2.1 Homological grading conventions

In this article, we will work with chain complexes. We view cochain complexes as chain complexes by negating the grading. In particular, the Khovanov complex was originally defined as a cochain complex [Reference Khovanov31], but we will view it as a chain complex. So, our homological gradings differ from Khovanov’s by a sign.

2.2 Multicategories

Multicategories, which model the notion of multilinear maps, are a common generalisation of a category (a multicategory in which only multimorphism sets with one input are nonempty) and an operad (a multicategory with one object). Multicategories were introduced by Lambek [Reference Lambek37] and Boardman–Vogt [Reference Boardman and Vogt7]. In Boardman–Vogt’s work and most modern algebraic topology, the multimorphism sets in multicategories are equipped with actions of the symmetric group; the definition we have given would be called a nonsymmetric multicategory. Some of our multicategories (notably $\underline {\mathscr {B}}$ , $\mathsf {Sets}/X$ and $\mathscr {S}$ ) are, in fact, symmetric multicategories. In particular, the multicategories $\mathsf {Sets}/X$ to which we apply Elmendorf–Mandell’s K-theory are symmetric multicategories.

A monoidal category $(\mathscr {C},\otimes )$ produces a multicategory, which we will denote $\underline {\mathscr {C}}$ , as follows. The objects of $\underline {\mathscr {C}}$ are the same as the objects of $\mathscr {C}$ and the multimorphism sets are given by

$$ \begin{align*} \operatorname{\mathrm{Hom}}_{\underline{\mathscr{C}}}(x_1,\dots,x_n;y)=\operatorname{\mathrm{Hom}}_{\mathscr{C}}(x_1\otimes\dots\otimes x_n;y) \end{align*} $$

(for any choice of how to parenthesise the tensor product). If the monoidal category happened to be a symmetric monoidal category, as in the case of abelian groups $\mathsf {Ab}$ , graded abelian groups $\mathsf {Ab}_*$ or chain complexes $\mathsf {Kom}$ , then the corresponding multicategory is a symmetric multicategory. (These are examples of Hu–Kriz–Kriz’s $\star $ -categories [Reference Hu, Kriz and Kriz25].)

Many of our multicategories will be enriched in groupoids. That is, the multimorphism sets will be groupoids (i.e., categories in which all the morphisms are invertible) and the composition maps are maps of groupoids (i.e., functors).

Most of our nonenriched multicategories will be rather simple, in a sense we make precise.

Generalising Definition 2.2, we have the following variant.

2.3 Linear categories and multifunctors to abelian groups

Many of the algebras that we will encounter in this article will come equipped with an extra structure, which we abstract below.

The following three notions are equivalent.

1. (1) A ring A (algebra over $\mathbb {Z}$ ) equipped with an orthogonal set of idempotents X.

2. (2) A linear category (category enriched over abelian groups $\mathsf {Ab}$ ) with objects a finite set X.

3. (3) A multifunctor from the shape multicategory M of a finite set X to the multicategory $\underline {\mathsf {Ab}}$ of abelian groups.

(A similar statement holds for algebras over any ring R; the corresponding linear category has to be enriched over R-modules and the corresponding multifunctor should map to the multicategory of R-modules.)

To see the equivalence, given a multifunctor $F\colon M\to \underline {\mathsf {Ab}}$ there is a corresponding linear category with objects X, $\operatorname {\mathrm {Hom}}(x,y)=F((x,y))$ , composition $\operatorname {\mathrm {Hom}}(y,z)\otimes \operatorname {\mathrm {Hom}}(x,y)\to \operatorname {\mathrm {Hom}}(x,z)$ is the image of the unique morphism $(x,y),(y,z)\to (x,z)$ and the identity $\operatorname {\mathrm {Id}}_x\in \operatorname {\mathrm {Hom}}(x,x)$ is the image of $1$ under the maps $\mathbb {Z}\to \operatorname {\mathrm {Hom}}(x,x)$ , which is the image under F of the unique morphism $\varnothing \to (x,x)$ . Given a linear category $\mathscr {C}$ with finitely many objects, we can form a ring $A_{\mathscr {C}}=\bigoplus _{x,y\in \operatorname {\mathrm {Ob}}(\mathscr {C})}\operatorname {\mathrm {Hom}}_{\mathscr {C}}(x,y)$ with multiplication given by composition (i.e., $a\cdot b := b\circ a$ ) when defined and $0$ otherwise; the ring $A_{\mathscr {C}}$ is equipped with the orthogonal set of idempotents $\{\operatorname {\mathrm {Id}}_x\mid x\in \operatorname {\mathrm {Ob}}(\mathscr {C})\}$ . From a ring A equipped with an orthogonal set of idempotents I, we obtain a map $F\colon M\to \underline {\mathsf {Ab}}$ by setting $F((x,y))=xAy$ and declaring that F sends the unique morphism $(x,y),(y,z)\to (x,z)$ to the multiplication map $xAy\otimes yAz\to xAz$ and that F respects composition and identity maps.

In a similar fashion, given linear categories $\mathscr {C}$ and $\mathscr {D}$ with finitely many objects, the following are equivalent notions for bimodules:

1. (1) A left- $A_{\mathscr {C}}$ right- $A_{\mathscr {D}}$ bimodule B.

2. (2) An enriched functor $F_A\colon \mathscr {C}^{\mathrm {op}}\times \mathscr {D}\to \mathsf {Ab}$ ; an enriched functor between linear categories is one for which the map on morphisms $\operatorname {\mathrm {Hom}}_{\mathscr {C}^{\mathrm {op}}\times \mathscr {D}}((c,d),(c^{\prime },d^{\prime }))\to \operatorname {\mathrm {Hom}}_{\mathsf {Ab}}(F_A(c,d),F_A(c^{\prime },d^{\prime }))$ is linear or, equivalently, $\operatorname {\mathrm {Hom}}_{\mathscr {C}^{\mathrm {op}}\times \mathscr {D}}((c,d),(c^{\prime },d^{\prime }))\times F_A(c,d)\to F_A(c^{\prime },d^{\prime })$ is bilinear.

3. (3) A multifunctor from the shape multicategory $M(\mathscr {C},\mathscr {D})$ of $(\operatorname {\mathrm {Ob}}(\mathscr {C}),\operatorname {\mathrm {Ob}}(\mathscr {D}))$ to $\underline {\mathsf {Ab}}$ , which restricts to the multifunctors corresponding to $\mathscr {C}$ , respectively $\mathscr {D}$ , (as defined above) on the subcategory of $M(\mathscr {C},\mathscr {D})$ which is the shape multicategory of $\operatorname {\mathrm {Ob}}(\mathscr {C})$ , respectively $\operatorname {\mathrm {Ob}}(\mathscr {D})$ .

Recall from Definition 2.3 that $M(\mathscr {C},\mathscr {D})$ consists of the following data:

• • Three kinds of objects:

  1. – Pairs $(x_1,x_2)\in \operatorname {\mathrm {Ob}}(\mathscr {C})\times \operatorname {\mathrm {Ob}}(\mathscr {C})$ .

  2. – Pairs $(y_1,y_2)\in \operatorname {\mathrm {Ob}}(\mathscr {D})\times \operatorname {\mathrm {Ob}}(\mathscr {D})$ .

  3. – Pairs $(x,y)$ where $x\in \operatorname {\mathrm {Ob}}(\mathscr {C})$ and $y\in \operatorname {\mathrm {Ob}}(\mathscr {D})$ . For notational clarity, we will write $(x,y)$ instead as $(x,[B],y)$ .

• • A single multimorphism in each of the following cases:

  1. – $(x_1,x_2),(x_2,x_3),\dots ,(x_{m-1},x_m)\to (x_1,x_m)$ where $x_1,\dots ,x_m\in \operatorname {\mathrm {Ob}}(\mathscr {C})$ .

  2. – $(y_1,y_2),(y_2,y_3),\dots ,(y_{n-1},y_n)\to (y_1,y_n)$ where $y_1,\dots ,y_n\in \operatorname {\mathrm {Ob}}(\mathscr {D})$ .

  3. – $(x_1,x_2),\dots ,(x_{m-1},x_m),(x_m,[B],y_1),(y_1,y_2),\dots ,(y_{n-1},y_n)\to (x_1,[B],y_n)$ where $x_1,\dots ,x_m\in \operatorname {\mathrm {Ob}}(\mathscr {C})$ and $y_1,\dots ,y_n\in \operatorname {\mathrm {Ob}}(\mathscr {D})$ .

The bimodule B defines a multifunctor $F_B\colon M(\mathscr {C},\mathscr {D})\to \underline {\mathsf {Ab}}$ as follows:

• • On objects, for $x_1,x_2\in \operatorname {\mathrm {Ob}}(\mathscr {C})$ and $y_1,y_2\in \operatorname {\mathrm {Ob}}(\mathscr {D})$ , $F_B(x_1,x_2)=\operatorname {\mathrm {Hom}}_{\mathscr {C}}(x_1,x_2)=\operatorname {\mathrm {Id}}_{x_1}A_{\mathscr {C}}\operatorname {\mathrm {Id}}_{x_2}$ , $F_B(y_1,y_2)=\operatorname {\mathrm {Hom}}_{\mathscr {D}}(y_1,y_2)=\operatorname {\mathrm {Id}}_{y_1}A_{\mathscr {D}}\operatorname {\mathrm {Id}}_{y_2}$ and $F_B(x_1,[B],y_1)=\operatorname {\mathrm {Id}}_{x_1}B\operatorname {\mathrm {Id}}_{y_1}$ .

• • On the first and second types of multimorphisms, $F_B$ is simply composition. For the third type, the map $F_B$ sends the multimorphism

  $$ \begin{align*} (x_1,x_2),\dots,(x_{m-1},x_m),(x_m,[B],y_1),(y_1,y_2),\dots,(y_{n-1},y_n)\to (x_1,[B],y_n) \end{align*} $$
  to the product
  $$ \begin{align*} &\operatorname{\mathrm{Id}}_{x_1}R_{\mathscr{C}}\operatorname{\mathrm{Id}}_{x_2}\otimes\cdots\otimes \operatorname{\mathrm{Id}}_{x_{m-1}}R_{\mathscr{C}}\operatorname{\mathrm{Id}}_{x_m} \otimes \operatorname{\mathrm{Id}}_{x_m}B\operatorname{\mathrm{Id}}_{y_1}\otimes \operatorname{\mathrm{Id}}_{y_1}R_{\mathscr{D}}\operatorname{\mathrm{Id}}_{y_2}\\&\qquad \otimes\cdots\otimes \operatorname{\mathrm{Id}}_{y_{n-1}}R_{\mathscr{D}}\operatorname{\mathrm{Id}}_{y_n}\to \operatorname{\mathrm{Id}}_{x_1}B\operatorname{\mathrm{Id}}_{y_n}. \end{align*} $$

Conversely, every multifunctor $M(\mathscr {C},\mathscr {D})\to \underline {\mathsf {Ab}}$ of the given form arises as $F_B$ for the bimodule $B=\bigoplus _{x\in \operatorname {\mathrm {Ob}}(\mathscr {C}),y\in \operatorname {\mathrm {Ob}}(\mathscr {D})}F_B(x,[B],y)$ .

Similarly, given a multifunctor $F_B\colon M(\mathscr {C},\mathscr {D})\to \underline {\mathsf {Ab}}$ , we can construct an enriched functor $F_A\colon \mathscr {C}^{\mathrm {op}}\times \mathscr {D}\to \mathsf {Ab}$ as follows:

• • On objects, $F_A(c,d)=F_B(c,[B],d)$ .

