The unit of the total décalage adjunction

Abstract

We consider the décalage construction ${{\,\mathrm{Dec}\,}}$ and its right adjoint $T$. These functors are induced on the category of simplicial objects valued in any bicomplete category ${\mathcal {C}}$ by the ordinal sum. We identify $T{{\,\mathrm{Dec}\,}}X$ with the path object $X^{\Delta [1]}$ for any simplicial object X. We then use this formula to produce an explicit retracting homotopy for the unit $X\rightarrow T{{\,\mathrm{Dec}\,}}X$ of the adjunction $({{\,\mathrm{Dec}\,}},T)$. When ${\mathcal {C}}$ is a category of objects of an algebraic nature, we then show that the unit is a weak equivalence of simplicial objects in ${\mathcal {C}}$.

1 Introduction

Let $\sigma :\Delta \times \Delta \rightarrow \Delta$ denote the ordinal sum functor on the simplex category $\Delta$, described on objects via $\sigma ([k],[l])=[k+1+l]$. The induced functor $\sigma ^*:{\mathcal {S}}\! et ^{\Delta ^{{{\,\mathrm{op}\,}}}} \rightarrow {\mathcal {S}}\! et ^{(\Delta \times \Delta )^{{{\,\mathrm{op}\,}}}}$ is sometimes called total décalage and denoted ${{\,\mathrm{Dec}\,}}$, going back to Illusie [13]. Informally speaking, it spreads out a simplicial set X “anti-diagonally” into a bisimplicial set ${{\,\mathrm{Dec}\,}}X$, which is levelwise described on objects by $({{\,\mathrm{Dec}\,}}X)_{k,l}=X_{k+1+l}$. The functor $\sigma ^*$ has a right adjoint $\sigma _*$.

This right adjoint often appears in the literature; it is known as ${\overline{W}}$, e.g. in [3, 4], or as the total simplicial set functor T, e.g. in [1], or as the Artin–Mazur codiagonal, e.g. in [26]. When composed with the levelwise nerve functor, the functor $\sigma _*$ yields a model for the classifying space of simplicial groups, which is also referred to as the ${\overline{W}}$-construction; see [26, Sect. 5] for further discussion.

The adjunction

$$\begin{aligned} {{\,\mathrm{Dec}\,}}=\sigma ^*:{\mathcal {S}}\! et ^{\Delta ^{{{\,\mathrm{op}\,}}}} \rightleftarrows {\mathcal {S}}\! et ^{(\Delta \times \Delta )^{{{\,\mathrm{op}\,}}}}:\sigma _*=T, \end{aligned}$$

plays a crucial role in work by Cegarra, Heredia, Remedios [3,4,5] and Stevenson [26], in particular due to its relation with Thomason’s homotopy colimit formula and Kan’s simplicial loop group functor, respectively. In [4, Prop. 7.1] and [26, Lemma 20] the two groups of authors prove that the unit $X\rightarrow T{{\,\mathrm{Dec}\,}}X$ is a weak equivalence for any simplicial set X. Both of their proofs rely on the homotopy-theoretical fact that the Artin–Mazur codiagonal of a bisimplicial set is weakly equivalent to its diagonal.

The aim of this article is to give a full and explicit description of the simplicial object $T{{\,\mathrm{Dec}\,}}X$ and of the unit $X\rightarrow T{{\,\mathrm{Dec}\,}}X$. This description is then used to construct an explicit retracting homotopy for the unit $X\rightarrow T{{\,\mathrm{Dec}\,}}X$, which is therefore a strong deformation retract.

We work in greater generality than in the above cited articles, and consider the adjunction

$$\begin{aligned} {{\,\mathrm{Dec}\,}}=\sigma ^*:{\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}} \rightleftarrows {\mathcal {C}}^{(\Delta \times \Delta )^{{{\,\mathrm{op}\,}}}}:\sigma _*=T, \end{aligned}$$

where ${\mathcal {C}}$ is any bicomplete category, rather than just the category ${\mathcal {S}}\! et$ of sets. Examples of interest include the category ${\mathcal {S}}\! et ^{\Delta ^{{{\,\mathrm{op}\,}}}}$ of simplicial sets (more generally any category of (pre)sheaves valued in a bicomplete category), the category of small categories, and the category of (abelian) groups. In this more general framework, the décalage construction ${{\,\mathrm{Dec}\,}}$ is the underlying bisimplicial space of the path construction from [2], and the adjoint T is closely related to the generalized$S_{\bullet }$-construction.

The category ${\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}$ is always cotensored over ${\mathcal {S}}\! et ^{\Delta ^{{{\,\mathrm{op}\,}}}}$, and in this paper we prove the following description of the unit, which will appear as Corollary 2.2.

Theorem

For any simplicial object X in ${\mathcal {C}}$, there is a natural isomorphism

$$\begin{aligned} T{{\,\mathrm{Dec}\,}}X\cong X^{\Delta [1]} \end{aligned}$$

and, under this isomorphism, the unit $X\rightarrow T{{\,\mathrm{Dec}\,}}X$ of the adjunction $({{\,\mathrm{Dec}\,}},T)$ is identified with the map $X^{\Delta [0]}\rightarrow X^{\Delta [1]}$ induced by $\Delta [1]\rightarrow \Delta [0]$.

The main ingredient, which appears as Theorem 2.1, is a careful analysis of the counit of the adjunction of $\sigma ^*$ and its left adjoint $\sigma _!$. An equivalent description of such counit in the case ${\mathcal {C}}={\mathcal {S}}\! et$ was already mentioned by Cordier–Porter in [6]. Similar combinatorics were also considered in [7, Remark 0.16], [9, Sect. 2.9], [10, Sect. 4], [14, Sect. 9.3], [17, Sect. 3], [20, Sect. 11.4], [25, Sect. II.5] and [29, Sect. 5.1].

Moreover, the category ${\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}$ is always enriched over ${\mathcal {S}}\! et ^{\Delta ^{{{\,\mathrm{op}\,}}}}$ (see e.g. [11, 21, 22]) and there is therefore a canonical notion of homotopy between maps in ${\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}$ (cf. [22, Sect. II.1.6] or [24, Sect. 3.8]). We then produce in Corollary 2.3 an explicit retracting homotopy for the unit with respect to this simplicial enrichment, obtaining the following.

Theorem

For any simplicial object X in ${\mathcal {C}}$, the unit $X\rightarrow T{{\,\mathrm{Dec}\,}}X$ of the adjunction $({{\,\mathrm{Dec}\,}},T)$ is strong deformation retract.

