Classically, the localization \( \mathcal{C} [\mathcal{W}^{-1}] \) of a (small) category \( \mathcal{C} \) at a subcategory \( \mathcal{W} \) is given by:
- the objects of \( \mathcal{C}[\mathcal{W}]^{-1} \) are objects of \( \mathcal{C} \),
- the morphisms \( \mathcal{C}[\mathcal{W}^{-1}] (X, Y) \) are diagrams \[ \displaystyle X \rightarrow X _ 1 \xleftarrow{f _ 1} X _ 2 \rightarrow X _ 3 \leftarrow \cdots \rightarrow X _ {2n-1} \xleftarrow{f _ n} X _ {2n} \rightarrow Y \] with \( f _ i \in \mathcal{W} \), quotiented out by the obvious relations, and
- composition is given by concatenation of diagrams.
This localization has the universal property that if \( F : \mathcal{C} \rightarrow \mathcal{D} \) sends all morphisms of \( \mathcal{W} \) to isomorphisms, then it factors uniquely through \( \mathcal{C} \rightarrow \mathcal{C}[\mathcal{W}^{-1}] \).
Our goal is to develop an analogue for \( (\infty,1) \)-categories, using simplicial categories. Given a simplicial category \( \mathcal{C} \) and a subcategory \( \mathcal{W} \), we would like to define a localization \( \mathcal{L}(\mathcal{C}, \mathcal{W}) \), but we not only invert the arrows but also the "higher homotopy data" of arrows. The construction of simplicial localization was first given by Dwyer and Kan [DK80].
1. Model structure on the category of simplicial categories#
There are different models for \( (\infty, 1) \)-categories, but we are going to use simplicial categories.
Definition 1. A simplicial category is a small category enriched over the category of simplicial sets \( \mathsf{sSet} \). Explicitly, this is a category \( \mathcal{C} \) such that each \( \mathcal{C}(X, Y) \) is a simplicial set and composition \( \mathcal{C}(X, Y) \times \mathcal{C}(Y, Z) \rightarrow \mathcal{C}(X, Z) \) is given as a map of simplicial sets. We denote by \( \mathsf{sCat} \) the category of simplicial categories.
The data of a simplicial category is equivalent to the data of a simplicial object \( \mathcal{C} _ \bullet \) in the category \( \mathsf{Cat} \) of categories, but such that all the face maps and degeneracy maps inducing isomorphisms on the set of objects. So for \( \mathcal{C} \in \mathsf{sCat} \), we may denote by \( \mathcal{C} _ p \in \mathsf{Cat} \) the category of \( p \)-simplices.
Definition 2. For \( \mathcal{C} \in \mathsf{sCat} \), we define its homotopy category \( \pi _ 0 \mathcal{C} \) as the category with the same objects and \[ \displaystyle (\pi _ 0 \mathcal{C})(X, Y) = \pi _ 0(\mathcal{C}(X, Y)), \] where we define \( \pi _ 0 \) of a simplicial set to be the set of connected components. (We may consider it as \( \pi _ 0 \) of the geometric realization.) We say that a morphism \( e \in \mathcal{C}(X, Y) _ 0 \) is a homotopy equivalence if it is an isomorphism in \( \pi _ 0 \mathcal{C} \).
Definition 3. We say that a functor \( f : \mathcal{C} \rightarrow \mathcal{D} \) is a Dwyer–Kan equivalence if for each \( X, Y \in \mathrm{ob}(\mathcal{C}) \), \[ \displaystyle \mathcal{C}(X, Y) \rightarrow \mathcal{D}(f(X), f(Y)) \] is a weak equivalence in the classical sense (i.e., the map on geometric realization is a homotopy equivalence), and the induced functor \( f _ 0 : \pi _ 0 \mathcal{C} \rightarrow \pi _ 0 \mathcal{D} \) is an equivalence of categories.
It turns out that there is a model structure on \( \mathsf{sCat} \) with Dwyer–Kan equivalences as weak equivalences.