• • On morphisms, $\operatorname {\mathrm {Hom}}_{\mathscr {C}^{\mathrm {op}}\times \mathscr {D}}((c,d),(c^{\prime },d^{\prime }))\otimes F_A(c,d)\to F_A(c^{\prime },d^{\prime })$ is the composition

  $$ \begin{align*} \operatorname{\mathrm{Hom}}_{\mathscr{C}^{\mathrm{op}}\times\mathscr{D}}((c,d),(c^{\prime},d^{\prime}))\otimes F_A(c,d) &=F_B(c^{\prime},c)\otimes F_B(c,[B],d)\otimes F_B(d,d^{\prime})\\&\quad \to F_B(c^{\prime},[B],d^{\prime}). \end{align*} $$

There are similar equivalences for the notions of differential $(A_{\mathscr {C}},A_{\mathscr {D}})$ -bimodules, enriched functors $\mathscr {C}^{\mathrm {op}}\times \mathscr {D}\to \mathsf {Kom}$ and multifunctors $M(\mathscr {C},\mathscr {D})\to \underline {\mathsf {Kom}}$ .

2.4 Trees and canonical groupoid enrichments

To define some enriched multicategories, we will first need some terminology about trees.

A planar, rooted tree is a tree with some number $n\geq 1$ of leaves, which has been embedded in $\mathbb {R}\times [0,1]$ so that $k\leq n-1$ of the leaves are embedded in $\mathbb {R}\times \{0\}$ , one leaf is embedded in $\mathbb {R}\times \{1\}$ and no other edges or vertices are mapped to $\mathbb {R}\times \{0,1\}$ . The vertices mapped to $\mathbb {R}\times \{0\}$ are called inputs of and the vertex mapped to $\mathbb {R}\times \{1\}$ is the output or root of . We call the remaining vertices of internal. We view planar, rooted trees as directed graphs, in which edges point away from the inputs and towards the output. In particular, given a valence m internal vertex p of , $(m-1)$ of the edges adjacent to p are input edges to p and one edge is the output edge of p and the inputs of p are ordered. We allow the case $m=1$ and call such $0$ -input $1$ -output internal vertices stump leaves. We view two planar, rooted trees as equivalent if there is an orientation-preserving self-homeomorphism of $\mathbb {R}\times [0,1]$ which preserves $\mathbb {R}\times \{0\}$ and $\mathbb {R}\times \{1\}$ and takes one tree to the other.

Given a tree , the collapse of is the result of collapsing all internal edges of , to obtain a tree with one internal vertex (i.e., a corolla).

2.4.1 Canonical groupoid enrichments

First, given a nonenriched multicategory $\mathscr {C}$ we can enrich $\mathscr {C}$ over groupoids trivially as follows. Given elements $f,g\in \operatorname {\mathrm {Hom}}_{\mathscr {C}}(x_1,\dots ,x_n;y)$ , define $\operatorname {\mathrm {Hom}}(f,g)$ to be empty if $f\neq g$ and to consist of a single element, the identity map, if $f=g$ .

Next we give a different way of enriching multicategories over groupoids, which provides a tool for turning lax multifunctors into strict ones (from a different source), though we will avoid ever actually defining or using the notion of a lax multifunctor or multicategory. Suppose $\mathscr {C}$ is an unenriched multicategory. The canonical thickening $\widetilde {\mathscr {C}}$ is the multicategory enriched in groupoids defined as follows. The objects of $\widetilde {\mathscr {C}}$ are the same as the objects of $\mathscr {C}$ . Informally, an object in $\operatorname {\mathrm {Hom}}_{\widetilde {\mathscr {C}}}(x_1,\dots ,x_n;y)$ is a sequence of composable multimorphisms starting at $x_1,\dots ,x_n$ and ending at y. The $2$ -morphisms record whether two sequences compose to the same multimorphism.

More precisely, an object of $\operatorname {\mathrm {Hom}}_{\widetilde {\mathscr {C}}}(x_1,\dots ,x_n;y)$ is a tree with n inputs, together with a labelling of each edge of by an object of $\mathscr {C}$ and each internal vertex of by a multimorphism of $\mathscr {C}$ , subject to the following conditions:

1. (1) The input edges of are labelled by $x_1,\dots ,x_n$ (in that order).

2. (2) The output edge of is labelled by y.

3. (3) At a vertex v, if the input edges to v are labelled $w_1,\dots ,w_k$ and the output edge is labelled z, then the vertex v is labelled by an element of $\operatorname {\mathrm {Hom}}_{\mathscr {C}}(w_1,\dots ,w_k;z)$ . In particular, stump leaves of are labelled by multimorphisms in $\operatorname {\mathrm {Hom}}_{\mathscr {C}}(\varnothing ;z)$ ; that is, by $0$ -input multimorphisms.

Given a morphism $f\in \operatorname {\mathrm {Hom}}_{\widetilde {\mathscr {C}}}(x_1,\dots ,x_n;y)$ , we can compose the multimorphisms labelling the vertices according to the tree to obtain a morphism $f^\circ \in \operatorname {\mathrm {Hom}}_{\mathscr {C}}(x_1,\dots ,x_n;y)$ . Given morphisms $f,g\in \operatorname {\mathrm {Hom}}_{\widetilde {\mathscr {C}}}(x_1,\dots ,x_n;y)$ , define $ \operatorname {\mathrm {Hom}}_{\widetilde {\mathscr {C}}}(f,g) $ to have one element if $f^\circ =g^\circ $ and to be empty otherwise. The unit in $\operatorname {\mathrm {Hom}}_{\widetilde {\mathscr {C}}}(x;x)$ is the tree with one input, one output, no internal vertices and edge labelled x. This completes the definition of the multimorphism groupoids in $\widetilde {\mathscr {C}}$ .

Composition of multimorphisms is simply gluing of trees. (When gluing together the external vertices, they disappear rather than creating new internal vertices.)

Lemma 2.5. This definition of composition extends uniquely to morphisms in the multimorphism groupoids.

Proof. This is immediate from the definitions.

Lemma 2.6. These definitions make $\widetilde {\mathscr {C}}$ into a multicategory enriched in groupoids.

Proof. At the level of objects of the multimorphism groupoids, associativity follows from associativity of composition of trees. At the level of morphisms of the multimorphism groupoids, associativity trivially holds. The unit axioms follow from the fact that gluing on a tree with no internal vertices has no effect.

We will call a multimorphism in $\widetilde {\mathscr {C}}$ basic if the underlying tree has only one internal vertex. Every object in the multimorphism groupoid $\operatorname {\mathrm {Hom}}_{\widetilde {\mathscr {C}}}(x_1,\dots ,x_n;y)$ is a composition of basic multimorphisms.

Example 2.7. Consider the canonical groupoid enrichment of the shape multicategory of some set X (cf. Definition 2.2). For any $x,y,z,w\in X$ , the multimorphism set $\operatorname {\mathrm {Hom}}((x,y),(y,z),(z,w);(x,w))$ consists of infinitely many elements since the underlying tree could contain an arbitrary number of internal vertices. However, there is exactly one multimorphism when the underlying tree has exactly one internal vertex, exactly 10 when the underlying tree has exactly two interval vertices (shown in Figure 2.1) and so on.

Figure 2.1 Some of the multimorphisms in $\operatorname {\mathrm {Hom}}((x,y),(y,z),(z,w);(x,w))$ from Example 2.7. The edges are labelled by the objects and the internal vertices are labelled by multimorphisms in the original multicategory (i.e., basic multimorphisms). Edges ending in a node are stumps. The original multicategory being the shape multicategory of a set, the vertex labels and the internal edge labels are forced and are not shown.

There is a canonical projection multifunctor $\widetilde {\mathscr {C}}\to \mathscr {C}$ which is the identity on objects and composes the multimorphisms associated to the vertices of a tree according to the edges. (Here, we view $\mathscr {C}$ as trivially enriched in groupoids.)

Lemma 2.8. The projection map $\widetilde {\mathscr {C}}\to \mathscr {C}$ is a weak equivalence.

(See [Reference Elmendorf and Mandell19, Definition 12.1] for the definition of a weak equivalence.)

Proof. We must check that projection induces an equivalence on the categories of components and that for each $x_1,\dots ,x_n,y$ the projection map gives a weak equivalence of simplicial nerves

(2.1) $$ \begin{align} \mathscr{N}\operatorname{\mathrm{Hom}}_{\widetilde{\mathscr{C}}}(x_1,\dots,x_n;y)\to \mathscr{N}\operatorname{\mathrm{Hom}}_{\mathscr{C}}(x_1,\dots,x_n;y). \end{align} $$

The first statement follows from the fact that the components of the groupoid $\operatorname {\mathrm {Hom}}_{\widetilde {\mathscr {C}}}(x_1,\dots ,x_n;y)$ correspond, under the projection, to the elements of $\operatorname {\mathrm {Hom}}_{{\mathscr {C}}} (x_1,\dots ,x_n;y)$ . The second statement follows from the fact that in each component of the multimorphism groupoid $\operatorname {\mathrm {Hom}}_{\widetilde {\mathscr {C}}}(x_1,\dots ,x_n;y)$ , every object is initial (and terminal), so $\mathscr {N}\operatorname {\mathrm {Hom}}_{\widetilde {\mathscr {C}}}(x_1,\dots ,x_n;y)$ is contractible.

A related construction is strictification.

Definition 2.9. Given a multicategory $\mathscr {C}$ enriched in groupoids there is a strictification $\mathscr {C}^0$ of $\mathscr {C}$ , which is an ordinary multicategory, with objects $\operatorname {\mathrm {Ob}}(\mathscr {C}^0)=\operatorname {\mathrm {Ob}}(\mathscr {C})$ and multimorphism sets $\operatorname {\mathrm {Hom}}_{\mathscr {C}^0}(x_1,\dots ,x_n;y)$ the set of isomorphism classes (path components) in the groupoid $\operatorname {\mathrm {Hom}}_{\mathscr {C}}(x_1,\dots ,x_n;y)$ . If we view $\mathscr {C}^0$ as trivially enriched in groupoids, then there is a projection multifunctor $\mathscr {C}\to \mathscr {C}^0$ .

Strictification is a left inverse to thickening; that is, for any nonenriched multicategory $\mathscr {C}$ ,

$$ \begin{align*} ({\widetilde{\mathscr{C}}})^0\cong\mathscr{C}. \end{align*} $$

A more general notion than a multicategory enriched in groupoids is a simplicial multicategory; that is, a multicategory enriched in simplicial sets. Given a multicategory enriched in groupoids $\mathscr {C}$ , replacing each $\operatorname {\mathrm {Hom}}$ groupoid $\operatorname {\mathrm {Hom}}_{\mathscr {C}}(x,y)$ by its nerve gives a simplicial multicategory. One can also strictify a simplicial multicategory $\mathscr {D}$ by replacing each $\operatorname {\mathrm {Hom}}$ simplicial set by its set of path components. If $\mathscr {D}$ came from a multicategory $\mathscr {C}$ enriched in groupoids by taking nerves, then the strictification $\mathscr {C}^0$ of $\mathscr {C}$ and the strictification $\mathscr {D}^0$ of $\mathscr {D}$ are naturally equivalent. Our main reason for introducing simplicial multicategories is that some of the background results we use are stated in that more general language. For instance, spectra form a simplicial multicategory.

2.5 Homotopy colimits

In this section we will discuss homotopy colimits in the categories of simplicial sets and chain complexes.

Given an index category I and a functor F from I to the category $\mathsf {SSet}_*$ of based simplicial sets, there is a based homotopy colimit denoted by $\operatorname {\mathrm {hocolim}}_I F$ : it is a quotient of the space

$$ \begin{align*} \coprod_{p \geq 0} \coprod_{i_0 \to i_1 \to \dots \to i_p} F(i_0) \wedge (\Delta^p)_+ \end{align*} $$

by an equivalence relation induced by simplicial face and degeneracy operations [Reference Bousfield and Kan9, XII.2]. Similarly, if instead we are given a functor F from I to the category $\mathsf {Kom}$ of complexes, there is a homotopy colimit $\operatorname {\mathrm {hocolim}}_I F$ (denoted $\coprod _* F$ in [Reference Bousfield and Kan9]): it is a quotient of the complex

$$ \begin{align*} \bigoplus_{p \geq 0} \bigoplus_{i_0 \to i_1 \to \dots \to i_p} F(i_0) \otimes C_*(\Delta^p), \end{align*} $$

where $C_*$ is the normalised chain functor on simplicial sets. (More explicit chain-level descriptions can be given.) In particular, the natural commutative and associative Eilenberg–Zilber shuffle pairing $\widetilde C_*(X) \otimes \widetilde C_*(Y) \to \widetilde C_*(X \wedge Y)$ , applied to the above constructions, gives rise to a natural transformation $\operatorname {\mathrm {hocolim}} (\widetilde C_* \circ F) \to \widetilde C_* (\operatorname {\mathrm {hocolim}} F)$ .