To give further homotopical meaning to this strong deformation retract, it is convenient to have a model structure on ${\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}$ that is compatible with the simplicial enrichment. For the categories of sets and of many objects of algebraic nature that have a well-behaved forgetful functor to sets (such as groups, modules, or rings), such a model structure was introduced by Quillen (see [22, Sect. II.4] or [11, Sect. II.5]). With respect to this model structure, we prove the following corollary, that will appear as Corollary 2.3.

Theorem

Let ${\mathcal {C}}$ be a category with a suitable forgetful functor to ${\mathcal {S}}\! et$. For any simplicial object X in ${\mathcal {C}}$, the unit $X\rightarrow T{{\,\mathrm{Dec}\,}}X$ of the adjunction $({{\,\mathrm{Dec}\,}},T)$ is a weak equivalence.

When specializing to ${\mathcal {C}}={\mathcal {S}}\! et$, the theorem strengthens and provides a more transparent proof of the known fact that in the classical context the unit of the adjunction $({{\,\mathrm{Dec}\,}},T)$ is a weak equivalence (see e.g. [3]).

2 The main result and applications

Let ${\mathcal {C}}$ be any bicomplete category. By [22, Sect. II.1], [11, Theorem II.2.5] or [21, Prop. 92], the category ${\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}$ of simplicial objects in ${\mathcal {C}}$ is simplicially enriched.¹ The mapping spaces assemble into a bifunctor

$$\begin{aligned} {\text {Map}}_{{\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}}(-,-):({\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}})^{{{\,\mathrm{op}\,}}}\times {\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}\rightarrow {\mathcal {S}}\! et ^{\Delta ^{{{\,\mathrm{op}\,}}}}. \end{aligned}$$

With respect to this simplicial enrichment the category ${\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}$ is moreover tensored and cotensored over ${\mathcal {S}}\! et ^{\Delta ^{{{\,\mathrm{op}\,}}}}$, in that there are functors

$$\begin{aligned} (-)\boxtimes (-):{\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}\times {\mathcal {S}}\! et ^{\Delta ^{{{\,\mathrm{op}\,}}}}\rightarrow {\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}\text { and }(-)^{(-)}:{\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}\times ({{\mathcal {S}}\! et ^{\Delta ^{{{\,\mathrm{op}\,}}}}})^{{{\,\mathrm{op}\,}}}\rightarrow {\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}, \end{aligned}$$

with the property that $X \boxtimes -$ is left adjoint to ${\text {Map}}_{{\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}}(X,-)$ and $(-) \boxtimes K$ is left adjoint to $(-)^K$. For future reference, we recall that the tensor $X\boxtimes K$ of a simplicial object X with a simplicial set K is given levelwise by the formula

$$\begin{aligned} (X\boxtimes K)_k=\coprod \limits _{K_k} X_k, \end{aligned}$$

and there is a canonical map $X\boxtimes K\rightarrow X$.

When ${\mathcal {C}}$ is the category of sets, groups, rings, or modules over a fixed ring, the structure described above recovers the familiar simplicial enrichments (together with the corresponding tensors and cotensors) for the categories of simplicial sets, simplicial groups, simplicial rings, simplicial modules and simplicial groupoids, which were considered e.g. in [11, Ex. 6.2].

The restriction along the ordinal sum functor $\sigma$ induces a functor

$$\begin{aligned} \sigma ^*: {\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}} \rightarrow {\mathcal {C}}^{(\Delta \times \Delta )^{{{\,\mathrm{op}\,}}}}. \end{aligned}$$

that can be computed componentwise as

$$\begin{aligned} (\sigma ^*X)_{m,n}= X_{m+1+n}. \end{aligned}$$

By a standard argument (cf. e.g. [18, Sect. X.3]), the functor $\sigma ^*$ has both a left adjoint $\sigma _!$ and a right adjoint $\sigma _*$, given by left and right Kan extensions of a bisimplicial object along $\sigma$. As usual in this type of constructions, $\sigma _!$ is determined by sending representables to representables according to the formula

$$\begin{aligned} \sigma _!\Delta [k,l]=\Delta [k+1+l], \end{aligned}$$

and the value of $\sigma _*$ on a bisimplicial object Y satisfies

$$\begin{aligned} (\sigma _*Y)_{n}={{\,\mathrm{Map}\,}}_{{\mathcal {C}}^{(\Delta \times \Delta )^{{{\,\mathrm{op}\,}}}}}(\sigma ^*\Delta [n],Y). \end{aligned}$$

As mentioned in the introduction, when ${\mathcal {C}}={\mathcal {S}}\! et$ the functor $\sigma ^*={{\,\mathrm{Dec}\,}}$ is the classical décalage construction and $\sigma _*=T$ is the classical $T$-construction.

We are interested in describing the functor $\sigma _*\sigma ^*$ and the unit of the adjunction

$$\begin{aligned} \sigma ^*:{\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}} \rightleftarrows {\mathcal {C}}^{(\Delta \times \Delta )^{{{\,\mathrm{op}\,}}}}:\sigma _*. \end{aligned}$$

For this, we start by describing in general the functor $\sigma _!\sigma ^*$ and the counit of the adjunction

$$\begin{aligned} \sigma _!:{\mathcal {C}}^{(\Delta \times \Delta )^{{{\,\mathrm{op}\,}}}} \rightleftarrows {\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}:\sigma ^*. \end{aligned}$$

Theorem 2.1

Let ${\mathcal {C}}$ be a bicomplete category. For any simplicial object X in ${\mathcal {C}}$, there is a natural isomorphism

$$\begin{aligned} \sigma _!\sigma ^*X\cong X\boxtimes \Delta [1] \end{aligned}$$

and, under this isomorphism, the counit $\sigma _!\sigma ^*X\rightarrow X$ is identified with the canonical map $X\boxtimes \Delta [1]\rightarrow X$.

Before proving Theorem 2.1, we discuss a few direct consequences. First, the theorem can be used to produce an explicit description for the unit of the adjunction $(\sigma ^*,\sigma _*)$.

Corollary 2.2

Let ${\mathcal {C}}$ be a bicomplete category. For any simplicial object X in ${\mathcal {C}}$, there is a natural isomorphism

$$\begin{aligned} \sigma _*\sigma ^*X\cong X^{\Delta [1]} \end{aligned}$$

and, under this isomorphism, the unit $X\rightarrow \sigma _*\sigma ^*X$ is identified with the map $X^{\Delta [0]}\rightarrow X^{\Delta [1]}$ induced by $\Delta [1]\rightarrow \Delta [0]$.

