Contents

category theory

# Contents

## Idea

The Segal condition is a condition on a simplicial object $X_\bullet \in \mathcal{C}^{\Delta^{op}}$ which says that each component $X_{n \geq 2} \in \mathcal{C}$ is obtained from $X_1 \stackrel{\overset{\partial_0}{\to}}{\underset{\partial_1}{\to}} X_0$ by the $n$-fold fiber product of $X_1$ over $X_0$ which “glues” $n$ copies of $X_1$ end-to-end.

Hence if one thinks of $X_0$ as a collection of objects and of $X_1$ as a collection of morphisms, then $X_\bullet$ satisfies the Segal condition precisely if each $X_{n \geq 2}$ can be interpreted as the collection of sequences of composable morphisms of length $n$. The precise formulation is below in Definition – For simplicial objects.

Accordingly, if $\mathcal{C} =$ Set is the category of sets, then the Segal condition characterizes precisely those simplicial sets which are the nerve of a (strict) small category, theorem below. This is the observation due to Grothendieck 1961 as recalled in Segal 1968), which today gives the Segal condition its name. Sometimes this statement also called the nerve theorem (no relation to what is called nerve theorem in homotopy theory).

It is useful to decompose this statement into its constituents as follows:

A small category $C$ may be thought of as a directed graph $U(C)$ equipped with a unital associative composition operation. This corresponds to a sequence of inclusions of sites

$(1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0) \hookrightarrow \Delta_0 \to \Delta$

into the simplex category, where $\Delta_0$ is the category of finite non-empty directed linear graphs:

• a directed graph is equivalently a presheaf on $(1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0)$;

• a presheaf on $\Delta$, hence a simplicial set encodes via its face and degeneracy maps a kind of associative and unital composition – but not necessarily “of composable morphisms” if $X_{n\geq 2}$ is not given in the above fashion.

In terms of this we can say that equipping a directed graph with the structure of a category is equivalent to asking for its pushforward along $(1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0) \hookrightarrow \Delta_0$ (which encodes all the collections of sequences of composable edges) to be equipped with a lift to a simplicial object through the pullback along $\Delta_0 \to \Delta$. Conversely, the simplicial objects obtained as such lifts are precisely the simplicial objects that satisfy the Segal condition.

Formulated in this way one sees that the Segal condition has a large variety of generalizations to structures with richer kinds of composition operations, such as globular operads. This is made precise below in Definition – For cellular objects.

## Definition

### For simplicial objects

Let $\mathcal{C}$ be a category with pullbacks.

###### Definition
$X : \Delta^{op} \to \mathcal{C}$

is said to satisfy the Segal condition if it sends the colimits in the simplex category to limits, hence if the Segal maps exhibit equivalences

$X_n \stackrel{\simeq}{\longrightarrow} \underbrace{ X_1 \times_{X_0} \cdots \times_{X_0} X_1 }_{n\; factors}$

for all $n \in \mathbb{N}$.

More in detail:

###### Definition

For all $n \in \mathbb{N}$, consider $\Delta[n]$ naturally as a cocone in the simplex category under the diagram

$\array{ \Delta && && \Delta && && \Delta &&& \cdots \\ & {}_{\mathllap{d_1}}\searrow && {}^{\mathllap{d_0}}\swarrow && \searrow^{\mathrlap{d_1}} && {}_{d_0}\swarrow && \cdots \\ && \Delta &&&& \Delta && \cdots }$

with $n$ copies of $\Delta$ at the bottom, such that the cocone injection of the $k$th copy is $\Delta \simeq (k-1,k) \hookrightarrow (0,1,2, \cdots, n) \simeq \Delta[n]$.

A simplicial object $X \colon \Delta^{op} \to \mathcal{C}$ satisfies the Segal conditions if it sends these cocones to limit cones in $\mathcal{C}$.

###### Remark

This definition immediately generalizes to (∞,1)-category theory where $\mathcal{C}$ is an (∞,1)-category and $X$ is a simplicial object in an (∞,1)-category. Then for $X$ to satisfy the Segal conditions means that it sends the cocones of def. to (∞,1)-limit cones in $\mathcal{C}$.

