Complete Segal objects

$ \newcommand{\Sp}{\mathrm{Sp}} \newcommand{\Map}{\mathrm{Map}} \newcommand{\Ani}{\mathrm{Ani}} \newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\N{\mathrm{N}} \newcommand\R{\mathrm{R}} \newcommand\s{\mathrm{s}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Iscr{\mathcal{I}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Gpd}{\mathrm{Gpd}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand{\Dec}{\mathrm{Dec}} \newcommand{\CMon}{\mathrm{CMon}} \newcommand{\CGp}{\mathrm{CGp}} \newcommand{\bS}{\mathbf{S}} \newcommand{\Fin}{\mathrm{Fin}} \newcommand{\part}{\mathrm{part}} $

One model for $\infty$-categories is the theory of complete Segal objects in anima, due to Rezk [5]. The internal theory of complete Segal anima is shown by Lurie [3] to be equivalent to $\Cat_\infty$. In general, one can use complete Segal objects to speak of category objects in quite general $\infty$-categories. In this post, I explain that, for example, complete Segal objects in $\CMon(\Ani)$ are equivalent to symmetric monoidal $\infty$-categories. Here, $\CMon(\Ani)$ is the $\infty$-category of commutative monoid objects in $\Ani$, also known as the homotopy theory of $\bE_\infty$-spaces. This material is well-known to experts and I claim no originality.

Segal objects

Let $\Cscr$ be an $\infty$-category with finite limits and let $\s\Cscr$ denote the $\infty$-category $\Fun(\Delta^\op,\Cscr)$ of simplicial objects in $\Cscr$. For $n\geq 1$ and $1\leq i\leq n$, let $\rho_i\colon[1]\rightarrow[n]$ be the function $\rho_i(0)=i-1$ and $\rho_i(1)=i.$ Given an object $X_\bullet\in\s\Cscr$, there are induced maps $\rho_i\colon X_n\rightarrow X_1$. Moreover, since $\rho_i\circ\partial_0=\rho_{i+1}\circ\partial_1$, the $\rho_i$ assemble into a map \(X_n\rightarrow X_1\times_{X_0}X_1\times_{X_0}\cdots\times_{X_0}X_1.\) The Segal condition is that these maps are equivalences for $n\geq 2$. A Segal object, or category object, in $\Cscr$ is a simplicial object satisfying the Segal condition. The $\infty$-category of Segal objects in $\Cscr$ is the full subcategory $\Cat(\Cscr)\subseteq\s\Cscr$ on the Segal objects.

Suppose that $X_\bullet\in\Cat(\Ani)$ is a Segal anima. The connection between Segal objects and category theory is that $X_0$ is supposed to be anima of objects, $X_1$ is the anima of morphisms, and the Segal condition says that composition is well-defined up to coherent homotopy. For example, given two points $x,y\in X_0$, the mapping anima $\Map_\Cscr(x,y)$ will be obtained as the fiber of $X_1$ over the source and target maps $X_1\xrightarrow{\partial_1\times\partial_0} X_0\times X_0$.

Groupoids

Again, suppose that $\Cscr$ admits finite limits. Let $X=X_\bullet\in\s\Cscr$ be a simplicial object. Suppose that $S\cup T=[n]$ is a partition of $[n]$ and that $S\cap T$ is a singleton ${i}$. Then, there is a natural map \(X([n])\rightarrow X(S)\times_{X(\{i\})}X(T).\) One says that $X_\bullet$ is a groupoid object of $\Cscr$ if this map is an equivalence for all $n\geq 2$ and all such partitions. Let $\Gpd(\Cscr)\subseteq\s\Cscr$ denote the full subcategory of groupoid objects.

The definition here is a form of the Kan condition.

The notion of groupoid objects is also important in theory of $\infty$-topoi. Indeed, the $\infty$-categorical generalization of Giraud’s axioms for an $\infty$-topos includes the axiom that every groupoid object $X_\bullet$ be effective. This means that the geometric realization $|X_\bullet|$ exists and that $X_\bullet$ is equivalent to the Čech complex of the natural map $X_0\rightarrow|X_\bullet|$.

Example. Constant simplicial objects are groupoids and groupoid objects are category objects. A groupoid object is said to be constant if it is in the image of the constant simplicial object functor $\Cscr\rightarrow\Gpd(\Cscr)\subseteq\Cat(\Cscr)\subseteq\s\Cscr$.