Proof

The natural isomorphism follows from the fact $\sigma _*\sigma ^*$ and $(-)^{\Delta [1]}$ are the right adjoints of the functors $\sigma _!\sigma ^*$ and $(-)\boxtimes \Delta [1]$, which are isomorphic by Theorem 2.1, and by the uniqueness of right adjoints (dual to [18, Corollary IV.1.1]). $\square$

Given that the category ${\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}$ is always enriched, tensored and cotensored over ${\mathcal {S}}\! et ^{\Delta ^{{{\,\mathrm{op}\,}}}}$, there is a canonical notion of homotopy between maps (cf. [22, Sect. II.1.6] or [24, Sect. 3.8]) and therefore of strong deformations retracts. Relying on the explicit description of the unit, we can exploit the homotopy theory of simplicial sets and prove the following.

Corollary 2.3

Let X be a simplicial object in ${\mathcal {C}}$. The unit $X\rightarrow \sigma _*\sigma ^*X$ is a strong deformation retract.

Proof

Thanks to Corollary 2.2, it is enough to prove that the map $X\cong X^{\Delta [0]}\rightarrow X^{\Delta [1]}$ is a strong deformation retract. To see this, consider the maps

$$\begin{aligned} d^1:\Delta [0]\rightarrow \Delta [1]\text { and }s^0:\Delta [1]\rightarrow \Delta [0] \end{aligned}$$

which satisfy $s^0\circ d^1={{\,\mathrm{id}\,}}_{\Delta [0]}$ and $d^1\circ s^0\simeq _l{{\,\mathrm{id}\,}}_{\Delta [1]}$, where the symbol $\simeq _l$ denotes the (non-symmetric) relation of left homotopy of simplicial maps. They induce maps

$$\begin{aligned} d_1:X^{\Delta [1]}\rightarrow X^{\Delta [0]}\text { and }s_0:X^{\Delta [0]}\rightarrow X^{\Delta [1]} \end{aligned}$$

which satisfy the relations $d_1\circ s_0={{\,\mathrm{id}\,}}_{X^{\Delta [0]}}$ and $s_0\circ d_1\simeq _l{{\,\mathrm{id}\,}}_{X^{\Delta [1]}}$. This completes the proof. $\square$

Having in mind categories of an algebraic nature (such as those of groups, rings and modules), Quillen identifies certain conditions on a category ${\mathcal {C}}$ so that the category ${\mathcal {C}}^{{\Delta }^{{{\,\mathrm{op}\,}}}}$ supports a model structure that is compatible with the simplicial enrichment. Quillen’s original result is [22, Theorem II.4.4], but we here recall the formulation from [11, Theorems 5.1, 5.4].

Theorem 2.4

Let ${\mathcal {C}}$ be a bicomplete category, and $U:{\mathcal {C}}^{{\Delta }^{{{\,\mathrm{op}\,}}}}\rightarrow {\mathcal {S}}\! et ^{{\Delta }^{{{\,\mathrm{op}\,}}}}$ a functor that admits a left adjoint and that respects filtered colimits. Then the category ${\mathcal {C}}^{{\Delta }^{{{\,\mathrm{op}\,}}}}$ supports a simplicial model structure in which a morphism $f:X\rightarrow X'$ is a fibration (resp. weak equivalence) if and only if $Uf:UX\rightarrow UX'$ is a fibration (resp. weak equivalence) in the Kan–Quillen model structure, provided that every map with the left lifting property with respect to all fibrations is a weak equivalence.

In particular, this model structure recovers the Kan–Quillen model structure, and the usual model structure for simplicial objects of algebraic nature, as recalled in [11, Ex. 6.2]. The model structure for simplicial commutative rings and for simplicial commutative algebras over a commutative ring are used e.g. in [19, 28].

With respect to this model structure, we obtain the following corollary.

Corollary 2.5

For any simplicial object X, the unit $X\rightarrow \sigma _*\sigma ^*X$ is a weak equivalence in ${\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}$.

Proof

The statement follows from the fact that in a simplicial model structure all deformation retracts are weak equivalences by [12, Prop. 9.5.16]. $\square$

When ${\mathcal {C}}={\mathcal {S}}\! et$ and the category ${\mathcal {S}}\! et ^{\Delta ^{{{\,\mathrm{op}\,}}}}$ is endowed with the Kan–Quillen model structure, the corollary specializes to the well-known [3, Prop. 7.1], and it seems to be new in its generality.

3 The proof of the main result

To prove the theorem, we will use the relation of $\Delta$ with the category $\Delta _a$, which is the category $\Delta$ with an additional initial object $[-1]=\varnothing$. The ordinal sum $\sigma _a$ makes sense as a functor $\sigma _a:\Delta _a\times \Delta _a\rightarrow \Delta _a$ and endows $\Delta _a$ with a monoidal structure, whose unit object is the new object $[-1]$. We will denote the inclusion of $\Delta$ into $\Delta _a$ by $\iota :\Delta \rightarrow \Delta _a$.

Remark 3.1

Given a map $\beta :[l]\rightarrow [k]$ in $\Delta _a$, either $l\ne -1$ and $\beta$ actually lives in $\Delta$, or $l=-1$ and $\beta$ can be written as a composite of the unique map $[-1]\rightarrow [0]$ with a map $[0]\rightarrow [k]$ in $\Delta$.

In particular, a presheaf $X\in {\mathcal {C}}^{\Delta _a^{{{\,\mathrm{op}\,}}}}$ is an augmented simplicial object in ${\mathcal {C}}$, and to specify the structure of X it is enough to specify its structure as a simplicial object $X\in {\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}$, together with an extra face map $d_0:X_0\rightarrow X_{-1}$ that coequalizes all the other structure maps, i.e., it satisfies the extra simplicial identity $d_0d_0=d_0d_1:X_1\rightarrow X_{-1}$.

Similarly, to specify the structure map of a presheaf $Y\in {\mathcal {C}}^{(\Delta _a\times \Delta _a)^{{{\,\mathrm{op}\,}}}}$ it is enough to specify its structure as a bisimplicial object $Y\in {\mathcal {C}}^{(\Delta \times \Delta )^{{{\,\mathrm{op}\,}}}}$, together with the additional structure maps for the objects that involve $[-1]$.

