Abstract
We introduce the notion of weighted limit in an arbitrary quasicategory, suitably generalizing ordinary limits in a quasicategory, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal’s approach: we identify a meaningful construction for the quasicategory of weighted cones over a diagram in a quasicategory, whose terminal object is the weighted limit of the considered diagram. We then show that each weighted limit can be expressed as an ordinary limit. When the quasicategory arises as the homotopy coherent nerve of a category enriched over Kan complexes, we generalize an argument by RiehlVerity to show that the weighted limit agrees with the homotopy weighted limit in the sense of enriched category theory, for which explicit constructions are available. When the quasicategory is complete, tensored and cotensored over the quasicategory of spaces, we discuss a possible comparison of our definition of weighted limit with the approach by GepnerHaugsengNikolaus.
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Notes
The nature of the model of quasicategories as \((\infty ,1)\)categories is such that strict notions do not make sense, and all constructions are automatically derived. For instance, the only kind of limit that makes sense in a quasicategory is already a homotopy limit. For the same reason, it only makes sense to introduce homotopy weighted limits in a quasicategory, and we will just refer to them as weighted limits.
In general, the comma quasicategory for any two given functors with the same target appears itself as weighted limit in the simplicial category \( q {\mathcal {C}}\! at \), and is reviewed in Example 3.11.
To avoid size issues, all simplicial categories are assumed to be small unless otherwise specified.
The reader should be aware that this expression contains an abuse of notation, as the left hand side is define up to isomorphism and the right hand side is only defined up to equivalence. It should be read as: the equivalence class of the strict weighted limit represents the homotopy weighted limit.
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Acknowledgements
The author is grateful to Emily Riehl for suggesting this project and for sharing some of her work in progress with Dominic Verity, as well as for useful discussions on the subject and valuable feedback. This paper also benefited from conversations with Viktoriya Ozornova and from the referee’s feedback. The author was partially funded by the Swiss National Science Foundation, grant P2ELP2_172086.
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A Necklaces in a weighted join
A Necklaces in a weighted join
In this section, we give a description of the homcategories of a homotopy coherent realization \({\mathfrak {C}}[J]\) in terms of flagged necklaces, and then specialize the result to understand \(\mathfrak C[I\star ^{p}J]\).
Notation A.1
Given \(\mathbf {n}:=(n_1,\dots ,n_k)\), denote by \(\varDelta [\mathbf {n}]\) the headtotail wedge of standard simplices
We slightly revisit the standard terminology of necklaces from [8, §1.1] and [28, §2].
Definition A.2
Let J be a simplicial set.

A necklace in J consists of a map \(\tau :\varDelta [\mathbf {n}]\rightarrow J\). The necklace is totally nondegenerate if each bead \(\tau _{\varDelta [n_k]}:\varDelta [n_k]\rightarrow J\) represents a non degenerate simplex of J. We denote by \(V(\tau ):=\tau (\varDelta [\mathbf {n}])_0\) the set of all vertices of \(\tau (\varDelta [\mathbf {n}])\subset J\) and by \(J(\tau )\) the subset of all the vertices of \(\tau (\varDelta [\mathbf {n}])\) coming from vertices of \(\varDelta [\mathbf {n}]\) along which the beads of \(\tau \) have been glued.