Complete Segal objects

If $\Cscr$ admits finite limits, the inclusion $\Gpd(\Cscr)\subseteq\Cat(\Cscr)$ admits a right adjoint $(-)^\we$. When $\Cscr=\Ani$, this adjoint can be described as follows. Given a category object $X_\bullet\in\Cat(\Ani)$ there is a homotopy category $\Ho(X_\bullet)$ whose set of objects is $X_0$ and whose morphisms are built out of homotopy classes of elements of $X_1$. One can define $X_\bullet^\we\rightarrow X_\bullet$ by declaring that in simplicial degree $0$ it is $X_0$ and in simplicial degree $n\geq 1$ it corresponds to the components of $X_n$ corresponding to $n$ composable morphisms each of which is invertible in the homotopy category.

One can use Yoneda to extend this construction to more general $\Cscr$. To do so, one uses the univalence property. (This perspective was explained to me by Peter Haine.) Let $\Delta^3$ be the $3$-simplex on objects ${0,1,2,3}$. Let $K$ be the quotient of $\Delta^3$ obtained by collapsing the $1$-simplices $\Delta^{{0,2}}$ and $\Delta^{{1,3}}$ to separate points. Let $K^0\subseteq K$ be the image of $\Delta^{{1,2}}$. Since $\Cscr$ admits finite limits, given $X\in\s\Cscr$, it makes sense to evaluate $X$ on $K$ and on $K^0$.

The main thing we need about the right adjoint $X\mapsto X^\we$ is that the span $X^\we(K^0)\leftarrow X^\we(K)\rightarrow X(K)$ consists of equivalences and that $X^\we(\Delta^0)\rightarrow X(\Delta^0)$ is an equivalence for every $X\in\Cat(\Cscr)$.

A category object $X\in\Cat(\Cscr)$ is a complete Segal object in $\Cscr$ if the groupoid object $\Cscr^\we$ is constant (in which case it is constant on $X_0$).

Theorem (Lurie). There is a fully faithful functor $\Cat_\infty\rightarrow\s\Ani$ whose essential image consists of the complete Segal anima.

In more detail, there is a full subcategory of $\Cat_\infty$ consisting of the objects $\Delta^n=[n]$ for $n\geq 0$, where $[n]={0<\cdots<n}$. These assemble into a cosimplicial ($\infty$-)category. The restricted Yoneda along $\Delta^\bullet\colon\Delta\rightarrow\Cat$ induces a functor $\Cat_\infty\rightarrow\s\Ani$. It is this functor which appears in the theorem. This means in particular that if $\Cscr$ is an $\infty$-category with associated complete Segal anima $X_\bullet$, then $X_0=\Map_{\Cat_\infty}(\Delta^0,\Cscr)$, the underlying anima of $\Cscr$ and $X_1=\Map_{\Cat_\infty}(\Delta^1,\Cscr)$, the anima of arrows in $\Cscr$.

Symmetric monoidal $\infty$-categories

These too admit a description as certain functor objects. Recall from Section 2.4.2 of [4] that a symmetric monoidal $\infty$-category is the same as a commutative monoid object in $\Cat_\infty$. In particular, the $\infty$-category of symmetric monoidal $\infty$-categories and symmetric monoidal functors is equivalent to $\CMon(\Cat_\infty)$. These admit a compact description as follows. Let $\Fin^\part$ denote the category of finite sets and partially defined functions. The objects can be indexed on sets $\langle n\rangle = {1,\ldots,n}$ for $n\geq 0$. The morphisms $\langle m\rangle\rightarrow\langle n\rangle$ are functions to ${1,\ldots,n}$ defined on a subset of ${1,\ldots,m}$.

Let $\Cscr$ be an $\infty$-category with finite products. Given a functor $Y\colon\Fin^{\part}\rightarrow\Cscr$, there are partially defined functions $\gamma_i\colon{1,\ldots,n}\dashrightarrow{1}$ for $1\leq i\leq n$ sending $i$ to $1$ and undefined elsewhere. These induce a map $Y(\langle n\rangle)\rightarrow\prod_{i=1}^n Y(\langle 1\rangle)$.

