Every 2-Segal space is unital

TL;DR

This paper proves that all 2-Segal spaces inherently possess a unital structure, establishing a fundamental property in their theory.

Contribution

It demonstrates that every 2-Segal space is unital, a previously unproven property in the study of these spaces.

Findings

All 2-Segal spaces are unital.

Unital property holds universally for 2-Segal spaces.

Provides a foundational result in higher category theory.

Abstract

We prove that every 2-Segal space is unital.

Equations

$$X_{n+1}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X_{n+1}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$P\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$Q\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$B\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Delta Q\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

EVERY $2$-SEGAL SPACE IS UNITAL

Matthew Feller

University of Virginia

  

Richard Garner

Macquarie University

  

Joachim Kock

Univ. Autònoma de Barcelona

May U. Proulx

University of Leicester

  

Mark Weber

Macquarie University

Introduction

$2$-Segal spaces were introduced by Dyckerhoff and Kapranov [1] for applications in representation theory, homological algebra, and geometry, motivated in particular by Waldhausen’s $S$-construction and Hall algebras. A $2$-Segal space is a simplicial space $X$ such that for every triangulation $T$ of every convex plane $n$-gon (for $n\geqslant 2$), we have $X_{n}\simeq\lim_{t\in T}X(t)$. Independently, a little later, Gálvez-Carrillo, Kock, and Tonks [2] introduced the notion of decomposition space for applications in combinatorics, in connection with Möbius inversion. A decomposition space is a simplicial space $X\colon\mathbbe{\Delta}^{\text{{\rm{op}}}}\rightarrow\mathcal{S}$ for which all pushouts of active maps along inert maps in $\mathbbe{\Delta}$ are sent to pullbacks in $\mathcal{S}$. Here, the inert maps in $\mathbbe{\Delta}$ are generated by the outer coface maps, while the active maps are generated by the codegeneracy and inner coface maps. The condition satisfied by $X$ with respect to pushouts of outer coface maps against inner ones is precisely equivalent to the $2$-Segal condition. For Dyckerhoff and Kapranov, the condition for pushouts of outer cofaces against codegeneracies is a further axiom which they call unitality [1, Definition 2.5.2]. Thus, decomposition spaces are the same thing as unital $2$-Segal spaces. While the $2$-Segal axiom is expressly the condition required in order to induce a (co)associative (co)multiplication on the linear span of $X_{1}$, the unitality condition ensures that this (co)multiplication is (co)unital, which is an important property in many applications.

The present note shows that the unitality condition is actually automatic, by proving:

Theorem. Every $2$-Segal space is unital.

This result is unexpected, as it is not so common in mathematics for (co)associativity to imply (co)unitality.

1 Definitions and theorem

In order to cover all flavours of $2$-Segal space that appear in the literature, we give a proof which applies both to $2$-Segal objects in an $\infty$-category with finite limits and to $2$-Segal objects in a Quillen model category. From now on, $\mathcal{C}$ will denote either an $\infty$-category with finite limits or a Quillen model category. In the latter case, “pullback” will mean a (strictly commuting) homotopy pullback.

Definition**.**

(cf. [1], [2]) A simplicial object $X\colon\mathbbe{\Delta}^{\text{{\rm{op}}}}\to\mathcal{C}$ is called $2$-Segal when the commuting squares that express the simplicial identities between inner and outer face maps of $X$ are pullback squares. More precisely, for all $0<i<n$ we have pullbacks

[TABLE]

We say that $X$ is upper $2$-Segal when only squares as to the left are required to be pullbacks, and lower $2$-Segal when this is only required for squares as to the right. 111For our purposes, splitting into upper $2$-Segal and lower $2$-Segal is just for economy; in the theory of higher Segal spaces [5] ($k$-Segal spaces for $k>2$), the distinction between upper and lower becomes essential.

Definition**.**

A $2$-Segal space $X$ is called unital if for all $0\leqslant i\leqslant n$ the following squares are pullbacks:

[TABLE]

We call an upper $2$-Segal space upper unital when only the pullbacks on the left are required, and call a lower $2$-Segal space lower unital when only the pullbacks on the right are required.

Theorem**.**

Every $2$-Segal space is unital. More precisely, every upper $2$-Segal space is upper unital, and every lower $2$-Segal space is lower unital.

2 The proof

By symmetry, it is enough to prove:

Proposition 2.1**.**

If $X\colon\mathbbe{\Delta}^{\text{{\rm{op}}}}\to\mathcal{C}$ is upper $2$-Segal, then it is also upper unital.

We do so using two lemmas, which are standard both in $\infty$-category theory and model category theory.

Lemma 2.2** (Prism Lemma).**

Given a commuting diagram

[TABLE]

(formally a $\Delta^{1}\times\Delta^{2}$-diagram in the $\infty$-category case), suppose the right-hand square is a pullback. Then the outer rectangle is a pullback if and only if the left-hand square is a pullback.

Proof.

For the $\infty$-category version, see (the dual of) [4, Lemma 4.4.2.1]. The model category version is proven in the right-proper case in [3, Proposition 13.3.9]; we give the general case in the appendix. ∎

Lemma 2.3**.**

Pullback squares are stable under retract.

Proof.

The $\infty$-category version follows from (the dual of) [4, Lemma 5.1.6.3]. The model category version is known to experts, but since we do not know of any reference, we give a proof in the appendix. ∎

Proof of Proposition 2.1.