Since ${\mathcal {C}}$ is cocomplete, the functors

$$\begin{aligned} (\iota \times \iota )^*:{\mathcal {C}}^{\Delta _a^{{{\,\mathrm{op}\,}}}\times \Delta _a^{{{\,\mathrm{op}\,}}}}\rightarrow {\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}\times \Delta ^{{{\,\mathrm{op}\,}}}} \text { and }\sigma _a^*:{\mathcal {C}}^{\Delta _a^{{{\,\mathrm{op}\,}}}}\rightarrow {\mathcal {C}}^{\Delta _a^{{{\,\mathrm{op}\,}}}\times \Delta _a^{{{\,\mathrm{op}\,}}}} \end{aligned}$$

both admit left adjoints $(\iota \times \iota )_!$ and $(\sigma _a)_!$, which can be used to describe $\sigma _!$. Indeed, it is pointed out in [27, Sect. 2] that for every simplicial object X there is a natural isomorphism

$$\begin{aligned} \sigma _!(X)\cong \iota ^*(\sigma _a)_!(\iota \times \iota )_!(X). \end{aligned}$$

We give an explicit formula for $(\iota \times \iota )_!$ that makes use of the construction $\pi _0$, a generalization of the set of connected components of a simplicial set.

Notation 3.2

Let ${\mathcal {C}}$ be a category with coequalizers. For a simplicial object X in ${\mathcal {C}}$, we denote by

$$\begin{aligned} \pi _0(X):={{\,\mathrm{colim}\,}}\left( X_1\rightrightarrows X_0\right) \end{aligned}$$

the coequalizer of the face maps $d_1,d_0:X_1\rightarrow X_0$. This defines a functor

$$\begin{aligned} \pi _0:{\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}\rightarrow {\mathcal {C}}. \end{aligned}$$

We can now give a formula for $(\iota \times \iota )_!$, which was known to experts. Note that, for any bisimplicial object Y in ${\mathcal {C}}$, the assignment $[k]\mapsto \pi _0(Y_{k,-})$ defines a simplicial object in ${\mathcal {C}}$, to which $\pi _0$ can be applied again. The result is denoted by $\pi _0\pi _0(Y)$.

Proposition 3.3

Let ${\mathcal {C}}$ be a category with coequalizers. The left adjoint

$$\begin{aligned} (\iota \times \iota )_!:{\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}\times \Delta ^{{{\,\mathrm{op}\,}}}}\rightarrow {\mathcal {C}}^{\Delta _a^{{{\,\mathrm{op}\,}}}\times \Delta _a^{{{\,\mathrm{op}\,}}}} \end{aligned}$$

of the precomposition functor $(\iota \times \iota )^*$ is given levelwise on objects by

$$\begin{aligned} ((\iota \times \iota )_!Y)_{i,i'}={\left\{ \begin{array}{ll} \pi _0(\pi _0(Y)), &{}\quad \text{ if } i=i'=-1,\\ \pi _0(Y_{i,-}), &{}\quad \text{ if } i>-1, i'=-1,\\ \pi _0(Y_{-,i'}),&{}\quad \text{ if } i=-1, i'>-1,\\ Y_{i,i'}, &{}\quad \text{ else. } \end{array}\right. } \end{aligned}$$

The bisimplicial structure of $(\iota \times \iota )_!Y$ is inherited from Y, and the additional structure maps (in the sense of Remark 3.1) are given by the quotient maps

$$\begin{aligned} Y_{i,0}\rightarrow \pi _0(Y_{i,-})\text { and }Y_{0,i}\rightarrow \pi _0(Y_{-,i}) \end{aligned}$$

and the maps induced on $\pi _0$.

The proof of the proposition needs two preliminary lemmas. We start by recording the following description of $\iota _!$, the left adjoint of the restriction functor $\iota ^*:{\mathcal {C}}^{\Delta _a^{{{\,\mathrm{op}\,}}}}\rightarrow {\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}$.

Lemma 3.4

([27, Sect. 2.2]) Let ${\mathcal {C}}$ be a category with coequalizers. The left adjoint

$$\begin{aligned} \iota _!:{\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}}\rightarrow {\mathcal {C}}^{\Delta _a^{{{\,\mathrm{op}\,}}}} \end{aligned}$$

of the restriction functor $\iota ^*$ is given levelwise on objects by

$$\begin{aligned} (\iota _!X)_k={\left\{ \begin{array}{ll} \pi _0(X),&{}\quad \text{ if } k=-1,\\ X_k, &{}\quad \text{ else. } \end{array}\right. } \end{aligned}$$

The simplicial structure of $\iota _!X$ is inherited from X, and the additional structure map (in the sense of Remark 3.1) of $\iota _!X$ is given by the quotient map $X_0\rightarrow \pi _0(X)$.

The following lemma is straightforward.

Lemma 3.5

Let $I_1, I_2, J$ be small categories, $f:I_1\rightarrow I_2$ a functor, and ${\mathcal {C}}$ a cocomplete category. Then the functors

$$\begin{aligned} \begin{aligned} (f\times {{\,\mathrm{id}\,}})_! :{\mathcal {C}}^{I_1\times J} \rightarrow {\mathcal {C}}^{I_2\times J}\ \text{ and } \ f_! :({\mathcal {C}}^{J})^{I_1}\rightarrow ({\mathcal {C}}^{J})^{I_2} \end{aligned} \end{aligned}$$

exist and can be identified modulo the natural equivalence of categories

$$\begin{aligned} {\mathcal {C}}^{I_i\times J} \simeq ({\mathcal {C}}^{J})^{I_i}. \end{aligned}$$

We can now prove the formula for $(\iota \times \iota )_!$.

Proof of Proposition 3.3

Given a bisimplicial object Y, the desired formula for $(\iota \times \iota )_!(Y)$ is a straightforward computation, using the identification

$$\begin{aligned} (\iota \times \iota )_!(Y)=((\iota \times {{\,\mathrm{id}\,}})\circ ({{\,\mathrm{id}\,}}\times \iota ))_!(Y)=(\iota \times {{\,\mathrm{id}\,}})_!\circ ({{\,\mathrm{id}\,}}\times \iota )_!(Y), \end{aligned}$$

together with Lemmas 3.5 and 3.4. $\square$

We now proceed to the study of $({\sigma _a})_!$. The following construction (and the observation that it is always possible) will be crucial for the rest of the argument. Note that the object $[-1]$ plays an essential role, and the same construction would not make sense when replacing $\Delta _a$ with $\Delta$.

