nLab simplicial groupoid

Idea

The notion of simplicial groupoids pairs the concepts of groupoids and simplicial sets. Via the Dwyer-Kan loop groupoid functor [Dwyer & Kan (1984)] their homotopy theory is equivalent to the classical homotopy theory of simplicial sets/Kan complexes (both being models for $\infty$-groupoids).

Definition

A priori, the term simplicial groupoid may refer to simplicial objects in the (1-)category Grpd of (small) strict groupoids.

However (cf. discussion at simplicial category), in applications (notably in homotopy theory) one is interested only in those simplicial objects in Grpd whose underlying simplicial sets of objects are simplicially constant. This more restrictive notion — equivalent to sSet-enriched groupoids — is traditionally still referred to as “simplicial groupoids” [e.g. Dwyer & Kan (1984), §1.2.(ii), Goerss & Jardine (2009), V.7]; but for the moment let us call these instead “simplicial DK-groupoids”, for clarity, and let us denote the full subcategory they form by

$$sSet\text{-}Grpd \;\simeq\; sGrpd_{DK} \overset{\phantom{---}}{\hookrightarrow} Func(\Delta^{op}, Grpd) \,.$$

So given such a simplicial DK-groupoid $\mathcal{C}_\bullet \,\in\, sGrpd_{DK}$ we have:

1. a plain set of objects $Obj(\mathcal{C}_\bullet) \,\in\, Sets$;

2. for every pair $x, y \,\in\, Obj(\mathcal{C}_\bullet)$ of objects the degree-wise hom-sets form a simplicial set

   $$\mathcal{C}_\bullet(x,y) \;\coloneqq\; Hom_{\mathcal{C}_\bullet}(x,y) \;\in\; sSet \,;$$

3. where the degreewise composition operations constitutes an sSet-enriched composition operation

   $$\mathcal{C}_\bullet(x,y) \,\times\, \mathcal{C}_\bullet(y,z) \overset{\circ}{\longrightarrow} \mathcal{C}_\bullet(x,z)$$

4. making $\mathcal{C}_\bullet$ an sSet-enriched category.

Conversely, simplical DK-groupoids are therefore in bijection to those sSet-enriched categories whose degreewise categories formed by the $n$-cell morphisms for any $n \,\in\, \mathbb{N}$ are groupoids.

This makes the category $sGrpd_{DK}$ be a full subcategory of the category of (small) sSet-enriched categories:

$$sGrpd_{DK} \overset{\phantom{---}}{\hookrightarrow} sSet Cat \,.$$

Remarks

• Simplicially enriched groupoids are related to simplicial sets via an adjunction found independently by Dwyer & Kan and Joyal & Tierney; see Dwyer-Kan loop groupoid. This adjunction gives an equivalence of homotopy categories so that simplicially enriched groupoids model all homotopy types.

  • This is a groupoidal version of the equivalence between quasi-categories and simplicially enriched categories via the homotopy coherent nerve.

• A simplicially enriched groupoid having exactly one object is essentially the same as a simplicial group. Notationally however it is often important to distinguish a simplicial group form the corresponding single object simplicially enriched groupoid.

• Many constructions on simplicial groups, such as that of its Moore complex carry over to simplicialy enriched groupoids without difficulty.

Properties

Relation to simplicial groups

Example

(simplicial delooping groupoid)
For $\mathcal{G} \,\in\, Grp(sSet)$ a simplicial group, we obtain its simplicial delooping groupoid $\mathbf{B}\mathcal{G}$ by setting

• $Obj(\mathbf{B}\mathcal{G}) \,\coloneqq\, \ast$ (the singleton set),

• $(\mathbf{B}\mathcal{G})(\ast,\ast) \,\coloneqq\, \mathcal{G}$

• with composition given by the group operation of $\mathcal{G}$.

Conversely, for $\mathcal{X} \,\in\, sSet\text{-}Grpd$ and $x \,\in\, Obj(\mathcal{X})$, the endomorphism-hom-object at $x$ canonically carries the structure of a simplicial group:

$$\mathcal{X}(x,x) \,\in\, Grp(sSet) \,.$$

Definition

(some notation)
For $\mathcal{X} \,\in\, sSet\text{-}Grpd$ a DK-simplicial groupoid, write:

• $\mathcal{X}_0 \,\in\, Grpd$ for its underlying plain (i.e. Set-enriched) groupoid (its change of enrichment along the terminal functor $sSet \to \ast$)

• $\pi_0(\mathcal{X}) \,\coloneqq\, \pi_0(\mathcal{X}_0) \,\in\, Set$ for the set of isomorphism classes of objects, hence for the set of connected components of $\mathcal{X}$,

• $\mathcal{X}_{[x]} \hookrightarrow \mathcal{X}$ for the enriched full subcategory on all objects in the connected component $[x] \,\in\, \pi_0(\mathcal{X})$.