In the following, we use the shorthand equivalence to denote both a weak equivalence of simplicial sets and a quasi-isomorphism of chain complexes.

In particular, these combine to give a natural quasi-isomorphism

$$ \begin{align*} (\operatorname{\mathrm{hocolim}}_I F) \otimes (\operatorname{\mathrm{hocolim}}_J G) \to \operatorname{\mathrm{hocolim}}_{I \times J} (F \otimes G) \end{align*} $$

which is compatible with associativity (but not commutativity) of the tensor product.

Homotopy colimits in the category $\mathsf {Kom}$ are closely related to left derived functors. In the following, we view $\mathsf {Ab}$ as a subcategory of $\mathsf {Kom}$ , given by the chain complexes concentrated in degree zero.

Proposition 2.12 [Reference Bousfield and Kan9, XII.5.6]

Suppose $\Delta $ denotes the category of finite ordinals and order-preserving maps and $A\colon \Delta ^{\mathrm {op}} \to \mathsf {Kom}$ represents a simplicial chain complex $A_{\bullet }$ . Then the chain complex $\operatorname {\mathrm {hocolim}}_{\Delta ^{\mathrm {op}}} A$ is quasi-isomorphic to the total complex of the double complex

$$ \begin{align*} \cdots \to A_2 \to A_1 \to A_0 \to 0, \end{align*} $$

where the ‘horizontal’ boundary maps are given by the standard alternating sum of the face maps of $A_{\bullet }$ .

Proposition 2.14. Suppose I is a category and we have a natural transformation $\phi \colon F \to G$ of functors $I \to \mathsf {Kom}$ . Let P denote the category $\{* \leftarrow 0 \rightarrow 1\}$ and define a functor $C\phi \colon P \times I \to \mathsf {Kom}$ on objects by

$$ \begin{align*} C\phi(x,y) = \begin{cases} 0 &\text{if }x = *,\\ F(y) &\text{if }x = 0,\\ G(y) &\text{if }x = 1 \end{cases} \end{align*} $$

with morphisms determined by F, G and $\phi $ . Then the chain complex $\operatorname {\mathrm {hocolim}}_{P\times I} (C\phi )$ is quasi-isomorphic to the standard mapping cone of the map of chain complexes $\operatorname {\mathrm {hocolim}}_I F \to \operatorname {\mathrm {hocolim}}_I G$ induced by $\phi $ .

Using the previous two propositions to iterate a mapping cone construction gives the following result for cube-shaped diagrams.

Corollary 2.15. Let P denote the category $\{* \leftarrow 0 \to 1\}$ and $\underline {2}^{}$ denote the subcategory $\{0 \to 1\}$ . Given a functor $F\colon \underline {2}^{n} \to \mathsf {Ab}$ , its totalisation is defined to be the chain complex

(2.2) $$ \begin{align} \bigoplus_{\substack{v\in\underline{2}^{n}\\|v|=0}}F(v)\to\bigoplus_{\substack{v\in\underline{2}^{n}\\|v|=1}}F(v)\to \dots\to\bigoplus_{\substack{v\in\underline{2}^{n}\\|v|=n}}F(v), \end{align} $$

graded so that $\bigoplus _{\substack {v\in \underline {2}^{n}\\|v|=i}}F(v)$ is in grading $n-i$ (where $|v|$ denotes the number of $1$ s in v), and the differential counts the sum of the edge maps of F with standard signs. Let $\widetilde F\colon P^n\to \mathsf {Ab}$ be the extended functor given by

$$ \begin{align*} \widetilde F(v) = \begin{cases} F(v)&\text{if }v \in \underline{2}^{n},\\ 0&\text{otherwise.} \end{cases} \end{align*} $$

Then the complex $\operatorname {\mathrm {hocolim}}_{P^n} \tilde F$ is quasi-isomorphic to the totalisation of F.

2.6 Classical spectra

In this section we will review some of the models for the category of spectra and some of the properties we will need.

For us, a classical spectrum X (sometimes called a sequential spectrum) is a sequence of based simplicial sets $X_n$ , together with structure maps $\sigma _n\colon X_n \wedge S^1 \to X_{n+1}$ . A map $X \to Y$ is a sequence of based maps $f_n\colon X_n \to Y_n$ such that the diagrams

(2.3)

all commute. The structure maps produce natural homomorphisms on homotopy groups $\pi _k(X_n) \to \pi _{k+1}(X_{n+1})$ and (reduced) homology groups $\widetilde H_k(X_n) \to \widetilde H_{k+1}(X_{n+1})$ , allowing us to define homotopy and homology groups

$$ \begin{align*} \pi_k(X) &= \operatorname{\mathrm{colim}}_n \pi_{k+n} X_n & H_k(X) &= \operatorname{\mathrm{colim}}_n \widetilde H_{k+n} X_n \end{align*} $$

for all $k \in \mathbb {Z}$ that are functorial in X. A map of classical spectra $X \to Y$ is defined to be a weak equivalence if it induces an isomorphism $\pi _* X \to \pi _* Y$ and the stable homotopy category is obtained from the category of classical spectra by inverting the weak equivalences. The functors $\pi _*$ and $H_*$ both factor through the stable homotopy category. (This description is due to Bousfield and Friedlander [Reference Bousfield and Friedlander8], and they show that it gives a stable homotopy category equivalent to the one defined by Adams [Reference Adams1]. It has the advantage that maps of spectra are easier to describe but the disadvantage that maps $X \to Y$ in the stable homotopy category are not defined as homotopy classes of maps $X \to Y$ .)

Classical spectra X and Y have a handicrafted smash product given by

$$ \begin{align*} (X \wedge Y)_n = \begin{cases} X_k \wedge Y_k &\text{if }n=2k,\\ (X_k \wedge Y_k)\wedge S^1 &\text{if }n=2k+1. \end{cases} \end{align*} $$

The structure map $(X \wedge Y)_n \wedge S^1 \to (X \wedge Y)_{n+1}$ is the canonical isomorphism when n is even and is obtained from the structure maps of X and Y when n is odd. This smash product is not associative or unital, but it induces a smash product functor that makes the stable homotopy category symmetric monoidal. There is a Künneth formula for homology: there is a multiplication pairing $H_p(X) \otimes H_q(Y) \to H_{p+q}(X \wedge Y)$ that is part of a natural exact sequence

$$ \begin{align*} 0 \to \bigoplus_{p+q=n} H_p(X) \otimes H_q(Y) \to H_n(X \wedge Y) \to \bigoplus_{p+q=n-1} \operatorname{\mathrm{Tor}}_1^{\mathbb{Z}}(H_p(X), H_q(Y)) \to 0 \end{align*} $$

that can be obtained by applying colimits to the ordinary Künneth formula. In particular, this multiplication pairing is an isomorphism if the groups $H_*(X)$ or $H_*(Y)$ are all flat over $\mathbb {Z}$ .

Given a functor F from I to the category of classical spectra, there is a homotopy colimit $\operatorname {\mathrm {hocolim}}_I F$ obtained by applying homotopy colimits levelwise. Homotopy colimits preserve weak equivalences and the handicrafted smash product preserves homotopy colimits in each variable. There is also a derived functor spectral sequence

$$ \begin{align*} \mathbb{L}_p \operatorname{\mathrm{colim}}_I(H_q \circ F) \Rightarrow H_{p+q}(\operatorname{\mathrm{hocolim}}_I F) \end{align*} $$

for calculating the homology of a homotopy colimit. (In fact, this spectral sequence exists for stable homotopy groups $\pi _*$ as well.)

The Hurewicz theorem for spaces translates into a Hurewicz theorem for spectra:

This induces a homology Whitehead theorem.

Spectra have suspensions and desuspensions.

Definition 2.19. For a spectrum X, there are suspension and loop functors, as well as formal shift functors, as follows:

$$ \begin{align*} (S^1 \wedge X)_n &= S^1 \wedge (X_n) & (\Omega X)_n &= \Omega(X_n)\\ \operatorname{\mathrm{sh}}(X)_n &= X_{n+1} & \operatorname{\mathrm{sh}}^{-1}(X)_n &= \begin{cases} X_{n-1}&\text{if }n> 0\\ \ast&\text{if }n=0 \end{cases} \end{align*} $$

Proposition 2.20. The pairs $(S^1 \wedge (-), \Omega )$ and $(\operatorname {\mathrm {sh}}^{-1}, \operatorname {\mathrm {sh}})$ are adjoint pairs and all unit and counit maps are weak equivalences.

In the stable homotopy category, there are isomorphisms

$$ \begin{align*} S^1 \wedge X &\cong \operatorname{\mathrm{sh}}(X) & \Omega X &\cong \operatorname{\mathrm{sh}}^{-1}(X) \end{align*} $$

In particular, the suspension functor and desuspension (i.e., loop) functor are inverse to each other.

Although it may look like there are natural maps $S^1 \wedge X \to \operatorname {\mathrm {sh}}(X)$ and $\Omega X\to \operatorname {\mathrm {sh}}^{-1}(X)$ that implement these equivalences, there are not: the apparent maps do not make Diagram (2.3) commute.

2.7 Symmetric spectra

Many of our constructions make use of Elmendorf–Mandell’s paper [Reference Elmendorf and Mandell19], which uses Hovey–Shipley–Smith’s more structured category of symmetric spectra [Reference Hovey, Shipley and Smith24]. In this section we review some details about symmetric spectra and their relationship to classical spectra.

A symmetric spectrum (which, in this article, we may simply call a spectrum) is a sequence of based simplicial sets $X_n$ , together with actions of the symmetric group $\mathfrak {S}_n$ on $X_n$ and structure maps $\sigma _n\colon X_n \wedge S^1 \to X_{n+1}$ . These are required to satisfy the following additional constraint. For any n and m, the iterated structure map

$$ \begin{align*} X_n \wedge S^m \cong X_n \wedge (S^1 \wedge S^1 \wedge \dots \wedge S^1) \to X_{n+m} \end{align*} $$

has actions of $\mathfrak {S}_n \times \mathfrak {S}_m$ on the source and target: via the actions on the two factors for the source and via the standard inclusion $\mathfrak {S}_n \times \mathfrak {S}_m \to \mathfrak {S}_{n+m}$ in the target. The structure maps are required to intertwine these two actions. A map of symmetric spectra consists of a sequence of based, $\mathfrak {S}_n$ -equivariant maps $f_n\colon X_n \to Y_n$ commuting with the structure maps. We write $\mathscr {S}$ for the category of symmetric spectra.

A symmetric spectrum can also be described as the following equivalent data. To a finite set S, a symmetric spectrum assigns a simplicial set $X(S)$ and this is functorial in isomorphisms of finite sets. To a pair of finite sets S and T, there is a structure map $X(S) \wedge \left (\bigwedge _{t \in T}S^1\right ) \to X(S \coprod T)$ , and this is compatible with isomorphisms in S and T as well as satisfying an associativity axiom in T. We recover the original definition by setting $X_n = X(\{1,2,\dots ,n\})$ .

Symmetric spectra also have a more rigid monoidal structure $\wedge $ , characterised by the property that a map $X \wedge Y \to Z$ is equivalent to a natural family of maps $X(S) \wedge Y(T) \to Z(S \coprod T)$ compatible with the structure maps in both variables. This makes the category of symmetric spectra symmetric monoidal closed.

Again, the constructions of homotopy colimits are compatible enough that they extend to symmetric spectra. Given a functor F from I to the category of symmetric spectra, there is a homotopy colimit $\operatorname {\mathrm {hocolim}}_I F$ obtained by applying homotopy colimits levelwise. Homotopy colimits preserve weak equivalences. The smash product also behaves well with respect to homotopy colimits, as follows.

The category of symmetric spectra has an internal notion of weak equivalence and a homotopy category of symmetric spectra. Both symmetric spectra and classical spectra have model structures [Reference Hovey, Shipley and Smith24, Reference Bousfield and Friedlander8] and we have the following results.