Construction 3.6

Let $\beta :[l] \rightarrow [k]$ be a morphism in $\Delta _a$, and let $-1\le i\le k$. Then there is a unique number $-1\le j_{\beta }(i) \le l$ and a unique pair of morphisms $(\beta _{1,i}, \beta _{2,i})$ in $\Delta _a\times \Delta _a$ with

$$\begin{aligned} \beta _{1,i}:[j_{\beta }(i)]\rightarrow [i]\quad \text { and } \quad \beta _{2,i}:[l-j_{\beta }(i)-1]\rightarrow [k-i-1] \end{aligned}$$

so that $\sigma _a(\beta _{1,i}, \beta _{2,i})=\beta$. Moreover, the value of $j_{\beta }(i)$ is given by

$$\begin{aligned} j_{\beta }(i)=\left\{ \begin{array}{lrc} \max \{j \in [l]\,|\,\beta (j)\le i\}&{}\quad \text {if }\{j \in [l]\,|\,\beta (j)\le i\}\ne \varnothing ,&{}\\ -1&{}\quad \text {if }\{j \in [l]\,|\,\beta (j)\le i\}=\varnothing .&{}\\ \end{array} \right. \end{aligned}$$

Indeed, if there exists $(\alpha _1,\alpha _2)$ in $\Delta _a\times \Delta _a$ with

$$\begin{aligned} \alpha _1:[l_1]\rightarrow [i]\quad \text { and } \quad \alpha _2:[l_2]\rightarrow [k-i-1] \end{aligned}$$

such that $\sigma _a(\alpha _1,\alpha _2)=\beta$, we necessarily have

$$\begin{aligned} \beta (j)\le i\text { for }j\le l_1\text { and }\beta (j)>i\text { for }j>l_1. \end{aligned}$$

This forces the value of $l_1$ to be

$$\begin{aligned} l_1=\left\{ \begin{array}{lrc} \max \{j \in [l]\,|\,\beta (j)\le i\}&{}\quad \text {if }\{j \in [l]\,|\,\beta (j)\le i\}\ne \varnothing ,&{}\\ -1&{}\quad \text {if }\{j \in [l]\,|\,\beta (j)\le i\}=\varnothing .&{}\\ \end{array} \right. \end{aligned}$$

If we define the maps

$$\begin{aligned} \beta _{i,1}:[j_{\beta }(i)]\rightarrow [i]\text { and }\beta _{i,2}:[l-j_{\beta }(i)-1]\rightarrow [k-i-1] \end{aligned}$$

by means of the assignments

$$\begin{aligned} r\mapsto \beta (r)\quad \text { and }\quad r\mapsto \beta (j_{\beta }(i)+1+r)-i-1,\text { respectively,} \end{aligned}$$

then one can check that $\sigma _a(\beta _{1,i}, \beta _{2,i})=\beta$ and the pair $(\beta _{i,1},\beta _{i,2})$ is unique with this property.

Example 3.7

The coface map $d^k :[n]\rightarrow [n+1]$ can be written as$\sigma _a({{\,\mathrm{id}\,}}_{[i]},d^{k-i-1})$ for $i<k$ and $\sigma _a(d^k, {{\,\mathrm{id}\,}}_{[n-i-1]})$ for $i\ge k$.

We can now give the formula for $(\sigma _a)_!$, cf. [15, Chapter 3] and [25, Lemma 5.1].

Proposition 3.8

Let ${\mathcal {C}}$ be a cocomplete category. The left adjoint

$$\begin{aligned} (\sigma _a)_! :{\mathcal {C}}^{\Delta _a^{{{\,\mathrm{op}\,}}}\times \Delta _a^{{{\,\mathrm{op}\,}}}}\rightarrow {\mathcal {C}}^{\Delta _a^{{{\,\mathrm{op}\,}}}} \end{aligned}$$

of $(\sigma _a)^*$ is given levelwise on objects by

$$\begin{aligned} ((\sigma _a)_!A)_k=\coprod _{i=-1}^{k} A_{i,k-i-1}. \end{aligned}$$

The structure map of the simplicial set $(\sigma _a)_!A$ induced by some $\beta :[l] \rightarrow [k]$ is given on the i-th summand via the structure map of A induced by $(\beta _{1,i}, \beta _{2,i})$,

$$\begin{aligned} A_{i,k-i-1}\rightarrow A_{j_{\beta }(i),l-j_{\beta }(i)-1}\rightarrow \coprod \limits _{j=-1}^{l} A_{j,l-j-1}= ((\sigma _a)_!A)_l, \end{aligned}$$

where $\beta _{1,i}:[j_{\beta }(i)]\rightarrow [i]$ and $\beta _{2,i}:[l-j_{\beta }(i)-1]\rightarrow [k-i-1]$ are the morphisms described in Construction 3.6 and uniquely determined by the condition $\sigma _a(\beta _{1,i}, \beta _{2,i})=\beta$.

The proof will make use of the following description of the slice category $[k]\downarrow \sigma _a$.

Remark 3.9

Observe that the set of objects of $[k]\downarrow \sigma _a$ can be canonically identified with

$$\begin{aligned} \{(l_1,l_2, \gamma )\ |\ l_1,l_2\ge -1,\gamma :[k]\rightarrow \sigma _a([l_1],[l_2])=[l_1+1+l_2]\text { in }\Delta _a\}. \end{aligned}$$

Modulo this identification, a morphism in $[k]\downarrow \sigma _a$ from $(l_1',l_2',\gamma ')$ to $(l_1, l_2, \gamma )$ consists of a pair of morphisms $(\alpha _1,\alpha _2)$ in $\Delta _a\times \Delta _a$ with

$$\begin{aligned} \alpha _1:[l_1']\rightarrow [l_1]\quad \text { and } \quad \alpha _2:[l_2']\rightarrow [l_2] \end{aligned}$$

such that $\sigma _a(\alpha _1, \alpha _2)\circ \gamma '=\gamma$.

For any $k\ge -1$, we regard the set $\{-1,0,1,\dots ,k\}$ as a discrete category.

Lemma 3.10

The functor

$$\begin{aligned} J:\{-1,0,1,\dots ,k\}\rightarrow [k]\downarrow \sigma _a\quad \text { given by }\quad j\mapsto (j, k-j-1, {{\,\mathrm{id}\,}}_{[k]}) \end{aligned}$$

is initial. Equivalently, its opposite functor

$$\begin{aligned} J:\{-1,0,1,\dots ,k\}\rightarrow ([k]\downarrow \sigma _a)^{{{\,\mathrm{op}\,}}}\cong \sigma _a^{{{\,\mathrm{op}\,}}}\downarrow [k] \end{aligned}$$

is final.

Proof of Lemma 3.10

Given an object $(i_1,i_2,\gamma )$ of $[k]\downarrow \sigma _a$, by Construction 3.6 there exists a unique pair of morphisms $(\alpha _1,\alpha _2)$ in $\Delta _a\times \Delta _a$ with

$$\begin{aligned} \alpha _1:[j_{\beta }(i_1)]\rightarrow [i_1]\quad \text { and } \quad \alpha _2:[k-j_{\beta }(i_1)-1]\rightarrow [i_2] \end{aligned}$$

such that $\sigma _a(\alpha _1, \alpha _2)=\gamma$. In other words, there exists a unique j, with $-1\le j\le k$, and a unique morphism in $[k]\downarrow \sigma _a$ to $(i_1,i_2,\gamma )$ from an element of the form $J(j)=(j, k-j-1, {{\,\mathrm{id}\,}}_{[k]})$. This proves that, for any object $(i_1,i_2,\gamma )$ of $[k]\downarrow \sigma _a$, the slice category $J\downarrow (i_1,i_2,\gamma )$ has precisely one object. In particular, the slice category $J\downarrow (i_1,i_2,\gamma )$ is connected and not empty. Thus

$$\begin{aligned} J:\{-1,0,1,\dots ,k\}\rightarrow [k]\downarrow \sigma _a \end{aligned}$$

is initial. $\square$

We can now prove the proposition.