Proposition

(simplicial groupoids adjoint equivalent to skeletons)
Assuming the axiom of choice, every sSet-enriched groupoid is sSet-enriched adjoint equivalent to a skeletal simplicial groupoid, namely to a disjoint union of simplicial delooping groupoids (Exp. ), one for each connected components, in that given a choice of representative objects $(i, x_i) \,\in\, \big(i \in \pi_0(\mathcal{X})\big) \times Obj(\mathcal{X}_i)$ of all the connected components, the inclusion enriched functor

(1)$$\iota \,\colon\, \left( \underset{i \in \pi_0(\mathcal{X})}{\amalg} \mathbf{B}\big(\mathcal{X}(x_i,\,x_i)\big) \right) \xhookrightarrow{\phantom{---}} \mathcal{X}$$

has a strict left inverse and a right inverse up to enriched natural isomorphism.

In other words, every sSet-enriched groupoid has a deformation retraction onto an sSet-skeletal groupoid.

Proof

This follows in direct enriched-analogy to the corresponding statement for plain groupoids (as for instance spelled out here) using (only) that the cosmos sSet is cartesian monoidal:

Choosing for each $x \in Obj(\mathcal{X})$ an element

$$\gamma_x \,\colon\, \ast \to \mathcal{X}(x_{[x]}, x)$$

induces

1. an enriched functor

   (2)$$\array{ \mathcal{X} &\xrightarrow{ \phantom{----} p \phantom{----} }& \underset{i \in \pi_0(\mathcal{X})}{\amalg} \mathbf{B}\big( \mathcal{X}(x_i,\,x_i) \big) \\ x &\mapsto& x_{[x]} \\ \mathcal{X}(x,y) &\xrightarrow{ \phantom{----} p_{x,y} \phantom{----} }& \mathcal{X}(x_{[x]}, x_{[y]}) \\ \mathllap{\simeq}\Big\downarrow && \Big\uparrow\mathrlap{\circ} \\ \ast \times \mathcal{X}(x,y) \times \ast & \underset{ \big((-)^{-1}\circ \gamma_y\big) \times id \times \gamma_x }{\longrightarrow} & \mathcal{X}(y,x_{[y]}) \times \mathcal{X}(x,y) \times \mathcal{X}(x_{[x]},x) }$$

2. an enriched natural transformation

   (3)$$\gamma \,\colon\, \iota \circ p \longrightarrow id_{\mathcal{X}}$$

   with components $\gamma_x$ (which satisfies its “enriched naturality square-condition” here essentially by construction of $p$).

Similarly there is an enriched transformation $id_{\mathcal{X}} \longrightarrow p \circ \iota$, and these make the unit and counit of an “enriched adjoint equivalence”. But if we choose $\gamma_{x_{[x]}} \coloneqq id_{x_{[x]}}$ — as we may — then there is already an equality $p \circ \iota = id$ (hence a retraction onto the skeleton).

Some consequences:

Remark

(simplicial presheaves on simplicial groupoids are equivalently products of simplicial group actions)
For

• $\mathcal{X}$ a DK-enriched groupoid regarded as an sSet-enriched category

• $\mathbf{C}$ any sSet-enriched category

the enriched functor category between the two is equivalent (as a plain locally small category) to a product of enriched functor categories on simplicial delooping groupoids (Exp. ), one for each connected component of $\mathcal{X}$,

$$sFunc(\mathcal{X},\,\mathbf{C}) \;\simeq\; \underset{ i \in \pi_0(\mathcal{X}) }{\prod} sFunc\big( \mathbf{B}\mathcal{X}(x_i,x_i) ,\, \mathbf{C} \big) \,.$$

This follows by observing that the functor of precomposion with the inclusion (1)

$$\iota^\ast \;\colon\; sFunc(\mathcal{X},\,\mathbf{C}) \longrightarrow \underset{ i \in \pi_0(\mathcal{X}) }{\prod} sFunc\big( \mathbf{B}\mathcal{X}(x_i,x_i) ,\, \mathbf{C} \big)$$

is inverse to $p^\ast$ (2) up to a natural isomorphism whose component at any enriched functor $F \colon \mathcal{X} \longrightarrow \mathbf{C}$ is the enriched natural transformation $p^\ast \iota^\ast F\longrightarrow F$ obtained as the horizontal composition (“whiskering” of enriched transformations, see there) by $F$ of the enriched transformation (3). Notice that this is already an adjoint equivalence.

