Weighted limits in an $(\infty ,1)$-category

Abstract

We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal’s approach: we identify a meaningful construction for the quasi-category of weighted cones over a diagram in a quasi-category, 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 quasi-category arises as the homotopy coherent nerve of a category enriched over Kan complexes, we generalize an argument by Riehl-Verity 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 quasi-category is complete, tensored and cotensored over the quasi-category of spaces, we discuss a possible comparison of our definition of weighted limit with the approach by Gepner-Haugseng-Nikolaus.

A Necklaces in a weighted join

In this section, we give a description of the hom-categories 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 head-to-tail wedge of standard simplices

$$\begin{aligned} \varDelta [\mathbf {n}]:=\varDelta [n_1]\vee \dots \vee \varDelta [n_k]. \end{aligned}$$

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 non-degenerate 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 non-degenerate) necklace of degree m consists of a pair $\left( \tau ,(T_i)_{i=0}^m\right)$ where $\tau$ is a (totally non-degenerate) necklace and $(T_i)_i$ is a flag of degree m for $\tau$. In particular, a flagged (totally non-degenerate) necklace of degree 0 consists of just of a (totally non-degenerate) 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 non-degenerate necklaces of degree m in J with source x and target $x'$, there is a well-defined concatenation

$$\begin{aligned}{}[-,-]:{{\,\mathrm{Nec}\,}}(J)_m(x,x')\times {{\,\mathrm{Nec}\,}}(J)_m(x',x'')\rightarrow {{\,\mathrm{Nec}\,}}(J)_m(x,x''), \end{aligned}$$

given by the assignment

$$\begin{aligned} \left( (\tau :\varDelta [\mathbf {n}]\rightarrow J,(T_i)_{i=0}^m),(\tau ':\varDelta [\mathbf {n}']\rightarrow J,(T'_i)_{i=0}^m)\right) \mapsto ([\tau ,\tau ']:\varDelta [\mathbf {n}]\vee \varDelta [\mathbf {n}']\rightarrow J,(T_i\cup T_i')_{i=0}^m). \end{aligned}$$

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 Dugger-Spivak [8].

Theorem A.5

( [28, Theorem 2.1]) For any simplicial set J and $n\ge 0$, there is a bijection

$$\begin{aligned} {{\,\mathrm{Map}\,}}_{{\mathfrak {C}}[J]}(x,x')_m\cong {{\,\mathrm{Nec}\,}}(J)_m(x,x') \end{aligned}$$

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 non-degenerate necklace of degree m in a fat join $I\star ^p J$

$$\begin{aligned} (\tau :\varDelta [\mathbf {n}]\rightarrow I\star ^p J,(T_i)_{i=0}^m) \end{aligned}$$

can be uniquely written as a concatenation of three consecutive flagged necklaces of the following form:

• 1. A flagged totally non-degenerate 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 non-degenerate 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 non-degenerate 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

$$\begin{aligned} {\overline{k}}:=\max \{k\ |\ \tau (\varDelta [n_k])\subset I\}. \end{aligned}$$

Note then that the bead

$$\begin{aligned} \tau |_{\varDelta [{\overline{k}}+1]}:\varDelta [{\overline{k}}+1]\rightarrow I\star ^p J \end{aligned}$$

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

$$\begin{aligned} \begin{array}{rcl} (I\star ^p J)_{{\overline{k}}+1}&{}:=&{}\left( I_{\overline{k}+1}\amalg \coprod _{i=1}^{{\overline{k}}}(I_{i}\times {\tilde{J}}_{\overline{k}-i})\amalg {\tilde{J}}_{{\overline{k}}+1}\right) \amalg _{\tilde{J}_{{\overline{k}}+1}}J_{{\overline{k}}+1}\\ &{}\cong &{} I_{{\overline{k}}+1}\amalg \coprod _{i=1}^{\overline{k}}(I_{i}\times {\tilde{J}}_{{\overline{k}}-i})\amalg J_{{\overline{k}}+1}. \end{array} \end{aligned}$$

Thus the simplex $\tau |_{\varDelta [{\overline{k}}+1]}$, which is not contained entirely in I nor ${\tilde{J}}$, can be uniquely written as

$$\begin{aligned} \sigma ^I\star \sigma ^{{\tilde{J}}}:\varDelta [\overline{k}+1]\cong \varDelta [i]\star \varDelta [{\overline{k}}-i]\rightarrow I\star {\tilde{J}}, \end{aligned}$$

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

$$\begin{aligned} (\tau ^I,\sigma ^I\star \sigma ^J,\tau ^J)\mapsto [\tau ^I,\sigma ^I\star \sigma ^{{\tilde{J}}},\tau ^J] \end{aligned}$$

can be checked to be an inverse for the decomposition

$$\begin{aligned} \tau \mapsto (\tau ^I,\sigma ^I\star \sigma ^{{\tilde{J}}},\tau ^J) \end{aligned}$$