Proof of Proposition 3.8

We deduce the formula for $((\sigma _a)_!A)_k$ by means of the pointwise left Kan extension formula from [18, Theorem X.3.1], the cofinality of J from Lemma 3.10, and the key property of cofinal functors from [18, Theorem IX.3.1]:

$$\begin{aligned} \begin{array}{rcl} ((\sigma _a)_!A)_k &{}\cong &{}{{\,\mathrm{colim}\,}}\left( \sigma _a^{{{\,\mathrm{op}\,}}}\downarrow [k]\cong ([k]\downarrow \sigma _a)^{{{\,\mathrm{op}\,}}}\rightarrow (\Delta _a\times \Delta _a)^{{{\,\mathrm{op}\,}}}\xrightarrow {A} {\mathcal {C}}\right) \\ &{}\cong &{}{{\,\mathrm{colim}\,}}\left( \{-1,0,\dots ,k\}\xrightarrow {J} (\sigma _a\downarrow [k])^{{{\,\mathrm{op}\,}}} \rightarrow (\Delta _a\times \Delta _a)^{{{\,\mathrm{op}\,}}} \xrightarrow {A} {\mathcal {C}}\right) \\ &{}\cong &{}\coprod \limits _{j=-1}^{k} A_{j,k-j-1}. \end{array} \end{aligned}$$

We now describe the structure map

$$\begin{aligned} ((\sigma _a)_!A)_{k}\rightarrow ((\sigma _a)_!A)_{l} \end{aligned}$$

of $(\sigma _a)_!A$ induced by a map $\beta :[l]\rightarrow [k]$ in $\Delta _a$ under the chain of isomorphisms above. By [18, Theorem X.3.1], the map induced by $\beta$

$$\begin{aligned} \underset{(i_1,i_2,\gamma )\in \sigma _a^{{{\,\mathrm{op}\,}}}\downarrow [k]}{{{\,\mathrm{colim}\,}}}A_{i_1,i_2}\rightarrow \underset{(j_1,j_2,\delta )\in \sigma _a^{{{\,\mathrm{op}\,}}}\downarrow [l]}{{{\,\mathrm{colim}\,}}}A_{j_1,j_2} \end{aligned}$$

identifies the copy of $A_{i_1,i_2}$ corresponding to the component $(i_1,i_2,\gamma )$ of the left-hand side with the copy of $A_{i_1,i_2}$ corresponding to the component $(i_1,i_2,\gamma \circ \beta )$ of the right-hand side. By reindexing the colimits according to [18, Theorem IX.3.1] and Lemma 3.10, one can check that the map induced by $\beta$ on the sums

$$\begin{aligned} \coprod \limits _{i=-1}^{k} A_{i,k-i-1}\cong \underset{i\in \{-1,0,\dots ,k\}}{{{\,\mathrm{colim}\,}}}A_{i,k-i-1}\rightarrow \underset{j\in \{-1,0,\dots ,l\}}{{{\,\mathrm{colim}\,}}}A_{j,l-j-1}\cong \coprod \limits _{j=-1}^{l} A_{j,l-j-1} \end{aligned}$$

is induced by $(\beta _{i,1},\beta _{i,2})$ on the i-th summand $A_{i,k-i-1}$. $\square$

We collect the insights so far to obtain a formula for $\sigma _!$.

Proposition 3.11

Let ${\mathcal {C}}$ be a cocomplete category. The left adjoint

$$\begin{aligned} \sigma _! :{\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}\times \Delta ^{{{\,\mathrm{op}\,}}}}\rightarrow {\mathcal {C}}^{\Delta ^{{{\,\mathrm{op}\,}}}} \end{aligned}$$

of $\sigma ^*$ is given levelwise on objects by

$$\begin{aligned} (\sigma _!Y)_k\cong \pi _0(Y_{-,k}) \sqcup \coprod _{i=0}^{k-1} Y_{i, k-i-1} \sqcup \pi _0(Y_{k,-}). \end{aligned}$$

The structure map of the simplicial object $\sigma _!Y$ induced by some $\beta :[l] \rightarrow [k]$ in $\Delta$ is given on the first summand of $(\sigma _!Y)_k$ by the maps induced on $\pi _0$, i.e.,

$$\begin{aligned} \pi _0(Y_{k,-})\rightarrow \pi _0(Y_{l,-})\rightarrow \pi _0(Y_{-,l}) \sqcup \coprod _{j=0}^{l-1} Y_{j, l-j-1} \sqcup \pi _0(Y_{l,-}), \end{aligned}$$

and is given on the last summand of $(\sigma _!Y)_k$ by the dual map induced on $\pi _0$.

To describe the structure map induced by the same $\beta :[l]\rightarrow [k]$ on the i-th summand $Y_{i,k-i-1}$ occurring in $(\sigma _!Y)_k$, decompose $\beta$ as a map of $\Delta _a$ using Construction 3.6 as $\beta =\sigma _a(\beta _{i,1},\beta _{i,2})$ with

$$\begin{aligned} \beta _{1,i}:[j_{\beta }(i)]\rightarrow [i]\text { and }\beta _{2,i}:[l-j_{\beta }(i)-1]\rightarrow [k-i-1]. \end{aligned}$$

If $-1<j_{\beta }(i)<l$, the map $\beta$ acts on $Y_{i,k-i-1}$ as the structure map of Y corresponding to $(\beta _{1,i},\beta _{2,i})$,

$$\begin{aligned} Y_{i,k-i-1}\rightarrow Y_{j_{\beta }(i),l-j_{\beta }(i)-1}\rightarrow \pi _0(Y_{-,l}) \sqcup \coprod _{j=0}^{l-1} Y_{j, l-j-1} \sqcup \pi _0(Y_{l,-}). \end{aligned}$$

If $j_{\beta }(i)=-1$, the map $\beta$ acts on $Y_{i,k-i-1}$ as the structure map of Y corresponding to $({{\,\mathrm{id}\,}}_{[i]},\beta _{2,i})$ composed with the map onto $\pi _0$,

$$\begin{aligned} Y_{i,k-i-1}\rightarrow Y_{i,l}\rightarrow Y_{0,l}\rightarrow \pi _0(Y_{-,l}) \rightarrow \pi _0(Y_{-,l}) \sqcup \coprod _{j=0}^{l-1} Y_{j, l-i-1} \sqcup \pi _0(Y_{l,-}). \end{aligned}$$

Dually, if $j_{\beta }(i)=k$, the map $\beta$ acts on $Y_{i,k-i-1}$ as the structure map of Y corresponding to $(\beta _{i,1},{{\,\mathrm{id}\,}}_{[k-i-1]})$ composed with the map onto $\pi _0$.