Theorem 4 (Bergner [Ber07]). There exists a model structure on the category \( \mathsf{sCat} \) of small simplicial categories such that:
- weak equivalences are Dwyer–Kan equivalences,
- fibrations are the maps \( f : \mathcal{C} \rightarrow \mathcal{D} \) such that \[ \displaystyle \mathcal{C}(X, Y) \rightarrow \mathcal{D}(f(X), f(Y)) \] is a Kan fibration, and any homotopy equivalence \( e : f(X) \rightarrow Y \) in \( \mathcal{D} \) can be lifted to some homotopy equivalence \( \tilde{e} : X \rightarrow Y^\prime \) such that \( f(\tilde{e}) = e \),
- the model structure is cofibrantly generated,
- the model structure is right proper, i.e., pullbacks of weak equivalences along fibrations are weak equivalences.
This allows us to define the (locally small) homotopy category \( \mathrm{Ho}(\mathsf{sCat}) \). Moreover, because \( \mathsf{sCat} \) is cofibrantly generated, there is a projective model structure on the arrow category \( \mathsf{sCat}^{[1]} \) given by
- a morphism from \( \mathcal{W} _ 1 \rightarrow \mathcal{C} _ 1 \) to \( \mathcal{W} _ 2 \rightarrow \mathcal{C} _ 2 \) is a weak equivalence if and only if \( \mathcal{W} _ 1 \rightarrow \mathcal{W} _ 2 \) and \( \mathcal{C} _ 1 \rightarrow \mathcal{C} _ 2 \) both are,
- a morphism from \( \mathcal{W} _ 1 \rightarrow \mathcal{C} _ 1 \) to \( \mathcal{W} _ 2 \rightarrow \mathcal{C} _ 2 \) is a fibration if and only if \( \mathcal{W} _ 1 \rightarrow \mathcal{W} _ 2 \) and \( \mathcal{C} _ 1 \rightarrow \mathcal{C} _ 2 \) both are.
Therefore it again makes sense to consider the homotopy category \( \mathrm{Ho}(\mathsf{sCat}^{[1]}) \), which is equivalent to \( \mathrm{Ho}(\mathsf{sCat})^{[1]} \) as categories.
2. Localization of simplicial categories#
We have discussed localization of an ordinary category at a subcategory. We first note that this can be generalized to localizing not only at a subcategory but at arbitrary morphisms.
Definition 5. For \( \mathcal{W} \rightarrow \mathcal{C} \) a functor between categories, we define \( \mathcal{C}[\mathcal{W}^{-1}] = \mathcal{C}[(\mathrm{im} \mathcal{W})^{-1}] \).
Note that \( \mathrm{im} \mathcal{W} \) is not necessarily a subcategory of \( \mathcal{C} \), but we can still formally invert morphisms of \( \mathrm{im} \mathcal{W} \). This satisfy the same universal property that any functor \( \mathcal{C} \rightarrow \mathcal{D} \) whose composition \( \mathcal{W} \rightarrow \mathcal{C} \rightarrow \mathcal{D} \) sends all arrows to isomorphisms factors uniquely through \( \mathcal{C}[\mathcal{W}^{-1}] \). This property can be equivalently phrased as \[ \displaystyle \mathsf{Fun} _ {\mathsf{Cat}}(\mathcal{C}[\mathcal{W}^{-1}], \mathcal{D}) \cong \mathsf{Fun} _ {\mathsf{Cat}^{[1]}}((\mathcal{W} \rightarrow \mathcal{C}), (\mathrm{iso}(\mathcal{D}) \hookrightarrow \mathcal{D})), \] where \( \mathrm{iso}(\mathcal{D}) \) is the subcategory of \( \mathcal{D} \) consisting of all objects but only isomorphisms.
We can then define localization for simplicial categories dimension-wise. Given \( p \) and \( k \), the face/degeneracy maps (either \( d _ {p,k} \) or \( s _ {p,k} \)) \( \mathcal{C} _ p \rightarrow \mathcal{C} _ {p \pm 1} \) and \( \mathcal{W} _ p \rightarrow \mathcal{W} _ {p \pm 1} \) canonically induce a functor \( \mathcal{C} _ p[\mathcal{W} _ p^{-1}] \rightarrow \mathcal{C} _ {p \pm 1}[\mathcal{W} _ {p \pm 1}^{-1}] \).