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 non-degenerate necklace of degree m in a fat join $\varDelta [0]\star ^p J$

$$\begin{aligned} (\tau :\varDelta [\mathbf {n}]\rightarrow \varDelta [0]\star ^p J,(T_i)_{i=0}^m) \end{aligned}$$

can be uniquely written as a concatenation of two flagged consecutive necklaces of the following form:

• 1. a flagged non-degenerate 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 non-degenerate 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

$$\begin{aligned} \varPhi _0:{{\,\mathrm{Map}\,}}_{{\mathfrak {C}}[I\star ^p J]}(a,b)_0\rightarrow {{\,\mathrm{Map}\,}}_{\mathfrak C[I\star \varDelta [0]]}(a,\top )_0\times {{\,\mathrm{Map}\,}}_{\mathfrak C[\varDelta [0]\star ^pJ]}(\bot ,b)_0, \end{aligned}$$

which, modulo the identifications from Theorem A.5 and Propositions A.6 and A.7, is described by

$$\begin{aligned} \tau =[\tau ^I,\sigma ^I\star \sigma ^{\tilde{J}},\tau ^J]\mapsto \left( [\tau ^I,\sigma ^I\star \bot ],[\top \star \sigma ^{{\tilde{J}}},\tau ^J]\right) . \end{aligned}$$

The map $\varPhi _0$ is bijective. Indeed, the map

$$\begin{aligned} \varPsi _0:{{\,\mathrm{Map}\,}}_{\mathfrak C[I\star \varDelta [0]]}(a,\top )_0\times {{\,\mathrm{Map}\,}}_{\mathfrak C[\varDelta [0]\star ^pJ]}(\bot ,b)_0\rightarrow {{\,\mathrm{Map}\,}}_{\mathfrak C[I\star ^pJ]}(a,b)_0, \end{aligned}$$

described, again using the identifications from Theorem A.5 and Propositions A.6 and A.7, by the assignment

$$\begin{aligned} (\nu ^I,\nu ^J)=\left( [\tau ^I,\sigma ^I\star \top ],[\bot \star \sigma ^{\tilde{J}},\tau ^J]\right) \mapsto [\tau ^I,\sigma ^I\star \sigma ^{\tilde{J}},\tau ^J] \end{aligned}$$

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 well-defined function

$$\begin{aligned} \varPhi _m:{{\,\mathrm{Map}\,}}_{{\mathfrak {C}}[I\star ^pJ]}(a,b)_m\rightarrow {{\,\mathrm{Map}\,}}_{\mathfrak C[I\star \top ]}(a,\top )_m\times {{\,\mathrm{Map}\,}}_{{\mathfrak {C}}[\bot \star J]}(\bot ,b)_m. \end{aligned}$$

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

$$\begin{aligned} \left( T_i\cap V(\sigma ^I\star \sigma ^{{\tilde{J}}})\cap V(I)\right) \cup \{\top \}\quad \text { and }\quad \left( T_i\cap V(\sigma ^I\star \sigma ^{{\tilde{J}}})\cap V(\tilde{J})\right) \amalg _{V({\tilde{J}})}V(J)\cup \{\bot \}. \end{aligned}$$

The map $\varPhi _m$ is bijective. Indeed, the function $\varPsi _0=\varPhi _0^{-1}$ can also be be promoted to a well-defined function

$$\begin{aligned} \varPsi _m:{{\,\mathrm{Map}\,}}_{{\mathfrak {C}}[I\star ^pJ]}(a,b)_m\rightarrow {{\,\mathrm{Map}\,}}_{\mathfrak C[I\star \top ]}(a,\top )_m\times {{\,\mathrm{Map}\,}}_{{\mathfrak {C}}[\bot \star J]}(\bot ,b)_m \end{aligned}$$

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

$$\begin{aligned} (S^I_i\setminus \{\top \})\cup (S^J_i\setminus \{\bot \})\amalg _{V({\tilde{J}})}V(J). \end{aligned}$$

Finally, a direct inspection should show the components of $\varPhi _m$,

$$\begin{aligned} {{\,\mathrm{Map}\,}}_{{\mathfrak {C}}[I\star ^pJ]}(a,b)_m\rightarrow {{\,\mathrm{Map}\,}}_{\mathfrak C[I\star \varDelta [0]]}(a,\top )_m\text { and }{{\,\mathrm{Map}\,}}_{\mathfrak C[I\star ^pJ]}(a,b)_m\rightarrow {{\,\mathrm{Map}\,}}_{{\mathfrak {C}}[\varDelta [0]\star ^pJ]}(\bot ,b)_m, \end{aligned}$$

is compatible with the simplicial structure, yielding the desired simplicial isomorphism. $\square$