Note that the forgetful functor U does not preserve weak equivalences except between certain symmetric spectra, the so-called semistable ones [Reference Hovey, Shipley and Smith24, Section 5.6]. Any fibrant symmetric spectrum is semistable, and any symmetric spectrum is weakly equivalent to a semistable one.

These results allow us to define homotopy and homology groups for a symmetric spectrum X as a composite: take the image of X in the homotopy category of symmetric spectra; apply the (right) derived functor of U to get an element in the homotopy category of classical spectra and then apply homotopy or homology groups. The homology groups of symmetric spectra therefore inherit the following properties from classical spectra.

Proposition 2.27. For symmetric spectra X and Y, there is a natural Künneth exact sequence

$$ \begin{align*} 0 \to \bigoplus_{p+q=n} H_p(X) \otimes H_q(Y) \to H_n(X \wedge Y) \to \bigoplus_{p+q=n-1} \operatorname{\mathrm{Tor}}_1^{\mathbb{Z}}(H_p(X), H_q(Y)) \to 0. \end{align*} $$

Proposition 2.28. For a diagram $F\colon I \to \mathscr {S}$ of symmetric spectra, there is a convergent derived functor spectral sequence

$$ \begin{align*} \mathbb{L}_p \operatorname{\mathrm{colim}}_I(H_q \circ F) \Rightarrow H_{p+q}(\operatorname{\mathrm{hocolim}}_I F). \end{align*} $$

It will be convenient for us to have a lift of these homology groups to a chain functor. Let L denote the reduced chain complex $\widetilde C_*(S^1)$ of the simplicial set $S^1$ . This is a complex with value $\mathbb {Z}$ in degree $1$ and zero elsewhere. For complexes C and D, let $\underline {\mathrm {Hom}}(C,D)$ be the function complex.

Definition 2.29. Fix a symmetric spectrum X. For an inclusion of finite sets $T \subset U$ , there is a natural map

$$ \begin{align*} \underline{\mathrm{Hom}}(L^{\otimes T}, \widetilde C_*X(T)) \stackrel{\sim}{\longrightarrow} \underline{\mathrm{Hom}}(L^{\otimes T} \otimes L^{\otimes U\setminus T}, \widetilde C_* X(T) \otimes L^{\otimes U \setminus T}) \to \underline{\mathrm{Hom}}(L^{\otimes U}, \widetilde C_*X(U)). \end{align*} $$

Now, given any set S (infinite or not), these maps make the complexes $\underline {\mathrm {Hom}}(L^{\otimes T}, \widetilde C_*X(T))$ into a directed system indexed by finite subsets $T\subset S$ . Define the chain complex

$$ \begin{align*} \widehat C_k(X)_S = \operatorname{\mathrm{colim}}_{T \subset S\text{ finite}} \underline{\mathrm{Hom}}(L^{\otimes T}, \widetilde C_kX(T)). \end{align*} $$

If S is finite of size n, $\widehat C_k(X)_S$ is isomorphic to the shift $\widetilde {C}_k X_n[-n]$ . More generally, these structure maps naturally make the system of chain groups and homology groups $\{H_{n+k}(X_n)\}$ into a functor from the category of finite sets and injections to the category of abelian groups (i.e., an $FI$ -module in the language of [Reference Church, Ellenberg and Farb13]).

There is a natural pairing

$$ \begin{align*} \widehat C_*(X)_S \otimes \widehat C_*(Y)_T \to \widehat C_*(X \wedge Y)_{S \coprod T}. \end{align*} $$

The construction of $\widehat C_*$ is also natural in injections $S \to S^{\prime }$ .

Definition 2.30. Let $\mathcal {M}$ be the category whose objects are the sets $\coprod ^k \mathbb {N}$ for $k \geq 1$ and whose morphisms are monomorphisms of sets. For a symmetric spectrum X, we define

$$ \begin{align*} C_*(X) = \operatorname{\mathrm{hocolim}}_{S \in \mathcal{M}} (\widehat C_*(X)_{S}). \end{align*} $$

Let M be the monoid of monomorphisms $\mathbb {N} \to \mathbb {N}$ . Since all objects in the category $\mathcal {M}$ are isomorphic to $\mathbb {N}$ , this homotopy colimit is quasi-isomorphic to the homotopy colimit over this one-object subcategory, which can be re-expressed as the derived tensor product $\mathbb {Z} \otimes ^{\mathbb {L}}_{\mathbb {Z}[M]} \widehat C_*(X)_{\mathbb {N}}$ . See [Reference Schwede53] and [Reference Schwede54, Exercise E.II.13] for a discussion of this functor.

Therefore, if $\underline {\mathscr {S}}$ denotes the associated multicategory of symmetric spectra, $C_*$ induces a multifunctor $\underline {\mathscr {S}}\to \underline {\mathsf {Kom}}$ . To a multimorphism in symmetric spectra realised by a map $X_1\wedge \cdots \wedge X_n\to Y$ , $C_*$ associates the chain map $C_*(X_1)\otimes \dots \otimes C_*(X_n)\to C_*(X_1\wedge \cdots \wedge X_n)\to C_*(Y)$ . This definition of $C_*$ respects multicomposition. (The multifunctor $C_*$ is not compatible with the symmetries interchanging factors, if we regard $\underline {\mathscr {S}}$ and $\underline {\mathsf {Kom}}$ as symmetric multicategories.)

If we defined homotopy and homology groups

$$ \begin{align*} \widehat\pi_k(X) &= \operatorname{\mathrm{colim}} \pi_{k+n}(X_n) & \widehat H_k(X) &= \operatorname{\mathrm{colim}} H_{k+n}(X_n) \end{align*} $$

using the same formula as for classical spectra, we obtain ‘naïve’ homotopy and homology groups of a symmetric spectrum X which are not preserved under weak equivalence. If we tensor with the sign representation of $\mathfrak {S}_n$ and take $\operatorname {\mathrm {colim}} H_{n+k}(X_n) \otimes sgn$ , the result is isomorphic to $\widehat H_k(X)_{\mathbb {N}}$ with its action of the monoid M of injections $\mathbb {N} \to \mathbb {N}$ [Reference Schwede53]. The natural map $\widehat H_k(X) \to H_k(X)$ to the true homology groups factors through the quotient by M. A similar action and factorisation hold relating the naïve homotopy groups $\widehat \pi _k(X)$ to the true homotopy groups $\pi _k(X)$ .

A similar warning holds for homotopy colimits. If F is a diagram of symmetric spectra, it is not the case that $U (\operatorname {\mathrm {hocolim}} F) \simeq \operatorname {\mathrm {hocolim}}(U \circ F)$ unless F is a diagram of semistable symmetric spectra. However, it is always possible to replace F with a weakly equivalent diagram $F^{\prime }$ of semistable symmetric spectra so that $\operatorname {\mathrm {hocolim}} F \simeq \operatorname {\mathrm {hocolim}} F^{\prime }$ and then $U (\operatorname {\mathrm {hocolim}} F^{\prime }) \simeq \operatorname {\mathrm {hocolim}}(U \circ F^{\prime })$ .

Symmetric spectra have suspension and desuspension (i.e., loop) functors.

Definition 2.32. For a symmetric spectrum X, there are suspension and loop functors, as well as formal shift functors, as follows:

$$ \begin{align*} (S^1 \wedge X)_n &= S^1 \wedge (X_n) & (\Omega X)_n &= \Omega(X_n)\\ \operatorname{\mathrm{sh}}(X)_n &= X_{1+n} & \operatorname{\mathrm{sh}}^{-1}(X)_n &= \begin{cases} (\mathfrak{S}_{1+m})_+ \wedge_{\mathfrak{S}_m} X_m&\text{if }n = 1+m\\ \ast&\text{if }n=0 \end{cases} \end{align*} $$

The notation $1+n$ in the shift functor $sh$ indicates that the $\mathfrak {S}_n$ -action on $X_{1+n}$ is via the inclusion $\mathfrak {S}_1 \times \mathfrak {S}_{n} \to \mathfrak {S}_{1+n}$ .

For example, the map $S^1 \wedge X \to \operatorname {\mathrm {sh}}(X)$ is the composite

$$ \begin{align*} S^1 \wedge X_n \cong X_n \wedge S^1 \to X_{n+1} \stackrel{\sigma}{\to} X_{1+n}, \end{align*} $$

where the final map $\sigma $ is a block permutation in $\mathfrak {S}_{n+1}$ : this is necessary to ensure that this commutes with the structure maps.

Proposition 2.34. The suspension functor $S^1 \wedge (-)$ and the formal shift functors preserve homotopy colimits. They also preserve smash products: there are natural isomorphisms

$$ \begin{align*} \operatorname{\mathrm{sh}}(X) \wedge Y &\to \operatorname{\mathrm{sh}}(X \wedge Y) & X \wedge \operatorname{\mathrm{sh}}(Y) &\to X \wedge \operatorname{\mathrm{sh}}(Y)\\ (S^1 \wedge X) \wedge Y &\to S^1 \wedge (X \wedge Y) & X \wedge (S^1 \wedge Y)&\to S^1 \wedge (X \wedge Y). \end{align*} $$

As with chain complexes, order matters in these identities. For example, the two isomorphisms for $(S^1 \wedge X) \wedge (S^1 \wedge Y)$ do not commute with each other but differ by a transposition of $(S^1 \wedge S^1)$ ; the two isomorphisms of $(\operatorname {\mathrm {sh}} X) \wedge (\operatorname {\mathrm {sh}} Y)$ with $\operatorname {\mathrm {sh}}(\operatorname {\mathrm {sh}}(X \wedge Y))$ differ by a transposition in $\mathfrak {S}_{2+n}$ .

In more standard notation, this implies that $C_*(\operatorname {\mathrm {sh}}(X)) \cong C_*(X)[1]$ and $C_*(\operatorname {\mathrm {sh}}^{-1}(X)) \cong C_*(X)[-1]$ . The isomorphism for $\operatorname {\mathrm {sh}}(X)$ is true before taking homotopy colimits for $\mathcal {M}$ , but the isomorphism for $\operatorname {\mathrm {sh}}^{-1}$ is not.

2.8 The Elmendorf–Mandell machine

A permutative category is a category $\mathscr {C}$ together with a $0$ -object, a strictly associative operation $\oplus \colon \mathscr {C}\times \mathscr {C}\to \mathscr {C}$ and a natural isomorphism $\gamma \colon a\oplus b\to b\oplus a$ satisfying certain coherence conditions (see [Reference Elmendorf and Mandell19, Definition 3.1]). An example is the category $\mathsf {Sets}/X$ of finite sets over X, with:

• • objects pairs $(Y,f\colon Y\to X)$ of a finite set Y and a map from Y to X,

• • morphisms $\operatorname {\mathrm {Hom}}((Y,f),(Z,g))=\{h\colon Y\to Z\mid f=g\circ h\}$ ,

• • zero object the pair $(\varnothing , \iota )$ (where $\iota $ is the unique map $\varnothing \to X$ ) and

• • sum $\oplus $ given by disjoint union.

The category $\mathsf {Sets}/X$ can be made small by requiring that all sets Y are elements of some chosen large set. For instance, the objects of $\mathsf {Sets}/X$ could be pairs $(n,S)$ where $n\in \mathbb {N}$ and S is a finite subset of $\mathbb {R}^n$ mapping to X. Moreover, the disjoint union operation may be made strictly associative by declaring objects to be finite sequences of such pairs $(n_1,S_1),\dots ,(n_k,S_k)$ (which morally represents their disjoint union), the actual sum $\oplus $ given by concatenation of sequences and the morphisms from $(n_1,S_1),\dots ,(n_k,S_k)$ to $(m_1,T_1),\dots ,(m_\ell ,T_\ell )$ are given by maps $\amalg _{i=1}^k S_i\to \amalg _{i=1}^\ell T_i$ (for any standard definition of disjoint union), respecting the maps to X. We will elide these points, but see [Reference Isbell27] for a more detailed account.