Definition 6. For \( \mathcal{W} \rightarrow \mathcal{C} \) a map of simplicial categories, we define \( \mathcal{C}[\mathcal{W}^{-1}] \) as a simplicial category with \[ \displaystyle (\mathcal{C}[\mathcal{W}^{-1}]) _ p = \mathcal{C} _ p[\mathcal{W} _ p^{-1}], \] with face/degeneracy maps induced from the corresponding maps in \( \mathcal{C} \) and \( \mathcal{W} \). We denote \[ \displaystyle L : \mathsf{sCat}^{[1]} \rightarrow \mathsf{sCat}; \quad (\mathcal{W} \rightarrow \mathcal{C}) \mapsto \mathcal{C}[\mathcal{W}^{-1}]. \]
Thus we have a candidate for simplicial localization. However this definition is incorrect from a homotopy-theoretical point of view. Weak equivalences in \( \mathsf{sCat}^{[1]} \) are not always sent to weak equivalences in \( \mathsf{sCat} \).
A natural way to resolve this problem is to take the derived functor of our naïve dimension-wise simplicial localization. However, for technical reasons, we need to restrict our attention to the category of simplicial categories on a fixed object set \( O \).
Definition 7. Fix a (small) set \( O \). We consider the subcategory \( \mathsf{sCat} _ O \) of \( \mathsf{sCat} \) consisting of simplicial categories on the object set \( O \). Morphisms in \( \mathsf{sCat} _ O \) are functors that are \( \mathrm{id} _ O \) on the level of objects.
It turns out that the restriction of the three classes of morphisms—cofibrations, fibrations, weak equivalences—in \( \mathsf{sCat} \) to \( \mathsf{sCat} _ O \) is associated with a model structure.
Definition 8. We define simplicial localization \[ \displaystyle \mathcal{L} = \mathbb{L} L : \mathrm{Ho}(\mathsf{sCat}^{[1]} _ O) \rightarrow \mathrm{Ho}(\mathsf{sCat} _ O) \] as the left derived functor of \( L : \mathsf{sCat} _ O^{[1]} \rightarrow \mathsf{sCat} _ O \). (By definition, this is the right Kan extension of \( \mathsf{sCat}^{[1]} _ O \rightarrow \mathsf{sCat} _ O \rightarrow \mathrm{Ho}(\mathsf{sCat} _ O) \) along the obvious functor \( \mathsf{sCat} _ O^{[1]} \rightarrow \mathrm{Ho}(\mathsf{sCat} _ O^{[1]}) \), and it is not clear at this point that it exists at all.)
Recall that \( \mathcal{C}[\mathcal{W}^{-1}] \) had a right adjoint \( \mathcal{D} \mapsto (\mathcal{D}, \mathrm{iso}(\mathcal{D})) \) for categories. By the same argument we get an adjunction \[ \displaystyle \mathsf{Fun} _ {\mathsf{sCat} _ O}(\mathcal{C}[\mathcal{W}^{-1}], \mathcal{D}) \cong \mathsf{Fun} _ {\mathsf{sCat} _ O^{[1]}}((\mathcal{W} \rightarrow \mathcal{C}), (\mathrm{iso}(\mathcal{D}) \hookrightarrow \mathcal{D})) \] for simplicial categories as well. Here, \( \mathrm{iso}(\mathcal{D}) \) is the dimension-wise isomorphisms \( \mathrm{iso}(\mathcal{D}) _ p = \mathrm{iso}(\mathcal{D} _ p) \), which is indeed a sub-simplicial category because face/degeneracy maps are functors and hence send isomorphisms to isomorphisms. Let us denote \( I : \mathcal{D} \mapsto (\mathrm{iso}(\mathcal{D}) \rightarrow \mathcal{D}) \) so that \( L \dashv I \).
It would be nice if this adjunction is a Quillen adjunction, in which case we could easily compute \( \mathcal{L} = \mathbb{L}L \) by taking a cofibrant resolution. But the functor \( I \) does not preserve fibrations.
