserves classifying discrete cocartesian fibrations:

######  Proposition 7.5.

Let $\mathfrak{G}_{\bullet}\to\mathfrak{G}$ be a univalent discrete
cocartesian fibration. Then
$\mathbb{Span}_{\mathfrak{P}^{\text{{inrt}}}}(\mathfrak{G}_{\bullet})\to\mathbb{Span}_{\mathfrak{P}^{\text{{inrt}}}}(\mathfrak{G})$
is a univalent discrete cocartesian fibration internally to
$\mathfrak{Seg}_{\mathfrak{P}^{\text{{inrt}}}}(\mathfrak{G})$. ∎

For an algebraic pattern with non-trivial active morphisms, the situation
becomes richer and goes beyond the $(\infty,2)$-topos theory of internal
categories in presheaf $(\infty,1)$-topoi. Indeed, 5.7 shows that, even when
restricting our attention as we are doing here from weak Segal fibrations to
strong ones, the morphisms of interest will be the lax morphisms, the maps of
underlying weak Segal fibrations. We will thus use a notion of lax univalence,
obtained by replacing strong morphisms by general lax morphisms of categorical
Segal $\mathfrak{P}$-$\infty$-groupoids in the definition of univalence for
fibrations in
$\mathfrak{Seg}_{\mathfrak{P}}(\mathfrak{\infty\textnormal{-}Grpd})$.

######  Conjecture 7.6.

Let $\mathfrak{G}_{\bullet}\to\mathfrak{G}$ be a univalent discrete
cocartesian fibration. Then
$\mathbb{Span}_{\mathfrak{P}}(\mathfrak{G}_{\bullet})\to\mathbb{Span}_{\mathfrak{P}}(\mathfrak{G})$
is a lax-univalent discrete cocartesian fibration internally to
$\mathfrak{Seg}_{\mathfrak{P}}(\mathfrak{G})$.

This section states that
$\mathbb{Span}_{\mathfrak{P}}(\mathfrak{G}_{\bullet})\to\mathbb{Span}_{\mathfrak{P}}(\mathfrak{G})$
classifies a class of discrete cocartesian fibrations. It remains to see that
_every_ such class is classified by a universal fibration of this form.

######  Conjecture 7.7.

Suppose
$\bigl{(}\mathfrak{G}^{(\kappa)}_{\bullet}\to\mathfrak{G}^{(\kappa)}\bigr{)}_{\kappa\in
K}$ is a sufficient family of univalent fibrations for
$\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}$. Then the family
$\bigl{(}\mathbb{Span}_{\mathfrak{P}}(\mathfrak{G}^{(\kappa)}_{\bullet})\to\mathbb{Span}_{\mathfrak{P}}(\mathfrak{G}^{(\kappa)})\bigr{)}_{\kappa\in
K}$ provides enough lax-univalent fibrations for
$\mathfrak{Cat}(\mathfrak{Seg}_{\mathfrak{P}}(\mathfrak{\infty\textnormal{-}Grpd}))$.

######  Corollary 7.8.

Let $\mathfrak{P}$ be a globally generated algebraic pattern and
$\mathfrak{X}\to\mathfrak{P}$ be a Segal $\mathfrak{P}$-fibration, and assume
that 7.6 and 7.7 hold. Then Segal $\mathfrak{X}$-objects in
$\mathfrak{\infty\textnormal{-}Grpd}$ are internal discrete cocartesian
fibrations over the straightening $\mathbb{X}$ of $\mathfrak{X}$.

###### Proof.

The key point is that, by [GK16, Proposition 3.8], if
$\mathfrak{G}^{(\kappa)}_{\bullet}\to\mathfrak{G}^{(\kappa)}$ is univalent
then $\mathfrak{G}^{(\kappa)}$ is a full sub-$(\infty,1)$-category of
$\mathfrak{\infty\textnormal{-}Grpd}$, from which it follows that lax
morphisms
$\mathfrak{X}\to\mathfrak{Span}_{\mathfrak{P}}(\mathfrak{G}^{(\kappa)})$ can
be seen as lax morphisms
$\mathfrak{X}\to\mathfrak{Span}_{\mathfrak{P}}(\mathfrak{\infty\textnormal{-}Grpd})$.
By the two conjectures, discrete cocartesian fibrations over $\mathbb{X}$ are
the same thing as lax morphisms
$\mathfrak{X}\to\mathfrak{Span}_{\mathfrak{P}}(\mathfrak{\infty\textnormal{-}Grpd})$
(factoring through some
$\mathfrak{Span}_{\mathfrak{P}}(\mathfrak{G}^{(\kappa)})$). At the same time,
by 5.7, the latter are the same thing as Segal $\mathfrak{X}$-objects in
$\mathfrak{\infty\textnormal{-}Grpd}$ (or, to be precise, in some
$\mathfrak{G}^{(\kappa)}$), which proves the result. ∎

######  Example 7.9 (Double fibrations and Segal fibrations).

For the algebraic pattern, 7.8 says explicitly that discrete cocartesian
fibrations of double $\infty$-categories correspond biunivocally to lax double
$\infty$-functors to the double $\infty$-category of spans (of
$\infty$-groupoids). This is precisely an $\infty$-categorical version of the
main construction of [Lam21]. This use of internal discrete fibrations is also
very similar to how [Ras21] deals with fibrations of Segal spaces (see also
[Lou23, §6.1.1] for the version for fibrations of $\omega$-categories).

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David Kern, KTH Royal Institute of Technology, Department of Mathematics,
SE-100 44 Stockholm, Sweden

Email address<EMAIL_ADDRESS>
_URL_ : https://dskern.github.io/