We first establish the pullback condition for $n\geqslant 1$ and $0\leqslant i\leqslant n$ by following the argument of [2, Proposition 3.5], exploiting that every degeneracy map except $s_{0}\colon X_{0}\rightarrow X_{1}$ is a section of an inner face map. Explicitly, if we choose $j\in\{i,i+1\}$ with $0<j\leqslant n$, then $s_{i}\colon X_{n}\rightarrow X_{n+1}$ is a section of the inner face map $d_{j}\colon X_{n+1}\rightarrow X_{n}$ and $s_{i+1}\colon X_{n+1}\rightarrow X_{n+2}$ is a section of $d_{j+1}$, forming the prism diagram to the left below. Here the outer square is a pullback since its top and bottom edges are the images of identity maps in $\mathbbe{\Delta}$, while the right-hand square is a pullback since $X$ is upper $2$-Segal and $d_{j}$ is an inner face map. So by Lemma 2.2, the left-hand square is a pullback as required.

[TABLE]

The remaining case, which is not covered by [2, Proposition 3.5], is the square with $n=i=0$. To see that this is a pullback, we exhibit it as a retract of the square for $n=i=1$, as displayed above right. Since we already know the $n=i=1$ square is a pullback, so is the $n=i=0$ square by Lemma 2.3. ∎

Appendix

We provide proofs of the two lemmas in the context of a model category $\mathcal{C}$. First we recall the notion of (strictly commuting) homotopy pullback. Writing $\Lambda$ for the cospan category $0\rightarrow 2\leftarrow 1$, we endow $\mathcal{C}^{\Lambda}$ with the injective model structure, whose weak equivalences and cofibrations are pointwise, and whose fibrant objects are cospans of fibrations between fibrant objects in $\mathcal{C}$. A commuting square in $\mathcal{C}$, as to the left in

[TABLE]

is a homotopy pullback if for some (equivalently, any) fibrant replacement in $\smash{\mathcal{C}^{\Lambda}}$ for its underlying cospan, as displayed to the right above, the induced map $\smash{P\rightarrow A_{0}^{\prime}\times_{A^{\prime}_{2}}A^{\prime}_{1}}$ into the strict pullback is a weak equivalence.

Proof of Lemma 2.2 in the model category case..

We first replace $D\rightarrow E\rightarrow F\leftarrow C$ by a diagram of fibrations between fibrant objects, as to the left in:

[TABLE]

By taking strict pullbacks we complete this to the diagram as to the right. Since the right-hand back face is assumed to be a homotopy pullback, $B\rightarrow B^{\prime}$ is a weak equivalence, and so $D^{\prime}\twoheadrightarrow E^{\prime}\twoheadleftarrow B^{\prime}$ is a fibrant replacement for $D\rightarrow E\leftarrow B$ in $\mathcal{C}^{\Lambda}$. Thus $A\rightarrow A^{\prime}$ is a weak equivalence exactly when the left-hand back face is a homotopy pullback. Since $D^{\prime}\twoheadrightarrow F^{\prime}\twoheadleftarrow C^{\prime}$ is a fibrant replacement for $D\rightarrow F\leftarrow C$, we also have that $A\rightarrow A^{\prime}$ is a weak equivalence exactly when the back rectangle is a homotopy pullback, as desired. ∎

Proof of Lemma 2.3 in the model category case..

Suppose given a homotopy pullback square in $\mathcal{C}$, together with a retract of it in the category of commutative squares in $\mathcal{C}$, as to the left in:

[TABLE]

We must show that the left (equally, the right) face of this diagram is also a homotopy pullback. Regarding $B_{0}\rightarrow B_{2}\leftarrow B_{1}$ and $A_{0}\rightarrow A_{2}\leftarrow A_{1}$ as objects of $\mathcal{C}^{\Lambda}$, and regarding $Q$ and $P$ as constant objects $\Delta Q$ and $\Delta P$, we obtain a retract diagram of arrows in $\mathcal{C}^{\Lambda}$ as displayed to the right above. We will now fibrantly replace $A$ and $B$ in $\mathcal{C}^{\Lambda}$ in such a way as to obtain a new retract diagram $B^{\prime}\to A^{\prime}\to B^{\prime}$. To this end, we first fibrantly replace $B$ via a trivial cofibration $B\mathrel{\stackrel{{\scriptstyle\scriptscriptstyle\sim}}{{\vphantom{a}\smash{\rightarrowtail}}}}B^{\prime}$. Now we factor the composite $A\rightarrow B\rightarrow B^{\prime}$ as a trivial cofibration $A\mathrel{\smash{\stackrel{{\scriptstyle\scriptscriptstyle\sim}}{{\vphantom{a}\smash{\rightarrowtail}}}}}A^{\prime}$ followed by a fibration $A^{\prime}\twoheadrightarrow B^{\prime}$. Finally, we take a lifting in the square

[TABLE]

Altogether, we now have a retract diagram in the category of composable pairs in $\mathcal{C}^{\Lambda}$, as to the left in:

[TABLE]

By forming the strict pullbacks $Q^{\prime}$ and $P^{\prime}$ of the cospans $B^{\prime}$ and $A^{\prime}$ we may complete this to the retract diagram as to the right above; note in particular that the map $Q\rightarrow Q^{\prime}$ in $\mathcal{C}$ is a retract of the map $P\rightarrow P^{\prime}$. Since $\Delta P\rightarrow A$ describes a homotopy pullback, $P\rightarrow P^{\prime}$ is a weak equivalence; so its retract $Q\rightarrow Q^{\prime}$ is also a weak equivalence, which is to say that $\Delta Q\rightarrow B$ also describes a homotopy pullback. ∎

Acknowledgments. R.G. was supported by Australian Research Council grants DP160101519 and FT160100393. J.K. was supported by grants MTM2016-80439-P (AEI/FEDER, UE) of Spain and 2017-SGR-1725 of Catalonia. M.W. was supported by Czech Science Foundation grant GA CR P201/12/G028.