Given a finite correspondence $A\colon X\to Y$ ; that is, a finite set A and a map $(\pi _X,\pi _Y)\colon A\to X\times Y$ , there is a corresponding functor of permutative categories

$$ \begin{align*} F_A&\colon \mathsf{Sets}/X\to \mathsf{Sets}/Y\\ F_A&(Z,f)=(A\times_X Z,\pi_Y)=\bigl(\{(a,z)\in A\times Z\mid \pi_X(a)=f(z)\},(a,z)\mapsto \pi_Y(a)\bigr). \end{align*} $$

The collection of all (small) permutative categories forms a simplicial multicategory $\mathsf {Permu}$ [Reference Elmendorf and Mandell19, Definition 3.2]. The category $\underline {\mathscr {S}}$ of symmetric spectra also forms a simplicial multicategory, and Elmendorf–Mandell construct an enriched multifunctor, K-theory,

$$ \begin{align*} K\colon \mathsf{Permu}\to\underline{\mathscr{S}}. \end{align*} $$

Their functor K takes the category $\mathsf {Sets}/X$ to $\bigvee _{x\in X}\mathbb {S}$ , a wedge of copies of the sphere spectrum. Further, given a correspondence A from X to Y, the induced map $K(A)\colon K(X)\to K(Y)$ sends $\mathbb {S}_x$ to $\mathbb {S}_y$ (for $x\in X$ , $y\in Y$ ) by a map of degree $\#\bigl (\pi _X^{-1}(x)\cap \pi _Y^{-1}(y)\bigr )$ . (This special case can be understood concretely using the Pontrjagin–Thom construction; see, for example, [Reference Lawson, Lipshitz and Sarkar40, Section 5].)

We note that K is invariant under equivalence in the following sense. Because K respects the enrichments of $\mathsf {Permu}$ and $\underline {\mathscr {S}}$ in simplicial sets, it takes natural isomorphisms between functors of permutative categories to homotopies between maps of K-theory spectra. Therefore, equivalent permutative categories give homotopy equivalent answers.

This concludes our general introduction to Elmendorf–Mandell’s K-theory machine. In the rest of this section, we discuss a precise sense in which multifunctors from different multicategories can be equivalent. This will be used in Subsection 4.1 to replace multifunctors from floppy multicategories (enriched in groupoids) with multifunctors from more rigid (unenriched) multicategories.

Definition 2.36 cf. [Reference Lurie45, 2.0.0.1]

Suppose I is a multicategory. The associated monoidal category $I^\otimes $ is the category defined as follows. An object of $I^\otimes $ is a (possibly empty) tuple $(i_1,\dots ,i_n)$ of objects of I. The maps $(i_1,\dots ,i_n) \to (j_1,\dots ,j_m)$ are given by

$$ \begin{align*} \coprod_{\alpha\colon \{1,\dots,n\} \to \{1,\dots,m\}} \prod_{k=1}^m\operatorname{\mathrm{Hom}}_I(\alpha^{-1}(j_k); j_k). \end{align*} $$

The monoidal structure on $I^\otimes $ is given by concatenation of tuples, with unit given by the empty tuple.

Definition 2.37. Given multicategories I and J and multifunctors $f\colon I\to J$ and $G\colon I\to \underline {\mathscr {S}}$ there is a map $f_*G\colon J\to \underline {\mathscr {S}}$ , the left Kan extension of G, defined on objects by

(2.4) $$ \begin{align} (f_*G)(j)=\operatorname{\mathrm{colim}}_{[(f(i_1),\dots,f(i_n)\to j]\in I^\otimes\downarrow j}G(i_1)\wedge \cdots\wedge G(i_n). \end{align} $$

(Here $I^\otimes \downarrow j$ denotes the overcategory of j.) Left Kan extension is functorial in G; that is, gives a functor of diagram categories $f_*\colon \underline {\mathscr {S}}^I\to \underline {\mathscr {S}}^J$ .

There is also a restriction map $f^*\colon \underline {\mathscr {S}}^J\to \underline {\mathscr {S}}^I$ and $f_*$ is left adjoint to $f^*$ .

Following Elmendorf–Mandell [Reference Elmendorf and Mandell19, Definition 12.1], a map $f\colon \mathcal {M}\to \mathcal {N}$ between simplicial multicategories is a (weak) equivalence if the induced map on the strictifications $f^0\colon \mathcal {M}^0\to \mathcal {N}^0$ is an equivalence of (ordinary) categories and for any $x_1,\dots ,x_n,y\in \operatorname {\mathrm {Ob}}(\mathcal {M})$ , the map $\operatorname {\mathrm {Hom}}_{\mathcal {M}}(x_1,\dots ,x_n;y)\to \operatorname {\mathrm {Hom}}_{\mathcal {N}}(f(x_1),\dots ,f(x_n);f(y))$ is a weak equivalence of simplicial sets.

A key technical result of Elmendorf–Mandell’s is the following:

Theorem 2.38. [Reference Elmendorf and Mandell19, Theorems 1.3 and 1.4]

Let $\mathcal {M}$ be a simplicial multicategory. Then the functor categories $\underline {\mathscr {S}}^{\mathcal {M}}$ and $\underline {\mathscr {S}}^{\mathcal {M}^0}$ are simplicial model categories with weak equivalences (respectively fibrations) the maps which are objectwise weak equivalences (respectively fibrations).

Further, suppose $\mathcal {N}$ is another simplicial multicategory and

$$ \begin{align*} f\colon \mathcal{M}\to \mathcal{N} \end{align*} $$

is an equivalence. Then there are Quillen equivalences

where $f_*$ is left Kan extension and $f^*$ is restriction.

For instance, in Theorem 2.38, $\mathcal {M}$ might be (the nerve of) a multicategory enriched in groupoids whose every component is contractible and $\mathcal {N}$ might be (the nerve of) its strictification $\mathcal {M}^0$ .

We will need some additional cofibrancy for the rectification results we apply (see Subsection 2.9). In particular, Elmendorf–Mandell also showed that $\underline {\mathscr {S}}^{\mathcal {M}}$ is cofibrantly generated [Reference Elmendorf and Mandell19, Section 11] and it is combinatorial in the sense of [Reference Lurie44, Definition A.2.6.1]. Using a small object argument, Chorny [Reference Chorny12] constructed functorial cofibrant factorisations that apply, in particular, to combinatorial model categories such as $\underline {\mathscr {S}}$ . So, his construction gives a cofibrant replacement functor

$$ \begin{align*} Q^{\mathcal{M}}\colon \underline{\mathscr{S}}^{\mathcal{M}}\to\underline{\mathscr{S}}^{\mathcal{M}}. \end{align*} $$

His construction satisfies the following property.

Proposition 2.39. Suppose $j\colon \mathcal {N}\hookrightarrow \mathcal {M}$ is a full subcategory such that $\operatorname {\mathrm {Hom}}(m_1,\dots , m_k;n)=\varnothing $ if $n\in \mathcal {N}$ and $m_i\not \in \mathcal {N}$ for some i. (That is, there are no arrows into $\mathcal {N}$ ; we call such a full subcategory $\mathcal {N}$ blockaded.) Then the small object argument is preserved by restriction: there is a natural isomorphism

$$ \begin{align*} j^*Q^{\mathcal{M}}\stackrel{\simeq}{\longrightarrow}Q^{\mathcal{N}}j^*. \end{align*} $$

We note that various operations preserve cofibrancy.

Proof. The functor $ev_x\colon \underline {\mathscr {S}}^{\mathcal {M}} \to \underline {\mathscr {S}}$ , given by $F \mapsto F(x)$ , has a right adjoint given by right Kan extension. Given a symmetric spectrum X, the value of this right Kan extension on an object y is

$$ \begin{align*} \prod_{n \geq 0} X^{\operatorname{\mathrm{Hom}}_{\mathcal{M}}(x,x,\dots,x;y)}. \end{align*} $$

In particular, any fibration $X \to Y$ becomes a fibration on applying right Kan extension. Therefore, $ev_x$ is a left Quillen functor and so preserves cofibrations and cofibrant objects.

2.9 Rectification

In the process of defining the arc algebras and tangle invariants, we will construct a number of cobordisms which are not equal but are canonically isotopic. The lax nature of the construction will be encoded by defining multifunctors from multicategories in which the $\operatorname {\mathrm {Hom}}$ sets are groupoids in which each component is contractible: the objects in the groupoids are mapped to the cobordisms and the morphisms in the groupoids are mapped to the isotopies, and contractibility of the groupoids encodes the fact that these isotopies are canonical. We then use the Khovanov–Burnside functor and the Elmendorf–Mandell machine to produce functors from these multicategories to spectra. At that point, we want to collapse the enriched multicategories to ordinary multicategories, to obtain simpler invariants. This collapsing is called rectification and is accomplished as follows.

Definition 2.42. Let $\mathcal {M}$ be a simplicial multicategory (e.g., the nerve of a multicategory enriched in groupoids), $\mathcal {M}^0$ the strictified (discrete) multicategory and $f\colon \mathcal {M}\to \mathcal {M}^0$ the projection. Given a functor $G\colon \mathcal {M}\to \underline {\mathscr {S}}$ , the rectification of G is the composite

$$ \begin{align*} f_*Q^{\mathcal{M}} G\colon \mathcal{M}^0\to\underline{\mathscr{S}}. \end{align*} $$

Proof. By definition of cofibrant replacement, the natural transformation $Q^{\mathcal {M}} G \to G$ is an equivalence of diagrams: for every object in $x \in \mathcal {M}$ the map $(Q^{\mathcal {M}} G)(x) \to G(x)$ is an equivalence. Thus, it suffices to show that the unit map from $Q^{\mathcal {M}} G$ to $f^* f_* Q^{\mathcal {M}} G$ is an equivalence.

By Theorem 2.38, the adjoint pair $f^*$ and $f_*$ forms a Quillen equivalence. This implies that for any fibrant replacement $f_* Q^{\mathcal {M}} G \to (f_* Q^{\mathcal {M}} G)_{\mathit {fib}}$ in $\underline {\mathscr {S}}^{\mathcal {M}^0}$ , the composite

$$ \begin{align*} Q^{\mathcal{M}} G \to f^* f_* Q^{\mathcal{M}} G \to f^* (f_* Q^{\mathcal{M}} G)_{\mathit{fib}} \end{align*} $$

is an equivalence. For every object $x \in \mathcal {M}$ the composite

$$ \begin{align*} (Q^{\mathcal{M}} G)(x) \to (f_* Q^{\mathcal{M}} G)(f(x)) \to (f_* Q^{\mathcal{M}} G)_{\mathit{fib}}(f(x)) \end{align*} $$

is therefore an equivalence. However, by definition of fibrant replacement, the map

$$ \begin{align*}(f_* Q^{\mathcal{M}} G)(y) \to (f_* Q^{\mathcal{M}} G)_{\mathit{fib}}(y)\end{align*} $$

is an equivalence for any $y \in \mathcal {M}^0$ and hence $Q^{\mathcal {M}}(G) \to f^* f_* Q^{\mathcal {M}} G$ is also an equivalence by the 2-out-of-3 property.

Lemma 2.44. Suppose that $j\colon \mathcal {N}\hookrightarrow \mathcal {M}$ is a blockaded subcategory and let $j^0\colon \mathcal {N}^0\to \mathcal {M}^0$ denote the strictification. For any functor $G\colon \mathcal {M}\to \underline {\mathscr {S}}$ , there is a natural isomorphism of rectifications

$$ \begin{align*} f^{\mathcal{N}}_*Q^{\mathcal{N}} j^*G\cong (j^0)^*f^{\mathcal{M}}_*Q^{\mathcal{M}} G. \end{align*} $$

Proof. There is a natural transformation $f^{\mathcal {N}}_*j^*G \to (j^0)^*f^{\mathcal {M}}_*G$ , the mate. Note that if $K\subset I$ is blockaded and $j\in K$ , then the colimit in Equation (2.4) only sees the objects of K. Thus, the mate is a natural isomorphism

$$ \begin{align*} f^{\mathcal{N}}_*j^*G \cong (j^0)^*f^{\mathcal{M}}_*G \end{align*} $$

(i.e., satisfies the Beck–Chevalley condition). So, the result follows from Proposition 2.39.