Example 1. Take \( \mathcal{D}, \mathcal{E} \) be one-object simplicial categories with \( \mathcal{D}(\ast, \ast) = \mathrm{Sing}(\mathbb{R} _ {\ge 0}^2) \) and \( \mathcal{E}(\ast, \ast) = \mathbb{R} \) with the monoid structure on the simplicial sets induced from the additive topological monoid structure on the spaces \( \mathbb{R} _ {\ge 0}^2 \) and \( \mathbb{R} \). Consider the functor \( F : \mathcal{D} \rightarrow \mathcal{E} \) induced by \( f : \mathbb{R} _ {\ge 0}^2 \rightarrow \mathbb{R} \) sending \( (x, y) \mapsto x-y \). Then \( F \) is a fibration \( f \) is a Serre fibration, but \( \mathrm{iso} \mathcal{D}(\ast, \ast) = \mathrm{Sing}(\{(0,0)\}) \) and \( \mathrm{iso} \mathcal{E}(\ast, \ast) = \mathrm{Sing}(\mathbb{R}) \) reveals that \( \mathrm{iso} \mathcal{D} \rightarrow \mathrm{iso} \mathcal{E} \) is not a fibration.
But to get a left derived functor, we don’t need the full power of a Quillen adjunction.
Definition 9. A left deformation is on a model category \( \mathcal{M} \) is a functor \( Q : \mathcal{M} \rightarrow \mathcal{M} \) with a natural transformation \( Q \rightarrow \mathrm{id} _ M \) such that \( QX \rightarrow X \) is an weak equivalence for all \( X \in M \).
Theorem 10 ([DHKS04]). Let \( F : \mathcal{M} \rightarrow \mathcal{N} \) be functors between model categories, and let \( Q \) be a left deformation on \( \mathcal{M} \). Suppose that \( F \) restricted to the full subcategory \( \{ QX \} \subseteq \mathcal{M} \) sends weak equivalences to weak equivalences. Then the left derived functor \( \mathbb{L}F \) exists and can be computed as \( \mathbb{L}F = F \circ Q \).
3. The free category#
The left deformation we use is the free category construction.
Definition 11. For a category \( \mathcal{C} \), we define its free category \( F\mathcal{C} \) as \[ \displaystyle F\mathcal{C}(X, Y) = \{ X \rightarrow X _ 1 \rightarrow \cdots \rightarrow X _ {n-1} \rightarrow Y : n \ge 0 \} \] where composition is formal concatenation.
Then there is an inclusion functor \( \iota : F \mathcal{C} \rightarrow F^2 \mathcal{C} \), and also a composition functor \( \mu : F\mathcal{C} \rightarrow C \).
Definition 12. We define \( F _ \bullet \mathcal{C} \in \mathsf{sCat} \) as \[ \displaystyle (F _ \bullet \mathcal{C}) _ p = F^{p+1} \mathcal{C}, \] where face maps and degeneracy maps are given as \[ \displaystyle d _ k : F^{p+1} \mathcal{C} \xrightarrow{F^k \mu F^{p-k}} F^p \mathcal{C}, \quad s _ k : F^{p+1} \mathcal{C} \xrightarrow{F^k \iota F^{p-k}} F^{p+2} \mathcal{C}. \]
Note that the object set of \( F _ \bullet \mathcal{C} \) is the same as the object set of \( \mathcal{C} \). Thus for \( \mathcal{C} \in \mathsf{sCat} _ O \), we have \( F _ \bullet \mathcal{C} \in \mathsf{sCat} _ O \). Concretely, \( p \)-simplices of \( F _ \bullet \mathcal{C} \) are described as \[ \displaystyle (F _ \bullet \mathcal{C}) _ p (X, Y) = \begin{Bmatrix} X \rightarrow X _ 1 \rightarrow \cdots \rightarrow X _ {n-1} \rightarrow X _ n = Y \text{ with a chain } \\ \{0, n\} \subseteq S _ {p} \subseteq \cdots \subseteq S _ 1 \subseteq S _ 0 = \{0, 1, \ldots, n\} \end{Bmatrix}, \] and face maps are removing one of the \( S _ i \)’s while degeneracy maps are inserting a same set.