Such a simplicial object is also called a pre-category object in an (∞,1)-category in $\mathcal{C}$.

### For cellular objects

A globular theory is a wide subcategory inclusion

$i_A \colon \Theta_0 \to \Theta_A$

of the globular site $\Theta_0$. There is an equivalence of categories

$\omega Grph \simeq Sh(\Theta_0) \,.$

of ∞-graphs and sheaves on the globular site. In particular for $\Theta_A = \Theta$ the cell category (Theta category) a presheaf on $\Theta$ is a cellular object.

The Segal condition on a cellular object $X \colon \Theta^{op} \to \mathcal{C}$ is that the restriction $i^* X \colon \Theta_0^{op} \to \Theta^{op} \to \mathcal{C}$ to the cellular site is a sheaf there.

The cellular objects that satisfy the Segal condition are precisely the ∞-category objects (Berger).

The cellular spaces/ cellular simplicial sets/cellular ∞-groupoids that satisfy the Segal condition as a weak homotopy equivalence/ equivalence of ∞-groupoids is a Theta_n-space an (∞,n)-category.

## Properties

### Characterization of nerves of (higher) categories

#### Of simplicial nerves of small categories

The archetypical role of the Segal condition is to make the following statement true.

###### Theorem

(nerve theorem)

A simplicial set is the nerve of a small category precisely if it satsfies the Segal conditions.

This is due to Grothendieck 61, recalled in Segal 1968.

###### Remark

There is an entirely unrelated theorem in homotopy theory also often called “the” nerve theorem. See there for more. Not to be confused with the discussion here.

#### Of complete Segal spaces

By refining the above result from sets to $\infty$-groupoids, one obtains the pre-category object in an (infinity,1)-category.

#### Of cellular nerves of strict $\omega$-categories

Similarly, a cellular set is the cellular nerve of a strict omega-category precisely if it satisfies the cellular Segal condition. (Berger).

#### Of cellular models of $(\infty,n)$-categories

See at Theta-space.

### In terms of sheaf conditions

We discuss an equivalent formulation of the Segal condition in terms of notions in topos theory/(∞,1)-topos theory. This perspective for instance lends itself more to a formulation of Segal conditions in terms of the internal language of toposes.

#### For simplicial objects and category objects

We characterize below in prop. the category of categories as the pullback of the topos of simplicial set along the inclusion of the topos of graphs into that of presheaves on finite linear graphs.

First we state some preliminaries.

###### Remark

The condition in def. superficially looks like a sheaf condition for coverings of $\Delta[n]$ by $n$ subsequent copies of $\Delta$. However, these coverings do not form a coverage on the simplex category $\Delta$: the refinement-of-covers-axiom is not satisfied:

For instance for $d_1 \colon \Delta \to \Delta$ the map that sends the single edge of $\Delta$ to the composite edge in $\Delta$ there is no way to “pull back” the cover

$\{\Delta \coprod \Delta \stackrel{((0,1),(1,2))}{\to} \Delta\}$

along this morphism, not even in the weak sense of coverage.

However, as this example also makes clear, the problem is precisely only with the morphisms in $\Delta$ that are no injective on generating edges.

###### Definition

Let

$j \colon \Delta_0 \hookrightarrow Graph$

be the full subcategory of that of directed graphs on the linear graphs $\{0 \to 1 \to \cdots \to n\}$ for $n \in \mathbb{N}$.

###### Remark

Morphisms in $\Delta_0$ have to send elementary edges to elementary edges. So there are

• precisely $n$ morphisms $\Delta_0 \to \Delta_0[n]$

• precisely $n$ morphisms $\Delta_0 \to \Delta_0[n+1]$

• precisely $n$ morphisms $\Delta_0 \to \Delta_0[n+2]$

• etc.

###### Definition

Write

$i \colon (1 \stackrel{\leftarrow}{\leftarrow} 0) \hookrightarrow \Delta_0$

for the full subcategory on the linear graphs with no edge and with one edge.