2.10 Khovanov invariants of tangles

See Figure 2.2 for examples of the actions of elements in $\operatorname {\mathrm {Diff}}^1$ and $\operatorname {\mathrm {Diff}}^2$ .

Figure 2.2 The actions of $\operatorname {\mathrm {Diff}}^1$ and $\operatorname {\mathrm {Diff}}^2$ . Left: two flat $(4,2)$ -tangles related by the action of $\operatorname {\mathrm {Diff}}^1\times \operatorname {\mathrm {Id}}$ and two flat $(0,0)$ -tangles related by the action of $\operatorname {\mathrm {Diff}}^1\times \operatorname {\mathrm {Id}}$ . Right: two flat $(0,0)$ -tangle cobordisms related by the action of $\operatorname {\mathrm {Diff}}^2\times \operatorname {\mathrm {Id}}$ .

By the $2n$ standard points in $(0,1)$ we mean $[2n]_{\mathrm {std}}=\{1/(2n+1),\dots ,2n/(2n+1)\}$ . A flat $(2m,2n)$ -tangle is an embedded cobordism in $[0,1]\times (0,1)$ from $\{0\}\times [2m]_{\mathrm {std}}$ to $\{1\}\times [2n]_{\mathrm {std}}$ . More generally, a $(2m,2n)$ -tangle is an embedded cobordism in $\mathbb {R}\times [0,1]\times (0,1)$ from $\{0\}\times \{0\}\times [2m]_{\mathrm {std}}$ to $\{0\}\times \{1\}\times [2n]_{\mathrm {std}}$ . We call flat tangles T and $T^{\prime }$ equivalent if there is a $\phi \in \operatorname {\mathrm {Diff}}^1$ so that $T^{\prime }=(\phi \times \operatorname {\mathrm {Id}}_{(0,1)})(T)$ . Similarly, tangles T and $T^{\prime }$ are equivalent if there is a $\phi \in \operatorname {\mathrm {Diff}}^1$ so that $T^{\prime }=(\operatorname {\mathrm {Id}}_{\mathbb {R}}\times \phi \times \operatorname {\mathrm {Id}}_{(0,1)})(T)$ .

Khovanov [Reference Khovanov32] associated an algebra $H^n$ to each integer n, an $(H^m,H^n)$ -bimodule $\mathcal {C}_{\mathit {Kh}}(T)$ to a flat $(2m,2n)$ -tangle T and, more generally, a chain complex of $(H^m,H^n)$ -bimodules to any $(2m,2n)$ -tangle. We will review Khovanov’s construction briefly. Because we reserve $H^n$ for singular cohomology, we will use the notation $\mathcal {H}^{n}$ for Khovanov’s algebra $H^n$ .

The constructions start from Khovanov’s Frobenius algebra $V=H^*(S^2)=\mathbb {Z}[X]/(X^2)$ with comultiplication $1\mapsto 1\otimes X + X\otimes 1,X\mapsto X\otimes X$ and counit $1\mapsto 0,X\mapsto 1$ .

Let ${}_{m}\widehat {\mathsf {B}}_n$ denote the collection of flat $(2m,2n)$ -tangles. Composition of flat tangles, followed by scaling $[0,2]\times (0,1)\to [0,1]\times (0,1)$ , is a map ${}_{m}\widehat {\mathsf {B}}_n\times {}_{n}\widehat {\mathsf {B}}_p\to {}_{m}\widehat {\mathsf {B}}_p$ , which we will write $(a,b)\mapsto ab$ . (This map is associative and has strict identities because we quotiented by $\operatorname {\mathrm {Diff}}^1$ .) Reflection is a map ${}_{m}\widehat {\mathsf {B}}_n\to {}_{n}\widehat {\mathsf {B}}_m$ , which we will write $a\mapsto \overline {a}$ .

The isotopy classes of ${}_{0}\widehat {\mathsf {B}}_n$ with no closed components are called crossingless matchings. For each crossingless matching a, we choose a namesake representative $a\subset [0,1]\times (0,1)$ in ${}_{0}\widehat {\mathsf {B}}_n$ so that the projection $a\to [0,1]$ to the x-coordinate is Morse with exactly n critical points with distinct critical values; therefore, we may view the set of crossingless matchings, ${\mathsf {B}}_{n}$ , as a subset of ${}_{0}\widehat {\mathsf {B}}_n$ .

Given a collection of disjoint, embedded circles Z in the plane, let $V(Z)=\bigotimes _{C\in \pi _0(Z)}V$ . As a $\mathbb {Z}$ -module, the ring $\mathcal {H}^{n}$ is given by

$$ \begin{align*} \mathcal{H}^{n}=\bigoplus_{a,b\in {\mathsf{B}}_{n}}V(a\overline{b}). \end{align*} $$

The product on $\mathcal {H}^{n}$ satisfies $xy=0$ if $x\in V(a\overline {b})$ and $y\in V(c\overline {d})$ with $b\neq c$ . To define the product $V(a\overline {b})\otimes V(b\overline {c})\to V(a\overline {c})$ , consider the representative $b\subset [0,1]\times (0,1)$ and let $\mu _1,\dots ,\mu _n$ be the critical points of the projection $b\to [0,1]$ , ordered according to the critical values. Define a sequence of $(2n,2n)$ -tangles $\gamma _i$ , $i=0,\dots , n$ , inductively by setting $\gamma _0=\overline {b}b$ and obtaining $\gamma _{i+1}$ by performing embedded surgery on $\gamma _i$ along an arc connecting $\overline {\mu _{i+1}}$ and $\mu _{i+1}$ . (See Figure 2.3.) Observe that $\gamma _n$ is canonically isotopic to the identity tangle on $2n$ strands. The Frobenius structure on V induces a map $V(a\gamma _i\overline {c})\to V(a\gamma _{i+1}\overline {c})$ ; define the product $V(a\overline {b})\otimes V(b\overline {c})\to V(a\overline {c})$ to be the composition

$$ \begin{align*} V(a\overline{b})\otimes V(b\overline{c})\cong V(a\gamma_0\overline{c})\to V(a\gamma_1\overline{c})\to\cdots\to V(a\gamma_n\overline{c})\cong V(a\overline{c}). \end{align*} $$

Figure 2.3 Flat tangles and the multiplication on $\mathcal {H}^{n}$ .

Given a flat $(2m,2n)$ -tangle $T\in {}_{m}\widehat {\mathsf {B}}_n$ , the bimodule $\mathcal {C}_{\mathit {Kh}}(T)$ is given additively by

$$ \begin{align*} \mathcal{C}_{\mathit{Kh}}(T)=\bigoplus_{(a,b)\in{\mathsf{B}}_{m}\times{\mathsf{B}}_{n}}V(aT\overline{b}). \end{align*} $$

The left action of $\mathcal {H}^{m}$ (respectively, the right action of $\mathcal {H}^{n}$ ) is defined similarly to the multiplication on $\mathcal {H}^{n}$ : multiplication sends $V(a\overline {b})\otimes V(cT\overline {d})$ to $0$ unless $b=c$ (respectively, sends $V(cT\overline {d})\otimes V(e\overline {f})$ to $0$ unless $d=e$ ) and the product $V(a\overline {b})\otimes V(bT\overline {c})\to V(aT\overline {c})$ (respectively, $V(bT\overline {c})\otimes V(c\overline {d})\to V(bT\overline {d})$ ) is defined by a sequence of merge and split maps, turning the tangle $\overline {b}b$ (respectively, $\overline {c}c$ ) into the identity tangle.

Now let ${}_{m}\mathsf {C}_n$ denote the collection of all $(2m,2n)$ -tangles in $\mathbb {R}\times [0,1]\times (0,1)$ , with each component oriented. Call such a tangle generic if its projection to $[0,1]\times (0,1)$ has no cusps, tangencies or triple points. A tangle diagram is a generic tangle along with a total ordering of its crossings (double points of the projection to $[0,1]\times (0,1)$ ). Let ${}_{m}\mathsf {D}_n$ be the set of all $(2m,2n)$ -tangle diagrams. (Forgetting the ordering of the crossings, followed by an inclusion, gives a map ${}_{m}\mathsf {D}_n\to {}_{m}\mathsf {C}_n$ .)

Given a $(2m,2n)$ -tangle diagram $T\in {}_{m}\mathsf {D}_n$ with N (totally ordered) crossings and any crossingless matchings $a\in {\mathsf {B}}_{m}$ and $b\in {\mathsf {B}}_{n}$ , there is a corresponding link $aT\overline {b}$ , which has an associated Khovanov complex $\mathcal {C}_{\mathit {Kh}}(aT\overline {b})$ . Additively, $\mathcal {C}_{\mathit {Kh}}(aT\overline {b})$ is a direct sum over the complete resolutions $T_v$ , $v\in \{0,1\}^N$ , of $V(aT_v\overline {b})$ . (Our conventions for resolutions are shown in Figure 2.4.) Thus,

$$ \begin{align*} \mathcal{C}_{\mathit{Kh}}(T):= \bigoplus_{\substack{(a,b)\in {\mathsf{B}}_{m}\times{\mathsf{B}}_{n}\\v\in\{0,1\}^N}}V(aT_v\overline{b}) \end{align*} $$

inherits the structure of a chain complex, as a direct sum over the a and b of $\mathcal {C}_{\mathit {Kh}}(aT\overline {b})$ and of a bimodule over $\mathcal {H}^{m}$ , and $\mathcal {H}^{n}$ , as a direct sum over v of $\mathcal {C}_{\mathit {Kh}}(T_v)$ .

Figure 2.4 Resolutions of a crossing.

These chain complexes of bimodules have the following TQFT property.

Proof Sketch of proof

Suppose $T_1$ has $N_1$ crossings and $T_2$ has $N_2$ crossings. Then the isomorphism

$$ \begin{align*} \mathcal{C}_{\mathit{Kh}}(T_1)\otimes_{\mathcal{H}^{n}}\mathcal{C}_{\mathit{Kh}}(T_2)\stackrel{\cong}{\longrightarrow} \mathcal{C}_{\mathit{Kh}}(T_1T_2) \end{align*} $$

identifies the summand of $\mathcal {C}_{\mathit {Kh}}(T_1)\otimes _{\mathcal {H}^{n}}\mathcal {C}_{\mathit {Kh}}(T_2)$ over the vertices $v\in \{0,1\}^{N_1}$ and $w\in \{0,1\}^{N_2}$ with the summand of $\mathcal {C}_{\mathit {Kh}}(T_1T_2)$ over $(v,w)\in \{0,1\}^{N_1+N_2}$ . For these flat tangles $T_{1,v}$ , $T_{2,w}$ and $(T_1T_2)_{(v,w)}$ , the gluing map

$$ \begin{align*} \mathcal{C}_{\mathit{Kh}}(T_{1,v})\otimes_{\mathcal{H}^{n}}\mathcal{C}_{\mathit{Kh}}(T_{2,w})\to \mathcal{C}_{\mathit{Kh}}((T_1T_2)_{(v,w)}) \end{align*} $$

is induced by the multisaddle cobordism (cf. Subsection 3.3) map

$$ \begin{align*} \mathcal{C}_{\mathit{Kh}}(aT_{1,v}\overline{b})\otimes_{\mathbb{Z}}\mathcal{C}_{\mathit{Kh}}(bT_{2,w}\overline{c})\to \mathcal{C}_{\mathit{Kh}}(a(T_1T_2)_{(v,w)}\overline{c}) \end{align*} $$

[Reference Khovanov32, Theorem 1].