Under the identification above, $\sigma _!$ acts on a bisimplicial map as a sum of the corresponding components together with the induced maps on $\pi _0$.

Proof

Given a bisimplicial object Y, we use the identification

$$\begin{aligned} \sigma _!(Y)\cong \iota ^*(\sigma _a)_!(\iota \times \iota )_!(Y) \end{aligned}$$

from [27, Sect. 2], together with Propositions 3.8 and 3.3, to obtain the isomorphisms for any $k\ge 0$

$$\begin{aligned} \sigma _!(Y)_k\cong \iota ^*(\sigma _a)_!(\iota \times \iota )_!(Y)_k\cong (\sigma _a)_!(\iota \times \iota )_!(Y)_k\cong \coprod _{i=-1}^{k} (\iota \times \iota )_!(Y)_{i,k-i-1}, \end{aligned}$$

where

$$\begin{aligned} (\iota \times \iota )_!(Y)_{i,k-i-1}=\left\{ \begin{array}{ll} \pi _0(Y_{i,-})=\pi _0(Y_{k,-}), &{} \text{ if } i=k,\\ \pi _0(Y_{-,k-i-1})=\pi _0(Y_{-,k}),&{} \text{ if } i=-1,\\ Y_{i,k-i-1} &{} \text{ if } -1<i<k. \end{array}\right. \end{aligned}$$

We now describe the structure map

$$\begin{aligned} \sigma _!(Y)_k\rightarrow \sigma _!(Y)_l \end{aligned}$$

of $\sigma _!(Y)$ induced by a map $\beta :[l]\rightarrow [k]$ in $\Delta$ under the chain of isomorphisms above, by saying how it acts on every summand of $\sigma _!(Y)_k$. By Propositions 3.11 and 3.13, the map $\beta ^*$ acts on the i-th summand of $\sigma _!(Y)_k$ as the structure map of $(\iota \times \iota )_!(Y)$ induced by $(\beta _{1,i},\beta _{2,i})$,

$$\begin{aligned} (\beta _{1,i},\beta _{2,i})^* :\left( (\iota \times \iota )_!Y\right) _{i, k-i-1}\rightarrow \left( (\iota \times \iota )_!Y\right) _{j_{\beta }(i), l-j_{\beta }(i)-1}, \end{aligned}$$

where $\beta _{1,i}:[j_{\beta }(i)]\rightarrow [i]$ and $\beta _{2,i}:[l-j_{\beta }(i)-1]\rightarrow [k-i-1]$ are so that $\sigma _a(\beta _{1,i},\beta _{2,i})=\beta$, as in Construction 3.6. To further rewrite these, we need to distinguish several cases. Factoring $(\beta _{1,i},\beta _{2,i})=(\beta _{1,i},{{\,\mathrm{id}\,}})\circ ({{\,\mathrm{id}\,}},\beta _{2,i})$, we may assume that $k-i=l-j_{\beta }(i)$ and $\beta _{2,i}={{\,\mathrm{id}\,}}_{[k-i-1]}$, since the two parts can be treated with similar arguments. By means of Proposition 3.3, we describe the map

$$\begin{aligned} (\beta _{1,i},{{\,\mathrm{id}\,}})^* :\left( (\iota \times \iota )_!Y\right) _{i, k-i-1}\rightarrow \left( (\iota \times \iota )_!Y\right) _{j_{\beta }(i), k-i-1}. \end{aligned}$$

by distinguishing several cases.

• If $-1<i<k$ and $-1<j_{\beta }(i)<l$, we obtain precisely the structure map of Y induced by $(\beta _{1,i},{{\,\mathrm{id}\,}})$,

  $$\begin{aligned} (\beta _{1,i},{{\,\mathrm{id}\,}})^* :Y_{i, k-i-1}\rightarrow Y_{j_{\beta }(i), k-i-1}. \end{aligned}$$

• If $i=-1$, then $j_{\beta }(i)=-1$ and $\beta ={{\,\mathrm{id}\,}}$. We therefore obtain the identity map,

  $$\begin{aligned} {{\,\mathrm{id}\,}}:\pi _0(Y_{-,k})\rightarrow \pi _0(Y_{-,k}). \end{aligned}$$

• If $i=k$, we obtain the map induced on $\pi _0$,

  $$\begin{aligned} \beta ^*:\pi _0(Y_{k,-})\rightarrow \pi _0(Y_{l,-}). \end{aligned}$$

• If $-1<i<k$ and $j_{\beta }(i)=-1$, we obtain the quotient map,

  $$\begin{aligned} Y_{i,k}\rightarrow \pi _0(Y_{-,k}). \end{aligned}$$

• If $-1<i<k$ and $j_{\beta }(i)=l$, we obtain the quotient map composed with $\beta ^*$,

  $$\begin{aligned} Y_{i,k}\rightarrow \pi _0(Y_{l,-}). \end{aligned}$$

$\square$

We will need the following property of simplicial objects, which is a variant of [29, Lemma 89].

Lemma 3.12

Let ${\mathcal {C}}$ be any category. For any a simplicial object X in ${\mathcal {C}}$ the diagram

is a coequalizer diagram in ${\mathcal {C}}$ for all $k\ge 2$ and all $0\le i\le k-1$.

Proof

As a consequence of the simplicial identities for X, the diagram

becomes a split fork (in the sense of [18, Sect. VI.6]) when considered together with the maps

$$\begin{aligned} X_k\xleftarrow {s_{i+1}} X_{k-1} \xleftarrow {s_i} X_{k-2} \end{aligned}$$

when $i<k-1$ and together with the maps

$$\begin{aligned} X_k\xleftarrow {s_{k-2}} X_{k-1} \xleftarrow {s_{k-2}} X_{k-2} \end{aligned}$$

when $i=k-1$. By [18, Lemma VI.6], the original diagram is therefore a split coequalizer, and in particular a coequalizer. $\square$

We now use this lemma to identify $\pi _0((\sigma ^*X)_{k,-})$ and $\pi _0((\sigma ^*X)_{-,k})$ with $X_{k}$.