###### Proposition

The category of directed graphs is equivalently the category of presheaves over $(1 \stackrel{\leftarrow}{\leftarrow} 0)$, def. :

$Graph(\mathcal{C}) \simeq PSh((1 \Leftarrow 0), \mathcal{C}) = \mathcal{C}^{(1 \Rightarrow 0)} \,.$
###### Definition

Write

$(i_! \dashv i^* \dashv i_*) \colon \mathcal{C}^{1 \Rightarrow 0} \stackrel{\overset{i_!}{\to}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}}} \mathcal{C}^{\Delta_0^{op}}$

for the adjoint triple induced on categories of presheaves by the inclusion $i$ of def. : $i^*$ is given by precomposition with $i$, $i_!$ is left and $i_*$ is right Kan extension along $i$.

###### Proposition

The functor $i_* \colon Graph(\mathcal{C}) \to \mathcal{C}^{\Delta_0^{op}}$ of def. sends a graph $X_1 \stackrel{\overset{\partial_1}{\to}}{\underset{\partial_0}{\to}} X_0$ to the presheaf $i_*(X)$ which on $n \in \mathbb{N}$ is given by th iterated pullback

$i_*(X) \colon n \mapsto \underbrace{X_1 \times_{X_0} \times \cdots \times_{X_0} X_1}_{n \; factors}$

and which sends an inclusion $\Delta[k] \simeq (j, \cdots, j+k) \hookrightarrow (0,\cdots, n)\simeq \Delta[n]$ to the corresponding projection map out of the pullback.

We may call $i_*(X)$ the nerve of the graph $X$.

###### Proof

Using the Yoneda lemma and the defining hom-isomorphisms of the adjunction as well as the fact that the hom functor sends colimits in the first argument to limits, we have

\begin{aligned} i_*(X)(n) & \simeq Hom(\Delta[n], i_*(X)) \\ & \simeq Hom(i^* \Delta[n], X) \\ & \simeq Hom( \underbrace{\Delta \coprod_{\Delta} \cdots \coprod_{\Delta} \Delta}, X ) \\ & \simeq \underbrace{ Hom(\Delta, X) \underset{Hom(\Delta, X)}{\times} \cdots \underset{Hom(\Delta, X)}{\times} Hom(\Delta, X) }_{n\;factors} \\ & \simeq \underbrace{ X \times_{X_0} \cdots \times_{X_0} X_1 }_{n\; factors} \end{aligned} \,.
###### Proposition

For $n \in \mathbb{N}$ declare a unique covering family of $\Delta[n] \in \Delta_0$ to be

$\left\{ \Delta\left[1\right] \simeq \left(k,k+1\right) \hookrightarrow \Delta\left[n\right] \right\}_{k = 0}^{n-1} \,.$

Then this is a coverage on $\Delta_0$.

###### Proposition

A presheaf $X \colon \Delta_0^{op} \to \mathcal{C}$ is a sheaf with respect to the coverage of def. precisely if it is in the essential image of the graph-nerve functor

$i_* \colon Graph(\mathcal{C}) \simeq \mathcal{C}^{(1 \stackrel{\to}{\to} 0)} \stackrel{}{\to} \mathcal{C}^{\Delta_0^{op}}$

of prop. . This yields an equivalence of categories

$Graph(\mathcal{C}) \simeq Sh(\Delta_0)$

with the category of sheaves on $\Delta_0$. The graph-nerve functor is a full and faithful functor

$i_* \colon Graph \simeq Sh(\Delta_0) \hookrightarrow PSh(\Delta_0) \,.$
###### Definition

Write

$(j_! \dashv j^* \dashv j_*) \colon \mathcal{C}^{\Delta_0^{op}} \stackrel{\overset{j_!}{\to}}{\stackrel{\overset{j^*}{\leftarrow}}{\underset{j_*}{\to}}} \mathcal{C}^{\Delta^{op}}$

for the adjoint triple induced on categories of presheaves by the inclusion $j$ of def. : $j^*$ is given by precomposition with $j$, $j_!$ is left and $j_*$ is right Kan extension along $j$.