Proposition 13. For \( \mathcal{C} \) any category considered as a simplicial category with \( F _ p \mathcal{C} = \mathcal{C} \), the natural map \( F _ \bullet \mathcal{C} \rightarrow \mathcal{C} \) induced by composition is a weak equivalence in \( \mathsf{sCat} \).
Proof. We show that \[ \displaystyle \mathcal{C}(X, Y) \hookrightarrow (F _ \bullet \mathcal{C})(X, Y), \quad (F _ \bullet \mathcal{C})(X, Y) \rightarrow \mathcal{C}(X, Y) \] are homotopy inverses to each other. (There is no canonical map \( \mathcal{C} \hookrightarrow F _ \bullet \mathcal{C} \) of simplicial categories but we still have this inclusion.) First it is clear that \( \mathcal{C}(X, Y) \rightarrow (F _ \bullet \mathcal{C})(X, Y) \rightarrow \mathcal{C}(X, Y) \) is the identity. We then see that the other composition is a strong deformation retract, because each \( p \)-simplex \( \{0, n\} \subseteq S _ p \cdots \subseteq S _ 0 \) can be considered as sitting inside the \( (p+1) \)-simplex \( \{0, n\} = S _ {p+1} \subseteq S _ p \subseteq \cdots \subseteq S _ 0 \), and then linear interpolation gives a strong deformation retract to \( S _ {p+1} \). ▨
Definition 14. For a simplicial category \( \mathcal{C} \in \mathsf{sCat} \), we define its free category as \[ \displaystyle (F _ \bullet \mathcal{C}) _ p = (F _ \bullet C _ p) _ p, \] which is the diagonal of the bisimplical category \( [p] \times [q] \mapsto (F _ \bullet \mathcal{C} _ p) _ q \).
Proposition 15. For \( \mathcal{C} \in \mathsf{sCat} \) any simplicial category, the natural map \( F _ \bullet \mathcal{C} \rightarrow \mathcal{C} \) induced by composition is a weak equivalence.
Proof. This follows from the fact that if \( K \rightarrow L \) is a map of bisimplicial sets such that \( K _ {i,\bullet} \rightarrow L _ {i,\bullet} \) is a weak equivalence of simplicial sets for all \( i \), then \( \mathrm{diag} K \rightarrow \mathrm{diag} L \) is a weak equivalence as well. ▨
The following lemma now justifies that we may use \( F _ \bullet \) as a left deformation.
Lemma 16 ([DK80], Lemma 6.2). Let \( \mathcal{U} _ 1 \rightarrow \mathcal{A} _ 1 \) and \( \mathcal{U} _ 2 \rightarrow \mathcal{A} _ 2 \) be maps of the form \( F _ \bullet \mathcal{W} \rightarrow F _ \bullet \mathcal{C} \) for \( \mathcal{W} \rightarrow \mathcal{C} \) a morphism in \( \mathsf{sCat} _ O \). If
Proof. This follows from Lemma 6.2 and Lemma 6.4 of [DK80]. First, \( \mathcal{A} _ j[\mathcal{U} _ j^{-1}] \) is weakly equivalent to \( F _ \bullet \mathcal{A} _ j[F _ \bullet \mathcal{U} _ j^{-1}] = F _ \bullet \mathcal{A} _ j[ F _ \bullet (\mathrm{im} \mathcal{U} _ j)^{-1}] = \mathcal{L}(\mathrm{im} \mathcal{U} _ j \hookrightarrow \mathcal{A} _ j) \). By Lemma 6.4, this is weakly equivalent to \( \mathcal{L}(\pi _ 0^{-1} (\mathrm{im} \pi _ 0 \mathcal{U} _ j) \hookrightarrow \mathcal{A} _ j) \). So the statement now follows from Lemma 6.2. ▨
Corollary 17. The left derived functor \( \mathcal{L} = \mathbb{L}L \) exists and can be computed as \( \mathcal{L}(\mathcal{W} \rightarrow \mathcal{C}) = F _ \bullet \mathcal{C}[F _ \bullet \mathcal{W}^{-1}] \).