Proposition 3.13

Let ${\mathcal {C}}$ be a cocomplete category. For any simplicial object X in ${\mathcal {C}}$ there are isomorphisms

$$\begin{aligned} \pi _0((\sigma ^*X)_{-,k}) \cong X_{k}\cong \pi _0((\sigma ^*X)_{k,-}), \end{aligned}$$

which are natural in X and k.

Proof

By definition of $\sigma ^*$, Lemma 3.12 and Notation 3.2, we have the following isomorphisms

which can be checked to be natural in X and k by direct inspection.

A similar argument applies to the second isomorphism. $\square$

We are now ready to prove Theorem 2.1, and describe the counit of the adjunction $(\sigma _!,\sigma ^*)$.

Proof of Theorem 2.1

For any simplicial object X in the bicomplete category ${\mathcal {C}}$, we start by identifying $\sigma _!\sigma ^*X$.

By means of Propositions 3.3, 3.11, 3.12, and 3.13, we obtain the identification

$$\begin{aligned} \sigma _!(\sigma ^*X)_k\cong \pi _0(\sigma ^*X_{-,k})\amalg \coprod _{i=0}^{k-1} \sigma ^*X_{i, k-i-1}\amalg \pi _0(\sigma ^*X_{k,-})\cong \coprod _{i=-1}^{k} X_k. \end{aligned}$$

(3.14)

We now describe the structure map

$$\begin{aligned} (\sigma _!(\sigma ^*X))_{k}\rightarrow (\sigma _!(\sigma ^*X))_{l} \end{aligned}$$

of $\sigma _!(\sigma ^*X)$ induced by a map $\beta :[l]\rightarrow [k]$ in $\Delta$ under the chain of isomorphisms above. By Propositions 3.11 and 3.13, the map induced by $\beta$ on

$$\begin{aligned} \coprod _{i=-1}^{k} X_k\rightarrow \coprod _{j=-1}^{l} X_l \end{aligned}$$

acts on the i-th summand $X_k$ occurring in $(\sigma _!(\sigma ^*X))_k$ as the structure map of X corresponding to $\beta$ with values in the $j_{\beta }(i)$-th copy of $X_l$ occurring in $(\sigma _!(\sigma ^*X))_l$.

We now describe the k-th component

$$\begin{aligned} \sigma _!\sigma ^*X_k\rightarrow X_k \end{aligned}$$

of the counit of the adjunction $(\sigma _!,\sigma ^*)$ at X under the following chain of isomorphisms

$$\begin{aligned} \sigma _!(\sigma ^*X)_k\cong (\sigma _a)_!(\iota \times \iota )_!(\sigma ^*X)_k\cong \coprod _{i=-1}^{k} (\iota \times \iota )_!(\sigma ^*X)_{i,k-i-1}\cong \coprod \limits _{i=-1}^{k} X_k, \end{aligned}$$

which gives an alternative but equivalent description of the identification (3.14) of $\sigma _!(\sigma ^*X)_k$ with the coproduct $\coprod _{i=-1}^{k} X_k$. We can expand $(\sigma _a)_!$ further using the pointwise Kan extension formula [18, Theorem X.3.1], obtaining

$$\begin{aligned} \sigma _!(\sigma ^*X)_k\cong \underset{(i_1,i_2,\gamma )\in \sigma _a^{{{\,\mathrm{op}\,}}}\downarrow [k]}{{{\,\mathrm{colim}\,}}}X_{i_1+1+i_2}\cong \underset{i\in \{-1,0,\dots ,k\}}{{{\,\mathrm{colim}\,}}}X_{i+1+k-i-1}\cong \coprod \limits _{i=-1}^{k} X_k. \end{aligned}$$

By [18, Theorem X.3.1], the k-th component of the counit as a map

$$\begin{aligned} \underset{(i_1,i_2,\gamma )\in \sigma ^{{{\,\mathrm{op}\,}}}\downarrow [k]}{{{\,\mathrm{colim}\,}}}X_{i_1+1+i_2}\rightarrow X_k \end{aligned}$$

acts on the $(i_1,i_2,\gamma )$-th component as the map $X_{i_1+1+i_2}\rightarrow X_k$ induced on X by $\gamma$. By reindexing the colimits according to [18, Theorem IX.3.1], one can check that the k-th component of the counit under this identification,

$$\begin{aligned} \coprod \limits _{i=-1}^{k} X_k\cong \underset{i\in \{-1,0,\dots ,k\}}{{{\,\mathrm{colim}\,}}}X_{i+1+k-i-1}\rightarrow X_k, \end{aligned}$$

is given by the folding map of $X_k$, as it acts on every copy of $X_k$ as the identity of $X_k$.

We need a further identification, which will allow us to see $\coprod _{i=-1}^{k} X_k$ as the k-th object of the tensor $X \boxtimes \Delta [1]$ and discuss the naturality in k of this identification. Note that there is bijection

$$\begin{aligned} \{-1,0,\dots ,k\}\cong \Delta [1]_k \text{ given } \text{ by } i\mapsto f_i, \end{aligned}$$

where

$$\begin{aligned} f_i:[k]\rightarrow [1]\text { for }i=-1,\dots ,k \text { is given by}\;\; r\mapsto {\left\{ \begin{array}{ll} 0, \quad r\le i, \\ 1, \quad \text{ else. } \end{array}\right. } \end{aligned}$$

This identification leads to the isomorphism

$$\begin{aligned} (\sigma _!(\sigma ^*X))_k\cong \coprod \limits _{i=-1}^{k} X_k \cong \coprod \limits _{\Delta [1]_k} X_k\cong (X\boxtimes \Delta [1])_k, \end{aligned}$$

which is seen to be natural in k by direct inspection. Moreover, with respect to this isomorphism, the folding map

$$\begin{aligned} \coprod _{i=-1}^{k} X_k\rightarrow X_k \end{aligned}$$

corresponds to the canonical map

$$\begin{aligned} (X\boxtimes \Delta [1])_k\rightarrow X_k\cong (X\boxtimes \Delta [0])_k. \end{aligned}$$

This proves the desired isomorphism of simplicial objects in ${\mathcal {C}}$

$$\begin{aligned} (\sigma _!(\sigma ^*X))\cong X\boxtimes \Delta [1], \end{aligned}$$

which also allows us to identify the counit with the canonical map

$$\begin{aligned} X\boxtimes \Delta [1]\rightarrow X\cong X\boxtimes \Delta [0] \end{aligned}$$

which is induced by the map $\Delta [1]\rightarrow \Delta [0].$$\square$