In terms of all this the nerve theorem says the following:

We have geometric morphisms of toposes

$Grph \simeq Sh(\Delta_0) \stackrel{\overset{i_!}{\to}}{ \stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\hookrightarrow}}} PSh(\Delta_0) \stackrel{\overset{j_!}{\to}}{\stackrel{\overset{j^*}{\leftarrow}}{\underset{j_*}{\to}}} PSh(\Delta) \simeq sSet$

which capture the Segal condition as follows.

###### Proposition
$\array{ Cat &\stackrel{N}{\hookrightarrow}& PSh(\Delta) \simeq sSet \\ \downarrow^{\mathrlap{U}} && \downarrow^{\mathrlap{j^*}} \\ Graph \simeq Sh(\Delta_0) &\stackrel{i_*}{\hookrightarrow}& PSh(\Delta_0) } \,,$

where

• $N$ forms the nerve of a category;

• $U$ is the forgetful functor that sends a category to its underlying graph

is a pullback.

###### Remark

The morphisms in the commuting diagram of prop. participate in further adjunctions, and in terms of these the Segal condition may further be reformulated as a restriction condition on algebras over an operad:

First of all the nerve has a left adjoint $\tau \colon PSh(\Delta) \to Cat$. With this the left adjoint $j_!$ of $j^*$ induces a left adjoint

$F \simeq \tau j_! i_*$

of $U$, which is the free category functor.

Moreover, with $U$ also $j^*$ is a monadic functor and the monad $U F \colon Grph \to Grph$ of which Cat is the category of algebras is the restriction of the monad $j^* \circ j_!$:

$i_* U F \simeq j^* j_! i_* \,.$

(All this is discussed in (Berger, p. 13), and actually in the further generality of cellular sets that we get to below.)

In summary we have a diagram of adjoint pairs of functors of the form

$\array{ Cat &\stackrel{\overset{\tau}{\leftarrow}}{\underoverset{N}{\bottom}{\hookrightarrow}}& PSh(\Delta) \simeq sSet \\ {}^{\mathllap{U}}\downarrow \vdash \uparrow^{\mathrlap{F}} && {}^{\mathllap{j^*}}\downarrow \vdash \uparrow^{\mathrlap{j_!}} \\ Graph \simeq Sh(\Delta_0) &\stackrel{\overset{i^*}{\leftarrow}}{\underoverset{i_*}{\bottom}{\hookrightarrow}}& PSh(\Delta_0) }$

where several (however not all) subdiagrams of functors commute, as discussed above. In terms of this the reformulation of the Segal condition as in prop. is now further reformulated as:

A category is equivalently an algebra over the monad $j^* j_!$ on $PSh(\Delta_0)$ which satisfies the Segal condition in that it is in the essential image of the functor $i_*$ of prop. .

#### For cellular objects and $\omega$-category objects

The immediate generalization of prop. from simplicial objects to cellular objects is the following.

Let

$j \colon \Theta_0 \to \Theta$

be the defining inclusion of the cellular site into the cell category.

###### Proposition

The category $Str\omega Cat$ of strict ∞-categories is the pullback

$\array{ Str\omega Cat &\underoverset{\simeq}{N}{\to}& Mod_\Theta &\hookrightarrow& PSh(\Theta) \\ \downarrow^{\mathrlap{U}} && \downarrow^{} && \downarrow^{\mathrlap{j^*}} \\ \omega Graph &\stackrel{\simeq}{\to}& Sh(\Theta_0) &\hookrightarrow& PSh(\Theta_0) } \,.$

See at globular theory for more.

The “Segal conditions” are originally due to:

and named after their mentioning in

The interpretation of the Segal condition as a sheaf condition is reviewed for instance in section 2 of

and discussed for strict infinity-categories in

• Clemens Berger, A cellular nerve for higher categories, Advances in Mathematics 169, 118-175 (2002) (pdf)

Based on that, an iterative and homotopy-theoretic formulation of the cellular Segal conditions is in section 5 of

• Charles Rezk, A cartesian presentation of weak $n$-categories Geom. Topol. 14 (2010), no. 1, 521–571 (arXiv:0901.3602)