A flag of degree m for the necklace \(\tau :\varDelta [\mathbf {n}]\rightarrow J\) consists of a sequence of nested sets \((T_i)_{i=0}^m\) such that
$$\begin{aligned} J(\tau )=T_0\subset \dots \subset T_m=V(\tau ). \end{aligned}$$ 
A flagged (totally nondegenerate) necklace of degree m consists of a pair \(\left( \tau ,(T_i)_{i=0}^m\right) \) where \(\tau \) is a (totally nondegenerate) necklace and \((T_i)_i\) is a flag of degree m for \(\tau \). In particular, a flagged (totally nondegenerate) necklace of degree 0 consists of just of a (totally nondegenerate) unflagged necklace \(\tau \).
Remark A.3
The collection \({{\,\mathrm{Nec}\,}}(J)_m\) of flagged necklaces in J of degree m is functorial in J.
Remark A.4
( [8, §1.1], [28, §2]) Any flagged necklace in J has a source and a target. If \({{\,\mathrm{Nec}\,}}(J)_m(x,x')\) denotes the collection of flagged totally nondegenerate necklaces of degree m in J with source x and target \(x'\), there is a welldefined concatenation
given by the assignment
The following result from [28] records how necklaces in J describe morphisms of \({\mathfrak {C}}[J]\), and is a variant of the characterization originally due to DuggerSpivak [8].
Theorem A.5
( [28, Theorem 2.1]) For any simplicial set J and \(n\ge 0\), there is a bijection
which is natural in J.
Aiming to understand the morphisms of \({\mathfrak {C}}[I\star ^p J]\), we look at necklaces in \(I\star ^p J\). Intuitively, a necklace in a weighted join must consist of three (possibly empty) parts: a necklace in I, followed by a single simplex in \(I\star {\tilde{J}}\), followed by a necklace in J. Pushing the investigation further, by nature of the join construction the simplex in the middle must consist of a simplex of I and a simplex of \({\tilde{J}}\).
The following proposition makes this precise.
Proposition A.6
Let \(p:{\tilde{J}}\rightarrow J\) be a map. Any flagged totally nondegenerate necklace of degree m in a fat join \(I\star ^p J\)
can be uniquely written as a concatenation of three consecutive flagged necklaces of the following form:

1.
A flagged totally nondegenerate necklace of degree m in I
$$\begin{aligned} (\tau ^I:\varDelta [\mathbf {n}^I]\rightarrow I,(T^I_i)_{i=0}^m) \end{aligned}$$whose source and target are vertices of I.

2.
A flagged nondegenerate bead of degree m in \(I\star ^p\tilde{J}\) of the form
$$\begin{aligned} (\sigma ^I\star \sigma ^{\tilde{J}}:\varDelta [N^I+N^J]\cong \varDelta [N^I]\star \varDelta [N^b]\rightarrow I\star {\tilde{J}},(S_i)_{i=0}^m) \end{aligned}$$whose source is a vertex of I and whose target is a vertex of \({\tilde{J}}\).

3.
A flagged totally nondegenerate necklace of degree m in J.
$$\begin{aligned} (\tau ^J:\varDelta [\mathbf {n}^J]\rightarrow J,(T^J_i)_{i=0}^m) \end{aligned}$$whose source and target are vertices of J.
Proof
First let
Note then that the bead
defines a simplex in \(I\star ^p J\) whose source is a vertex of I and whose target is a vertex of \({\tilde{J}}\). By unpacking the definition of the weighted join \(I\star ^p J\), the collection of \(({\overline{k}}+1)\)simplices of \(I\star ^pJ\) is given by
Thus the simplex \(\tau _{\varDelta [{\overline{k}}+1]}\), which is not contained entirely in I nor \({\tilde{J}}\), can be uniquely written as
where \(\sigma ^I:\varDelta [i]\rightarrow I\) is a simplex of I, and \(\sigma ^{{\tilde{J}}}:\varDelta [{\overline{k}}i]\rightarrow {\tilde{J}}\) is a simplex of \({\tilde{J}}\). We now build the three pieces.

1.
If we set \(\mathbf {n}^I:=(n_1,\dots ,n_{{\overline{k}}})\), we get a necklace in I
$$\begin{aligned} \tau ^I:=\tau _{\varDelta [\mathbf {n}^I]}:\varDelta [\mathbf {n}^I]\rightarrow I, \end{aligned}$$endowed with the flag \(T_i^I:=T_i\cap V(I)\).

2.
If we set \(N^I:=i\) and \(N^J:= {\overline{k}}i\), we get a simplex in \(I\star {\tilde{J}}\)
$$\begin{aligned} \sigma ^I\star \sigma ^{{\tilde{J}}}=\tau _{\varDelta [\overline{k}]}:\varDelta [{\overline{k}}+1]\cong \varDelta [i]\star \varDelta [\overline{k}i]\rightarrow I\star {\tilde{J}}, \end{aligned}$$endowed with the flag \(S_i:=T_i\cap V(\tau _{\varDelta [n_{\overline{k}}]})\).

