nLab Segal's category

This entry is about the small category named after Graeme Segal. Not to be confused with the model in higher category theory called Segal categories.

Definition

Definition

Segal’s category, denoted by $\Gamma$, is the category opposite to the skeleton of the category $FinSet^{*/}$ of pointed finite sets:

$$\Gamma^{op} \simeq FinSet^{*/}.$$

Equivalently, $\Gamma$ is the category opposite to the category $FinSetPart$ of finite sets and partially defined maps. Objects are finite sets. Morphisms $A\to B$ are pairs $(A',f)$, where $A'\subset A$ and $f\colon A'\to B$ is a map of sets. Identity morphisms are specified by $id_A=(A,id_A)$. Composition is defined as follows:

$$(A',f)\colon A\to B,\qquad (B',g)\colon B\to C,\qquad(B',g)\circ(A',f)=(A'\cap f^{-1}B',g\circ f|_{A'\cap f^{-1}B'}).$$

Equivalently (Segal, Definition 1.1), $\Gamma$ admits the following covariant description. Objects are finite sets. Morphisms $A\to B$ are maps $f\colon A\to P(B)$ such that $a_1\ne a_2$ implies $f(a_1)\cap f(a_2)=\emptyset$. Identity morphisms are specified by $id_A(a)=\{a\}$. Composition is defined as follows:

$$f\colon A\to P(B),\qquad g\colon B\to P(C),\qquad (g\circ f)(a)=\bigcup_{b\in f(a)}g(b).$$

Remark

The category $\Gamma$ is related to (infinity,1)-operads in a way similar to how the simplex category (non-empty and linearly ordered finite sets) is related to (∞,1)-categories.

Remark

The categories $FinSetPart$ and $FinSet_*$ are equivalent via the following pair of functors. The functor

$$FinSetPart\to FinSet_*$$

sends

$$A\mapsto A\sqcup\{*\},\qquad ((A',f)\colon A\to B)\mapsto(g\colon A\sqcup\{*\}\to B\sqcup\{*\}),\qquad g|_{A'}=f,\qquad g|_{\bar A'}(x)=*.$$

The functor

$$FinSet_* \to FinSetPart$$

sends

$$A\mapsto A\setminus\{*\},\qquad (g\colon A\to B)\mapsto(g^{-1}(B\setminus\{*\}),g|_{A\setminus\{*\}}).$$

Comparison to the simplex category

Recall the simplex category $\Delta$, whose objects are finite inhabited totally ordered sets and morphisms are order-preserving maps of sets. We set

$$[m]=\{0\lt\cdots\lt m\}.$$

Interpreting a simplex $A\in\Delta$ as a (posetal) category, a spine edge in $A$ is a morphism in $A$ that is not an identity morphism and cannot be decomposed as the composition of two nonidentity morphisms. That is to say, in a standard simplex $[m]$, a spine edge is a pair of the form $(i-1,i)$, $0\lt i\le m$, which encodes the morphism $(i-1)\to i$.

The canonical functor

$$\iota\colon\Delta\to\Gamma$$

sends a simplex to its set of spine edges:

$$[m]=\{0\lt1\lt2\lt\cdots\lt m\}\mapsto \{(0,1),(1,2),\ldots,(m-1,m)\}=\iota[m].$$

To define $\iota$ on morphisms, observe that a morphism of simplices $f\colon[m]\to[n]$ can be interpreted as a functor between posetal categories. This functor sends a spine edge in $[m]$ to the composition of a set of spine edges in $[n]$. More specifically, $f$ sends a spine edge $i-1\to i$ to the composition of spine edges

$$f(i-1)\to f(i-1)+1\to \cdots \to f(i)-1\to f(i).$$

Using Segal’s covariant description of $\Gamma$, we can define a morphism $\iota(f)\colon\iota[m]\to\iota[n]$ by sending a spine edge $e$ in $[m]$ to the set of spine edges in $[n]$ whose composition yields the image of $e$ under the functor $f\colon[m]\to [n]$.

In terms of the category FinSetPart, we can describe $\iota(f)$ as a partial map of finite sets $\iota[n]\to\iota[m]$ that sends $(j-1,j)\in\iota[n]$ to the unique pair $(i-1,i)\in\iota[m]$ such that $f(i-1)\lt j\le f(i)$. (If no such pair exists, then $\iota(f)$ is undefined on $(j-1,j)$.)