For comparison with our constructions later, note that each of the $1$ -manifolds $a\overline {b}$ in the construction of $\mathcal {H}^{n}$ lies in $(0,1)^2\subset [0,1]\times (0,1)$ , and so does each of the $1$ -manifolds $aT\overline {b}$ in the construction of $\mathcal {C}_{\mathit {Kh}}(T)$ for a flat tangle T. There is a disjoint union operation on embedded $1$ -manifolds in $(0,1)^2$ induced by the map

$$ \begin{align*} (0,1)^2\amalg (0,1)^2\to(0,1)^2 \end{align*} $$

which identifies the first copy of $(0,1)^2$ with $(0,1/2)\times (0,1)$ and the second copy of $(0,1)^2$ with $(1/2,1)\times (0,1)$ , by affine transformations. Since we have quotiented by the action of $\operatorname {\mathrm {Diff}}^1$ on the first $(0,1)$ -factor, this disjoint union operation is strictly associative. Further, we can view the maps inducing the multiplication on $\mathcal {H}^{n}$ , the actions on $\mathcal {C}_{\mathit {Kh}}(T)$ and the differential on $\mathcal {C}_{\mathit {Kh}}(T)$ when T is nonflat as induced by cobordisms embedded in $[0,1]\times (0,1)^2$ . For instance, the multiplication $V(a\overline {b})\otimes V(b\overline {c})\to V(a\overline {c})$ is induced by a cobordism in $[0,1]\times (0,1)^2$ from $\{0\}\times (a\overline {b}\amalg b\overline {c})$ to $\{1\}\times (a\overline {c})$ . For this section, only the abstract (not embedded) cobordisms are relevant, but for the stable homotopy refinement we will need the embedded cobordisms.

2.11 The Khovanov–Burnside 2-functor

Definition 2.56. Informally, the Burnside category $\mathscr {B}$ is the bicategory with objects finite sets X, $\operatorname {\mathrm {Hom}}(X,Y)$ the class of finite correspondences $A\colon X\to Y$ ; that is, diagrams of sets

and $\operatorname {\mathrm {2Hom}}(A,B)$ the set of isomorphisms of correspondences from A to B; that is, commutative diagrams

Composition of correspondences is fibre product: given $A\colon X\to Y$ and $B\colon Y\to Z$ , $B\circ A=A\times _YB$ . Note that one can think of a correspondence $A\colon X\to Y$ as a $(Y\times X)$ -matrix of sets; that is, for each $(y,x)\in Y\times X$ a set $A_{y,x}=s^{-1}(x)\cap t^{-1}(y)$ . Composition of correspondences then corresponds to multiplication of matrices, using the Cartesian product and disjoint union to multiply and add sets.

Note that, with this definition, composition is not strictly associative since $(A\times _Y B)\times _Z C$ is in canonical bijection with, but not equal to, $A\times _Y(B\times _Z C)$ . Composition also lacks strict identities since $A\times _X X$ is in canonical bijection with, but not equal to, A. There are many ways to rectify this; here is one.

Instead of correspondences, let $\operatorname {\mathrm {Hom}}(X,Y)$ denote the set of pairs of an integer n and a $(Y\times X)$ -matrix $(A_{y,x})_{x\in X,\ y\in Y}$ of finite subsets $A_{y,x}$ of $\mathbb {R}^n$ , with the following property:

1. (D) $A_{y,x}\cap A_{y^{\prime },x}=\varnothing $ if $y\neq y^{\prime }$ and $A_{y,x}\cap A_{y,x^{\prime }}=\varnothing $ if $x\neq x^{\prime }$ .

(A $(Y\times X)$ -matrix of subsets of $\mathbb {R}^n$ is a function $Y\times X\to 2^{\mathbb {R}^n}$ .) Given subsets $A\subset \mathbb {R}^n$ and $B\subset \mathbb {R}^m$ , $A\times B$ is a subset of $\mathbb {R}^{n+m}$ . Composition is defined by

$$ \begin{align*} (A_{z,y})_{y\in Y,\ z\in Z}\circ (A_{y,x})_{x\in X,\ y\in Y}=\left(\bigcup_{y\in Y}A_{z,y}\times A_{y,x}\right)_{x\in X,\ z\in Z}. \end{align*} $$

The condition that $A_{y,x}\cap A_{y^{\prime },x}=\varnothing $ whenever $y\neq y^{\prime }$ implies that the sets in the union are disjoint. Given $x\neq x^{\prime }$ , $(A_{z,y}\times A_{y,x})\cap (A_{z,y^{\prime }}\times A_{y^{\prime },x^{\prime }})$ is empty unless $y=y^{\prime }$ (by looking at the first factor) and thus is empty unless $x=x^{\prime }$ (by looking at the second factor). Similarly, $(A_{z,y}\times A_{y,x})\cap (A_{z^{\prime },y^{\prime }}\times A_{y^{\prime },x})=\varnothing $ if $z\neq z^{\prime }$ . Thus, the composition has Property (D). Composition is clearly strictly associative. The (strict) identity element of X is the $(X\times X$ )-diagonal matrix with diagonal entries the $1$ -element subset of $\mathbb {R}^0$ . A $2$ -morphism of correspondences $\phi \colon (A_{y,x})_{x\in X,\ y\in Y}\to (B_{y,x})_{x\in X,\ y\in Y}$ is a collection of bijections $(\phi _{y,x}\colon A_{y,x}\stackrel {\cong }{\longrightarrow } B_{y,x})_{x\in X,\ y\in Y}$ ; note that $2$ -morphisms ignore the embedding information.

Throughout, when we talk about the Burnside category we mean this latter, strict version of the category. Typically, however, the embedding data can be chosen arbitrarily and in those cases we will not specify it.

The free abelian group construction gives a functor $\mathscr {B}\to \mathsf {Ab}$ , by

$$ \begin{align*} \operatorname{\mathrm{Ob}}(\mathscr{B})\ni X&\mapsto \bigoplus_{x\in X}\mathbb{Z}\\ (A_{y,x})_{x\in X,\ y\in Y}&\mapsto (|A_{y,x}|)_{x\in X,\ y\in Y} \end{align*} $$

where $|A_{y,x}|$ denotes the number of elements of $A_{y,x}$ ; the right-hand side is a $(Y\times X)$ -matrix of nonnegative integers, specifying a homomorphism $\mathbb {Z}\langle X\rangle \to \mathbb {Z}\langle Y\rangle $ .

There is a forgetful map from the embedded cobordism category $\mathsf {Cob}_e=\mathsf {Cob}_e^{1+1}((0,1)^2)$ to the abstract $(1+1)$ -dimensional cobordism category $\mathsf {Cob}^{1+1}$ . So, any Frobenius algebra induces a functor $\mathsf {Cob}_e\to \mathsf {Ab}$ by composing the corresponding abstract $(1+1)$ -dimensional TQFT with the forgetful functor. (Here, we view the monoidal category $\mathsf {Ab}$ of abelian groups as a monoidal bicategory with only identity $2$ -morphisms.) In particular, the Khovanov Frobenius algebra $V=H^*(S^2)$ induces such a functor.

Hu–Kriz–Kriz [Reference Hu, Kriz and Kriz25] observed that the Khovanov functor $V\colon \mathsf {Cob}\to \mathsf {Ab}$ lifts to a lax $2$ -functor $V_{\mathit {HKK}}\colon \mathsf {Cob}_e\to \mathscr {B}$ :

(2.5)

In this section, we will describe this functor $V_{\mathit {HKK}}\colon \mathsf {Cob}_e\to \mathscr {B}$ , following the treatment in our earlier paper [Reference Lawson, Lipshitz and Sarkar40, Section 8.1].

On objects, for $C\in \operatorname {\mathrm {Ob}}(\mathsf {Cob}_e)$ a disjoint union of circles, $V_{\mathit {HKK}}(C)$ is the set of labellings of the circles in C by $1$ or X; that is, functions $\pi _0(C)\to \{1,X\}$ . Note that $\operatorname {\mathrm {Diff}}^1$ cannot interchange the components of C, so C, despite being a $\operatorname {\mathrm {Diff}}^1$ -equivalence class, still has a notion of components.

To define $V_{\mathit {HKK}}$ on morphisms, fix an embedded cobordism $\Sigma $ from $C_0$ to $C_1$ . Fix also a checkerboard colouring (2-colouring) of the complement of $\Sigma $ ; for definiteness, choose the colouring in which the region at $\infty $ (the region whose closure in $[0,1]\times (0,1)^2$ is noncompact) is coloured white.

The value of $V_{\mathit {HKK}}(\Sigma )$ is the product over the components $\Sigma ^{\prime }$ of $\Sigma $ of $V_{\mathit {HKK}}(\Sigma ^{\prime })$ (with respect to the checkerboard colouring of the complement of $\Sigma ^{\prime }$ that is induced from the checkerboard colouring of the complement of $\Sigma $ by declaring that the two colourings agree in a neighbourhood of $\Sigma ^{\prime }$ ) and the source and target maps respect this decomposition. (Once again, since $\Sigma $ has no closed components, $\operatorname {\mathrm {Diff}}^2$ cannot interchange components and so the notion of components descends to equivalence classes.)

So, to define $V_{\mathit {HKK}}(\Sigma )$ , we may assume $\Sigma $ is connected, but the checkerboard colouring is now arbitrary (that is, the region at $\infty $ need not be coloured white). Fix $x\in V_{\mathit {HKK}}(C_0)$ and $y\in V_{\mathit {HKK}}(C_1)$ . If $\Sigma $ has genus $>1$ , then $V_{\mathit {HKK}}(\Sigma )=\varnothing $ . If $\Sigma $ has genus $0$ , then we declare that $s^{-1}(x)\cap t^{-1}(y)\subset V_{\mathit {HKK}}(\Sigma )$ has $0$ or $1$ elements and so $V_{\mathit {HKK}}(\Sigma )$ is determined by Formula (2.5). If $\Sigma $ has genus $1$ , then $s^{-1}(x)\cap t^{-1}(y)\subset V_{\mathit {HKK}}(\Sigma )$ is empty unless x labels each circle in $C_0$ by $1$ and y labels each circle in $C_1$ by X.

In the remaining genus $1$ case, $V_{\mathit {HKK}}(\Sigma )$ has two elements, which we describe as follows. Let $S^2$ denote the one-point compactification of $(0,1)^2$ . Let $B(([0,1]\times S^2)\setminus \Sigma )$ denote the black region in the checkerboard colouring (possibly extended to the new points at infinity). Let $B((\{0,1\}\times S^2)\setminus \Sigma )= (\{0,1\}\times S^2)\cap B(([0,1]\times S^2)\setminus \Sigma )$ . Then $V_{\mathit {HKK}}(\Sigma )$ is the set of generators of

$$ \begin{align*} H_1(B(([0,1]\times S^2)\setminus \Sigma))/H_1(B((\{0,1\}\times S^2)\setminus \Sigma))\cong\mathbb{Z}. \\[-16pt]\end{align*} $$

To define $V_{\mathit {HKK}}$ on $2$ -morphisms, note that the definitions above are natural with respect to isotopies of the surface $\Sigma $ .

The composition $2$ -isomorphism is obvious except when composing two genus $0$ components $\Sigma _0$ , $\Sigma _1$ to obtain a genus $1$ component $\Sigma $ . In this nonobvious case, it again suffices to assume $\Sigma $ is connected. For any curve C on $\Sigma $ , let $C_b$ and $C_w$ be its push-offs into $B((\{1/2\}\times S^2)\setminus \Sigma )$ (the black region) and $(\{1/2\}\times S^2\setminus \Sigma )\setminus B((\{1/2\}\times S^2)\setminus \Sigma )$ (the white region), respectively. Now consider the unique component C of $(\partial \Sigma _0)\cap (\partial \Sigma _1)$ that is nonseparating in $\Sigma $ and is labelled $1$ and orient it as the boundary of $B((\{1/2\}\times S^2)\setminus \Sigma )$ . One of the two push-offs $C_b$ and $C_w$ is a generator of $H_1(([0,1]\times S^2)\setminus \Sigma )/H_1((\{0,1\}\times S^2)\setminus \partial \Sigma )\cong \mathbb {Z}^2$ and the other one is zero. If $C_b$ is the generator, label $\Sigma $ by $[C]$ . If $C_w$ is the generator, let D be a curve on $\Sigma $ , oriented so that the algebraic intersection number $D\cdot C=1$ (with $\Sigma $ being oriented as the boundary of the black region), and label $\Sigma $ by $[D_b]$ .