Definition. A commutative monoid object in $\Cscr$ is a functor $Y\colon\Fin^\part\rightarrow\Cscr$ such that for each $n\geq 0$, the map $Y(\langle n\rangle)\rightarrow\prod_{i=1}^n Y(\langle 1\rangle)$ defined above is an equivalence.

This definition is due to Segal who called these objects $\Gamma$-spaces in the special case they take values in an appropriate model for the $\infty$-category of anima.

Remark. In anima, the commutative monoid objects model $\bE_\infty$-anima, i.e., the homotopy theory of topological spaces equipped with an action of an appropriate $\bE_\infty$-operad.

The principal I want to highlight is summarized in the following proposition.

Proposition. There is a fully faithful functor $\CMon(\Cat_\infty)\rightarrow\s\CMon(\Ani)$ whose essential image consists of the complete Segal objects in $\bE_\infty$-anima.

Proof. Both sides admit embeddings into $\Fun(\Fin^\part\times\Delta^\op,\Ani)$. It suffices to check that the two sides match up. If we drop completeness, we see that $\Cat(\CMon(\Ani))\we\CMon(\Cat(\Ani))$, since limits in functor categories are computed pointwise. However, the same fact implies that an object of $\Cat(\CMon(\Ani))$ is complete if and only if the corresponding object of $\Fun(\Fin^\part,\Cat(\Ani))$ takes values in complete Segal anima. This completes the proof.

I like this description because it makes somewhat more transparent how the symmetric monoidal structure interacts with the category theory structure. The proposition says that to give a symmetric monoidal $\infty$-category, one must first of all specify an anima $X_0$ of objects together with a commutative monoid structure on $X_0$. Then, there must be an $\bE_\infty$-anima $X_1$ of morphisms. The identity morphism map $X_0\rightarrow X_1$ is a map of $\bE_\infty$-anima. The source and target maps $X_1\rightarrow X_0$ are maps of $\bE_\infty$-anima, and on and on and on.

Example. More generally, if $\Iscr$ is an algebraic theory, then we see that $\Mod_\Iscr(\Cat_\infty)$, the $\infty$-category of models of $\Iscr$ in $\Cat_\infty$, is equivalent to the $\infty$-category of complete Segal objects in $\Mod_\Iscr(\Ani)$.

Remark. The proposition is related to a result of Gepner–Groth–Nikolaus, specifically Proposition B-3 of [1], which implies in particular that $\CMon(\Cat_\infty)\we\Cat_\infty\otimes\CMon(\Ani)$, where the tensor product is taking place in $\Pr^\L$.

Grouplike symmetric monoidal $\infty$-categories

Recall that a commutative monoid $X$ is grouplike if the map $X\times X\rightarrow X\times X$ induced by summation and one of the projections is an equivalence. If $\Cscr$ has finite products, let $\CGp(\Cscr)\subseteq\CMon(\Cscr)$ be the full subcategory of grouplike commutative monoids.

Example. The $\infty$-category $\CGp(\Ani)$ is equivalent to the $\infty$-category $\Sp_{\geq 0}$ of connective spectra.

Lemma. Suppose that $\Cscr$ is a symmetric monoidal $\infty$-category. If $\Cscr$ is grouplike, then it is contained in $\CGp(\Ani)$.

Proof. Arguments such as we have already made imply that $\CGp(\Cat_\infty)$ is equivalent to the $\infty$-category of complete Segal objects in $\CGp(\Ani)$. I claim that any complete Segal object in $\CGp(\Ani)$ is in fact a groupoid and hence constant. This will imply that the entire $\infty$-category of complete Segal objects in $\CGp(\Ani)$ is equivalent to $\CGp(\Ani)$. For this it is enough to check that every morphism in a grouplike symmetric monoidal $\infty$-category is in fact an equivalence. This is a straightforward exercise left to the reader.

References

[1] Gepner, Groth, and Nikolaus, Universality of multiplicative infinite loop space machines, Alg. Geo. Top. 15 (2015), 3107-3153. [arXiv:1305.4550]

[2] Lurie, Higher topos theory, version dated April 2017.

[3] Lurie, $(\infty,2)$-categories and Goodwillie calculus, version dated 8 October 2009.

[4] Lurie, Higher algebra, version dated 18 September 2017.

[5] Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 35(3) (2001), 973–1007. [arXiv:math/9811037]

[6] Segal, Categories and cohomology theories, Topology 13 (1974), 293–312.