3.
If we set \(\mathbf {n}^J:=(n_{{\overline{k}}+2},\dots ,n_K)\), we get a necklace in J
$$\begin{aligned} \tau ^J:=\tau _{\varDelta [\mathbf {n}^J]}:\varDelta [\mathbf {n}^J]\rightarrow J, \end{aligned}$$endowed with the flag \(T_i^J:=T_i\cap V(J)\).
This proves the existence of the desired decomposition. For uniqueness, we observe that the assignment is in fact invertible and there is no loss of information. Indeed, each triple of concatenable flagged necklaces \((\tau ^I,\sigma ^I\star \sigma ^{{\tilde{J}}},\tau ^J)\) of the form (1), (2) and (3) (as in the statement of the proposition) can be concatenated to form a flagged necklace \([\tau ^I,\sigma ^I\star \sigma ^{{\tilde{J}}},\tau ^J]\) in \(I\star ^p J\). The operation
can be checked to be an inverse for the decomposition
that we described in the first part of this proof. \(\square \)
Specializing to the case \(I:=\varDelta [0]\), we obtain the following.
Proposition A.7
Let \(p:{\tilde{J}}\rightarrow J\) be a map. Any flagged totally nondegenerate necklace of degree m in a fat join \(\varDelta [0]\star ^p J\)
can be uniquely written as a concatenation of two flagged consecutive necklaces of the following form:

1.
a flagged nondegenerate bead of degree m in \(\varDelta [0]\star {\tilde{J}}\) of the form
$$\begin{aligned} (\top \star \sigma ^{\tilde{J}}:\varDelta [1+N]\cong \varDelta [0]\star \varDelta [N]\rightarrow \varDelta [0]\star \tilde{J},(S_i)_{i=0}^m) \end{aligned}$$from the cone boint \(\top \) and to a vertex of \({\tilde{J}}\);

2.
a flagged totally nondegenerate necklace of degree m in J.
$$\begin{aligned} (\tau ^J:\varDelta [\mathbf {n}^J]\rightarrow J,(T^J_i)_{i=0}^m) \end{aligned}$$whose source and target are vertices of J.
We are finally ready to prove Theorem 3.38, which describes the morphisms of \({\mathfrak {C}}[I\star ^b J]\) from a vertex a of I to a vertex b of J.
Proof of Theorem 3.38
In degree m, consider the map
which, modulo the identifications from Theorem A.5 and Propositions A.6 and A.7, is described by
The map \(\varPhi _0\) is bijective. Indeed, the map
described, again using the identifications from Theorem A.5 and Propositions A.6 and A.7, by the assignment
can be checked to be the inverse function.
By taking into account the structure of flags suitably, the assignment that defines \(\varPhi _0\) can be promoted to a welldefined function
If \((T_i)_i\) denotes the flag of \(\tau \), the flags for \(\tau ^I\) and \(\tau ^J\) are described in the proof of Proposition A.6, and we declare that the flags for \(\sigma ^I\star \bot \) and \(\top \star \sigma ^{{\tilde{J}}}\) are given for \(i=0,\dots ,m\) by
The map \(\varPhi _m\) is bijective. Indeed, the function \(\varPsi _0=\varPhi _0^{1}\) can also be be promoted to a welldefined function
by taking into account the flags as follows. If \((S^I_i)_i\) and \((S^J_i)_i\) are the flags of \(\nu ^I\) and \(\nu ^J\), the flag of \(\sigma ^I\star \sigma ^{{\tilde{J}}}\) is declared to be given for \(i=0,\dots ,m\) by
Finally, a direct inspection should show the components of \(\varPhi _m\),
is compatible with the simplicial structure, yielding the desired simplicial isomorphism. \(\square \)
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Rovelli, M. Weighted limits in an \((\infty ,1)\)category. Appl Categor Struct 29, 1019–1062 (2021). https://doi.org/10.1007/s1048502109643z
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DOI: https://doi.org/10.1007/s1048502109643z