If $\mathbf{C}$ is moreover tensored over sSet, then this, in turn, is equivalent to a product of categories of simplicial$\;$group actions on objects in $\mathbf{C}$:

$$sFunc(\mathcal{X},\,\mathbf{C}) \;\simeq\; \underset{ i \in \pi_0(\mathcal{X}) }{\prod} \Big( \big(\mathcal{X}(x_i,x_i)\big) Act(\mathbf{C}) \Big) \,.$$

Example

(Borel model structure on simplicial group actions over simplicial groupoids)
For $\mathcal{G} \,\in\, Grp(sSet)$ a simplicial group with sSet-enriched delooping groupoid denoted $\mathbf{B}\mathcal{G} \in sSet\text{-}Grpd$, an sSet-enriched functor $\mathbf{B}\mathcal{G} \longrightarrow sSet$ is equivalently a simplicial group action of $\mathcal{G}$.

Under this identification, the projective model structure on simplicial functors (this Prop.) is equivalently the Borel model structure on simplicial group actions, a context of Borel-equivariant homotopy theory:

$$sFunc\big( \mathbf{B}\mathcal{G} ,\, sSet \big)_{proj} \;\; = \;\; \mathcal{G}Act(sSet)_{Borel} \,.$$

More generally, for $\mathcal{X} \in sSet\text{-}Grpd$ an sSet-enriched groupoid (Dwyer-Kan simplicial groupoid) with a single connected component $\pi_0(\mathcal{X}) \simeq \{[x]\}$, so that the inclusion

$$\iota \,\colon\, \mathbf{B}(\mathcal{X}(x,x)) \xhookrightarrow{\phantom{---}} \mathcal{X}$$

is an sSet-enriched adjoint equivalence (see discussion there) the projective model structure on simplicial functors (from that Prop.) is transferred under the induced adjoint equivalence of sSet-enriched functor categories

$$sFunc\big( \mathcal{X} ,\, sSet \big)_{proj} \underoverset {\underset{\iota^\ast}{\longrightarrow}} {\overset{\iota_!}{\longleftarrow}} {\;\; \bot_{\simeq} \;\;} sFunc\Big( \mathbf{B}\big(\mathcal{X}(x,x)\big) ,\, sSet \Big)_{proj} \;=\; \big(\mathcal{X}(x,x)\big) Act(sSet)_{Borel}$$

By this example it follows that morphisms in all three classes $(\mathrm{W}, Fib, Cof)$ in $sFunc(\mathcal{X}, \, sSet)_{proj}$ are those which restrict on $x \in Obj(\mathcal{X})$ to the respective class in $\big(\mathcal{X}(x,x)\big) Act(sSet)_{Borel}$.

It follows that for $\mathcal{X} \,\in\, sSet\text{-}Grpd$ a simplicial groupoid with any set $\pi_0(\mathcal{X})$ of connected components, the projective model structure of simplicial functors over it is the product model structure of the Borel model structures of simplicial group actions, one for each connected component:

$$sFunc\big( \mathcal{X} ,\, sSet \big)_{proj} \underoverset {\underset{\iota^\ast}{\longrightarrow}} {\overset{\iota_!}{\longleftarrow}} {\;\; \bot_{\simeq} \;\;} \underset{i \in \pi_0(\mathcal{X})}{\prod} sFunc\Big( \mathbf{B}\big(\mathcal{X}(x_i,x_i)\big) ,\, sSet \Big)_{proj} \;=\; \underset{i \in \pi_0(\mathcal{X})}{\prod} \big(\mathcal{X}(x_i,x_i)\big) Act(sSet)_{Borel} \,.$$

The analogous statement holds (still by that Prop.) for the codomain sSet replaced by any combinatorial simplicial model category $\mathbf{C}$:

$$sFunc\big( \mathcal{X} ,\, \mathbf{C} \big)_{proj} \underoverset {\underset{\iota^\ast}{\longrightarrow}} {\overset{\iota_!}{\longleftarrow}} {\;\; \bot_{\simeq} \;\;} \underset{i \in \pi_0(\mathcal{X})}{\prod} sFunc\Big( \mathbf{B}\big(\mathcal{X}(x_i,x_i)\big) ,\, \mathbf{C} \Big)_{proj} \;=\; \underset{i \in \pi_0(\mathcal{X})}{\prod} \big(\mathcal{X}(x_i,x_i)\big) Act(\mathbf{C})_{Borel} \,.$$