Alternative description of the comparison functor

Given that the category $\Delta$ can be used to encode homotopy coherent associative monoids in the same manner that the category $\Gamma$ can be used to encode homotopy coherent commutative monoids, it is natural to wonder why the description of the comparison functor $\iota\colon \Delta\to\Gamma$ appears to be so asymmetric. The primary reason for this asymmetry is that the traditional description of the category $\Delta$ relies on the fact that elements are composed in a prescribed order, just like morphisms in a category. This enables us to define $\Delta$ as a full subcategory of the category of small categories. There is no analogue of such categorical structure for the category $\Gamma$.

However, it is possible to give an alternative definition of the category $\Delta$ resembling Segal’s covariant formulation and the category FinSetPart. In these formulations, objects encode sets of spine edges (as defined above) instead of sets of vertices.

In the first formulation, the category $\Delta$ is defined as the category opposite to the following category. Objects are finite totally ordered sets. Morphisms $A\to B$ are pairs $(I,f)$, where $I$ is an interval in $A$, defined as a triple $(X,Y,Z)$ of disjoint subsets of $A$ whose union equals $A$ and such that $X\lt Y\lt Z$, and $f\colon Y\to B$ is an order-preserving map. The set $Y$ uniquely determines $X$ and $Z$ unless $Y=\emptyset$. The identity morphism on $A$ is given by the pair $(I,id_A)$, where $I=(\emptyset,A,\emptyset)$. Given morphisms

$$(I_2,f_2)\colon A_1\to A_2,\qquad (I_1,f_1)\colon A_0\to A_1,$$

we define their composition as

$$(I_2,f_2)\circ(I_1,f_1)=(I,f_2\circ f_1|_{Y_1\cap f_1^{-1}Y_2}),$$

where

$$I=(X_1\cup f_1^{-1} X_2,Y_1\cap f_1^{-1} Y_2,Z_1\cup f_1^{-1} Z_2).$$

Now the forgetful functor $\Delta\to\Gamma=FinSetPart$ can be described as the functor that discards the total order on objects and discards the interval $I$ on morphisms.

There is also an analog of the covariant description of $\Gamma$ for the category $\Delta$. In this setting, $\Delta$ is the following category. Objects are finite totally ordered sets. Morphisms $A\to B$ are maps of sets $f\colon \{-\infty\}\sqcup A\sqcup\{\infty\}\to P(B)$ such that $a_1\lt a_2$ implies $f(a_1)\lt f(a_2)$ and we have $\bigcup_{a\in \{-\infty\}\sqcup A\sqcup\{\infty\}}f(a)=B$. The sets $f(-\infty)$ and $f(\infty)$ are uniquely determined by $\bigcup_{a\in A}f(a)$ unless the latter is empty. Morphisms are composed like in the category $\Gamma$.

Now the forgetful functor $\Delta\to\Gamma$ can be described as the functor that discards the total order on objects and discards the data of $f(-\infty)$ and $f(\infty)$ on morphisms.

Monoidal structure

The category $\Gamma$ is a symmetric monoidal category, induced from the cartesian monoidal structure on the category Set:

$$A_1\otimes A_2=A_1\times A_2,\qquad (A'_1,f_1)\otimes(A'_2,f_2)=(A'_1\times A'_2,f_1\times f_2).$$

The reader should keep in mind that $\times$ is not a product in the category FinSetPart, but only in Set.

In terms of the category $FinSet_*$, the monoidal structure is given by the smash product of pointed sets:

$$A_1\otimes A_2=(A_1\times A_2)/(\{*\}\times A_2\cup A_1\times \{*\}).$$

Applications

A Γ-object in a cartesian monoidal category (more generally, cartesian monoidal (∞,1)-category) $C$ is a functor

$$X\colon \Gamma^{op} \to C$$

(equivalently, a functor $FinSetPart \to C$) such that for all $A,B\in FinSetPart$ the map

$$X_{A\sqcup B}\to X_A\times X_B$$

induced by the morphisms

$$(A,id_A)\colon A\sqcup B\to A, \qquad (B,id_B)\colon A\sqcup B\to B$$

is an equivalence, as is the map $X_\emptyset\to1$. Here $A\sqcup B$ denotes the coproduct in the category Set (i.e., disjoint union). It does not denote the coproduct in the category FinSetPart.

Γ-objects in $C$ encode $\mathrm{E}_\infty$-monoids in $C$, i.e., homotopy coherent commutative monoids. See the article Γ-space for more information.

Related concepts

• category of operators

• Gamma space

References

Original reference:

• Graeme Segal, Categories and cohomology theories, Topology 13, 293–312, 1974 (doi:10.1016/0040-9383(74)90022-6, pdf)

For instance notation 2.0.0.2 in

• Jacob Lurie, Higher Algebra