4. Properties of simplicial localization#
Formally, because simplicial localization is a functor \( \mathrm{Ho}(\mathsf{sCat} _ O^{[1]}) \rightarrow \mathrm{Ho}(\mathsf{sCat} _ O) \), any morphism from \( \mathcal{W} _ 1 \rightarrow \mathcal{C} _ 1 \) to \( \mathcal{W} _ 2 \rightarrow \mathcal{C} _ 2 \) with \( \mathcal{W} _ 1 \rightarrow \mathcal{W} _ 2 \) and \( \mathcal{C} _ 1 \rightarrow \mathcal{C} _ 2 \) weak equivalences induces a weak equivalence \[ \displaystyle \mathcal{L}(\mathcal{W} _ 1 \rightarrow \mathcal{C} _ 1) \rightarrow \mathcal{L}(\mathcal{W} _ 2 \rightarrow \mathcal{C} _ 2). \] This is essentially Lemma 16. But we can say more than that. In the localization \( \mathcal{L}(\mathcal{W} \rightarrow \mathcal{C}) \), the weak equivalences \( \mathcal{W} \) only matter up to \( \pi _ 0 \)-data.
Proposition 18. The simplicial localization \( \mathcal{L}(\mathcal{W} \rightarrow \mathcal{C}) \) only depends on \( \mathcal{C} \) up to weak equivalence and \( \pi _ 0 \mathcal{C}[\pi _ 0 \mathcal{W}^{-1}] \). More precisely, if there is a commutative diagram
Proof. We have noted in the proof of Lemma 16 that \( \mathcal{L}(\mathcal{W} \rightarrow \mathcal{C}) \simeq F _ \bullet \mathcal{C} [F _ \bullet \mathcal{W}^{-1}] \) is weakly equivalent to \( \mathcal{L}(\pi _ 0^{-1} (\mathrm{im} \pi _ 0 F _ \bullet \mathcal{W}) \hookrightarrow F _ \bullet \mathcal{C}) \). By Lemma 6.4, this is determined only up to \( \pi _ 0 F _ \bullet \mathcal{C}[ \mathrm{im} \pi _ 0 F _ \bullet \mathcal{W}^{-1}] = \pi _ 0 \mathcal{C}[\pi _ 0 \mathcal{W}^{-1}] \). ▨
Finally, we remark that the classical localization can be recovered by taking \( \pi _ 0 \).
Proposition 19. If \( \mathcal{W} \rightarrow \mathcal{C} \) is a morphism of simplicial categories, then \[ \displaystyle \pi _ 0 \mathcal{L}(\mathcal{W} \rightarrow \mathcal{C}) \cong (\pi _ 0 \mathcal{C}) [ (\mathrm{im} \pi _ 0 \mathcal{W})^{-1}]. \]
Proof. According to our computation, \[ \displaystyle \pi _ 0 \mathcal{L}(\mathcal{W} \rightarrow \mathcal{C})(X, Y) = \mathcal{L}(\mathcal{W} \rightarrow \mathcal{C}) _ 0(X, Y) / \sim \] where \( \sim \) is the equivalence relation generated by \( 1 \)-simplices. But \[ \displaystyle \mathcal{L}(\mathcal{W} \rightarrow \mathcal{C}) _ 0 = F\mathcal{C} _ 0 [F\mathcal{W} _ 0^{-1}] = \{ X \rightarrow \leftarrow \rightarrow \cdots \leftarrow \rightarrow Y \} \] and the \( 1 \)-simplices allow us to identify (i) maps that are in the same connected component, (ii) a sequence of maps with its composite. ▨
References#
[Ber07] Julia E. Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc 359 (2007), no. 5, 2043–2058.
[DHKS04] William G. Dwyer, Philip S. Hirschorn, Daniel M. Kan, and Jeffery H. Smith, Homotopy limit functors on model categories and homotopical categories, Mathematical Surveys and Monographs, vol. 113, American Mathematical Society, Providence, RI, 2004.
[DK80] W. G. Dwyer and D. M. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980), no. 3, 267–284.