Moreover, if $I_{\mathbf{C}}, J_{\mathbf{C}} \,\subset\, \mathbf{C}$ denote classes of (acyclic) generating cofibrations of $\mathbf{C}$, then that Prop. gives generating cofibrations of $\mathcal{G}Act(\mathbf{C})_{Borel}$ to be

$$I^{\mathcal{G}}_{\mathbf{C}} \,\coloneqq\, \big\{ \mathcal{G} \cdot i \;\vert\; i \in I_{\mathbf{C}} \big\} \,,\;\;\;\; J^{\mathcal{G}}_{\mathbf{C}} \,\coloneqq\, \big\{ \mathcal{G} \cdot j \;\vert\; i \in J_{\mathbf{C}} \big\} \,.$$

Cartesian closure

Proposition

The category of sSet-enriched groupoids is cartesian closed.

Proof

The ambient category of sSet-enriched categories is cartesian closed (as any category of $\mathcal{V}$-categories for cartesian closed complete cosmos $\mathcal{V}$), given by forming enriched product categories and enriched functor categories. Therefore we just need to check that for $\mathbf{D},\,\mathbf{C} \,\in\, sSet\text{-}Grpd$ also the $sSet$-enriched product category $\mathbf{C} \times \mathbf{D}$ and the $sSet$-enriched functor category $[\mathbf{D},\,\mathbf{C}]$ are in fact enriched groupoids.

For the product this is immediate, the inversion-operation is given factorwise.

To see the statement for the internal hom, write $\mathbf{I}_S$ for the sSet-enriched category with set objects $\{0,1\}$ and hom objects given by:

• $\mathbf{I}_{S}(0,0) \,\coloneqq\, \ast$

• $\mathbf{I}_{S}(1,1) \,\coloneqq\, \ast$

• $\mathbf{I}_{S}(1,0) \,\coloneqq\, \varnothing$

• $\mathbf{I}_{S}(0,1) \,\coloneqq\, S$

(whence there is no non-trivial composition to be defined).

Now for $n \in \mathbb{N}$ the $sSet$-enriched functors

$$\mathbf{I}_{\Delta[n]} \longrightarrow [\mathbf{D},\,\mathbf{C}]$$

are in bijection with the $(n+1)$-morphisms in $[\mathbf{D},\,\mathbf{C}]$, and we need to exhibit an inversion operation on these.

But by the cartesian closure of sSet Cat these are also in natural bijection to enriched functors of the form

$$\mathbf{D} \times \mathbf{I}_{\Delta[n]} \longrightarrow \mathbf{C}$$

which are manifestly a simplicial diagram of natural transformations between the restrictions $\mathbf{D} \times \{0\} \to \mathbf{C}$ and $\mathbf{D} \times \{1\} \to \mathbf{C}$. The corresponding system of inverse natural transformations corresponds to the inverse $n+1$-morphism that we are after.

Related concepts

• model structure on simplicial groupoids

• simplicial category

  • simplicially enriched category

  • simplicial object in Cat

• relation between quasi-categories and simplicial categories

• model structure on presheaves of simplicial groupoids

References

Original discussion in the context of simplicial localization

• William Dwyer, Daniel Kan, §5.5 of: Simplicial localizations of categories, J. Pure Appl. Algebra 17 3 (1980), 267-284 [doi:10.1016/0022-4049(80)90049-3, pdf]

and in the context of a model structure on simplicial groupoids:

• William Dwyer, Daniel Kan, Homotopy theory and simplicial groupoids, Indagationes Mathematicae (Proceedings) 87 4 (1984) 379-385 [doi:10.1016/1385-7258(84)90038-6]

Textbook accounts

• Paul Goerss, J. F. Jardine, Section V.7 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) [doi:10.1007/978-3-0346-0189-4, webpage]

• John F. Jardine, §9.3 in: Local homotopy theory, Springer Monographs in Mathematics (2015) [doi:10.1007/978-1-4939-2300-7]

See also:

• Philip Ehlers, Simplicial groupoids as models for homotopy type, Master’s thesis (1991) [pdf]

• Philip J. Ehlers, Algebraic Homotopy in Simplicially Enriched Groupoids, PhD thesis (1993) [pdf]