3.1 A decoration with divides

The embedded cobordism category $\mathsf {Cob}_e$ has $2$ -morphisms which give nontrivial endomorphisms of $V_{\mathit {HKK}}(\Sigma )$ . For example, if $\Sigma $ consists of the connected sum of a cylinder and a torus, then rotating the torus by $\pi $ around the connect sum point exchanges the two elements of $V_{\mathit {HKK}}(\Sigma )$ . To define the tangle invariants, it is more convenient to be able to work with a multicategory where each $2$ -morphism space is empty or has a single element, so $V_{\mathit {HKK}}$ takes each $2$ -endomorphism to the identity map: this will save use from having to check many compatibility conditions.

So, let $\mathsf {Cob}_d$ be the following $2$ -category:

1. (1) An object of $\mathsf {Cob}_d$ is an equivalence class of the following data:

   • • A smooth, closed $1$ -manifold Z embedded in $(0,1)^2$ .

   • • A compact $1$ -dimensional submanifold-with-boundary $A\subset Z$ satisfying the following: If I denotes the closure of $Z\setminus A$ , then each of A and I is a disjoint union of closed intervals. We call components of A active arcs and components of I inactive arcs.

   We call $(Z,A)$ a divided $1$ -manifold. Two divided $1$ -manifolds $(Z,A)$ and $(Z^{\prime },A)$ are equivalent if there is an orientation-preserving diffeomorphism $\phi \in \operatorname {\mathrm {Diff}}^1$ so that $(\phi \times \operatorname {\mathrm {Id}}_{(0,1)})(Z,A)=(Z^{\prime },A^{\prime })$ .

   We may sometimes suppress A from the notation.

   See Figure 3.1 for some examples of divided $1$ -manifolds.

2. (2) A morphism from $(Z,A)$ to $(Z^{\prime },A^{\prime })$ is an equivalence class of pairs $(\Sigma ,\Gamma )$ where

   • • $\Sigma $ is a smoothly embedded cobordism in $[0,1]\times (0,1)^2$ from Z to $Z^{\prime }$ (satisfying Convention 2.45).

   • • $\Gamma \subset \Sigma $ is a collection of properly embedded arcs in $\Sigma $ (also satisfying Convention 2.45), with $(\partial A\cup \partial A^{\prime })= \partial \Gamma $ , and so that every component of $\Sigma \setminus \Gamma $ has one of the following forms:

     1. (I) A rectangle, with two sides components of $\Gamma $ and two sides components of $A\cup A^{\prime }$ .

     2. (II) A $(2n+2)$ -gon, with $(n+1)$ sides components of $\Gamma $ , one side a component of $I^{\prime }$ and the other n sides components of I. (The integer n is allowed to be zero.)

     We call the components of $\Gamma $ divides.

   The pairs $(\Sigma ,\Gamma )$ and $(\Sigma ^{\prime },\Gamma ^{\prime })$ are equivalent if there is a diffeomorphism $\phi \in \operatorname {\mathrm {Diff}}^2$ so that $(\phi \times \operatorname {\mathrm {Id}}_{(0,1)})(\Sigma )=\Sigma ^{\prime }$ and $(\phi \times \operatorname {\mathrm {Id}}_{(0,1)})(\Gamma )=\Gamma ^{\prime }$ .

   We will call a morphism in $\mathsf {Cob}_d$ a divided cobordism. Again, we will sometimes suppress $\Gamma $ from the notation.

   See Figure 3.2 for some examples of divided cobordisms.

3. (3) There is a unique $2$ -morphism from $(\Sigma ,\Gamma )$ to $(\Sigma ^{\prime },\Gamma ^{\prime })$ whenever (some representative of the equivalence class of) $(\Sigma ,\Gamma )$ is isotopic to (some representative of the equivalence class of) $(\Sigma ^{\prime },\Gamma ^{\prime })$ rel boundary.

4. (4) Composition of divided cobordisms is defined as follows. Given $(\Sigma ,\Gamma )\colon (Z,A)\to (Z^{\prime },A^{\prime })$ and $(\Sigma ^{\prime },\Gamma ^{\prime })\colon (Z^{\prime },A^{\prime })\to (Z^{\prime \prime },A^{\prime \prime })$ , choose a representative of the equivalence class of $(Z^{\prime },A^{\prime })$ and representatives of the equivalence classes $(\Sigma ,\Gamma )$ and $(\Sigma ^{\prime },\Gamma ^{\prime })$ which end/start at this representative of $(Z^{\prime },A^{\prime })$ . Define $(\Sigma ^{\prime },\Gamma ^{\prime })\circ (\Sigma ,\Gamma )$ to be $(\Sigma ^{\prime }\circ \Sigma ,\Gamma ^{\prime }\circ \Gamma )$ .

   It is not too hard to check that composition of divided cobordisms is indeed is a divided cobordism. To wit, Type (ii) regions compose to produce Type (ii) regions; in particular, since each divide has a Type (ii) region on one side, we do not get any closed components in the set of divides after composing. While composing Type (I) rectangles, we glue them along their active boundaries to get new Type (I) rectangles. We do not get any annuli by gluing together such rectangles since that would produce closed divides.

   It is also clear that the composition map extends uniquely to $2$ -morphisms.

Forgetting the divides does not immediately give a functor from the $2$ -category $\mathsf {Cob}_d$ to the $2$ -category $\mathsf {Cob}_e$ . Though we do get maps on the objects and the $1$ -morphisms, there are no immediate maps on the $2$ -morphisms. However, the following is a stand-in for extending the map to the 2-morphisms.

More precisely, there is an analogue $\widehat {\mathsf {Cob}_d}$ of $\mathsf {Cob}_d$ in which the set of $2$ -morphisms from $\Sigma _0$ to $\Sigma _1$ is the set of isotopy classes of isotopies of divided cobordisms from $\Sigma _0$ to $\Sigma _1$ . There are forgetful maps $\Pi _{\mathsf {Cob}_d}\colon \widehat {\mathsf {Cob}_d}\to \mathsf {Cob}_d$ (collapsing the $2$ -morphism sets) and $\Pi _{\mathsf {Cob}_e}\colon \widehat {\mathsf {Cob}_d}\to \mathsf {Cob}_e$ (forgetting the divides). Proposition 3.2 asserts that the map $V_{\mathit {HKK}}\circ \Pi _{\mathsf {Cob}_e}$ descends to a functor $\mathsf {Cob}_d\to \mathscr {B}$ , so that the following diagram commutes:

By a slight abuse of notation, we will let $V_{\mathit {HKK}}$ denote the induced functor $\mathsf {Cob}_d\to \mathscr {B}$ as well.

3.2 A meeting of multicategories

3.2.2 Shape multicategories

Recall from Subsection 2.10 that ${\mathsf {B}}_{n}$ denotes the set of crossingless matchings on $2n$ points. Define $\mathcal {S}_{n}^0$ to be the shape multicategory associated to ${\mathsf {B}}_{n}$ (Definition 2.2). Specifically, the multicategory $\mathcal {S}_{n}^0$ has one object for each pair $(a,b)$ of crossingless matchings of $2n$ points and a unique multimorphism

$$ \begin{align*} (a_1,a_2),(a_2,a_3),\dots,(a_{k-1},a_k)\to (a_1,a_k). \end{align*} $$

We will sometimes denote the unique morphism in $\operatorname {\mathrm {Hom}}((a_1,a_2),(a_2,a_3),\dots ,(a_{k-1},a_k); (a_1,a_k))$ by $f_{a_1,\dots ,a_k}$ . In particular, the special case $k=1$ of the zero-input multimorphism $\varnothing \to (a_1,a_1)$ is denoted $f_{a_1}$ .

Similarly, define ${}_{m}^{\vphantom {0}}\mathcal {T}_{n}^0$ to be the shape multicategory associated to the sequence of sets $({\mathsf {B}}_{m},{\mathsf {B}}_{n})$ (Definition 2.3). Specifically, the multicategory ${}_{m}^{\vphantom {0}}\mathcal {T}_{n}^0$ has three kinds of objects:

1. (1) Objects $(a,b)$ where $a,b$ are crossingless matchings on $2m$ points,

2. (2) objects $(a,b)$ where $a,b$ are crossingless matchings on $2n$ points and

3. (3) objects $(a,b)$ where a is a crossingless matching of $2m$ points and b is a crossingless matching of $2n$ points. For clarity we will write such objects instead as $(a,T,b)$ where T is just a notational placeholder.

There is a unique multimorphism

$$ \begin{align*} (a_1,a_2),(a_2,a_3),\dots,(a_{k-1},a_k)\to(a_1,a_k) \end{align*} $$

if $a_1,\dots ,a_k$ are crossingless matchings on $2m$ points. There is a unique multimorphism

$$ \begin{align*} (b_1,b_2),(b_2,b_3),\dots,(b_{\ell-1},b_\ell)\to(b_1,b_\ell) \end{align*} $$

if $b_1,\dots ,b_\ell $ are crossingless matchings on $2n$ points. There is a unique multimorphism

$$ \begin{align*} (a_1,a_2),\dots,(a_{k-1},a_k),(a_k,T,b_1),(b_1,b_2),\dots,(b_{\ell-1},b_\ell)\to (a_1,T,b_\ell) \end{align*} $$

if $a_1,\dots ,a_k$ are crossingless matchings on $2m$ points and $b_1,\dots ,b_\ell $ are crossingless matchings on $2n$ points. (The special cases $k=1$ and $\ell =1$ are allowed.)

Note that $\mathcal {S}_{m}^0$ and $\mathcal {S}_{n}^0$ are full sub-multi-categories of ${}_{m}^{\vphantom {0}}\mathcal {T}_{n}^0$ . Extending the notation $f_{a_1,\dots ,a_k}$ from $\mathcal {S}_{m}^0$ , we will sometimes denote the unique morphism in

$$ \begin{align*} \operatorname{\mathrm{Hom}}_{{}_{m}^{\vphantom{0}}\mathcal{T}_{n}^0}\bigl((a_1,a_2),\dots,(a_{k-1},a_k),(a_k,T,b_1),(b_1,b_2),\dots,(b_{\ell-1},b_\ell);(a_1,T,b_\ell)\bigr) \end{align*} $$

by $f_{a_1,\dots ,a_k,T,b_1,\dots ,b_\ell }$ .

Let $\mathcal {S}_{n}$ (respectively ${}_{m}\mathcal {T}_{n}$ ) be the canonical groupoid enrichment of $\mathcal {S}_{n}^0$ (respectively ${}_{m}^{\vphantom {0}}\mathcal {T}_{n}^0$ ) from Subsection 2.4.1. See in particular Example 2.7 for some of the multimorphisms that appear in the groupoid enriched categories.

Now recall, from Subsection 2.10, Khovanov’s arc algebra $\mathcal {H}^{n}$ and Khovanov’s tangle invariant $\mathcal {C}_{\mathit {Kh}}(T)$ , which is a dg $(\mathcal {H}^{m},\mathcal {H}^{n})$ -bimodule. The algebra $\mathcal {H}^{n}$ is equipped with an orthogonal set of idempotents (Definition 2.4), one for each crossingless matching $a\in {\mathsf {B}}_{n}$ , with the idempotent corresponding to a being the element of $V(a\overline {a})$ that labels each of the n circles by $1\in V$ . Therefore, via the equivalences from Subsection 2.3, we have the following.

Principle 3.4. The Khovanov arc algebra $\mathcal {H}^{n}$ may be viewed as a multifunctor $\mathcal {S}_{n}^0\to \underline {\mathsf {Ab}}$ . Composing with the inclusion $\underline {\mathsf {Ab}}\to \underline {\mathsf {Kom}}$ (which views an abelian group as a chain complex concentrated in grading $0$ ), we can also view the Khovanov arc algebra as a multifunctor from $\mathcal {S}_{n}^0$ to chain complexes. Similarly, Khovanov’s tangle invariant $\mathcal {C}_{\mathit {Kh}}(T)$ may be viewed as a multifunctor ${}_{m}^{\vphantom {0}}\mathcal {T}_{n}^0\to \underline {\mathsf {Kom}}$ which restricts to $\mathcal {S}_{m}^0$ and $\mathcal {S}_{n}^0$ as the arc algebra multifunctors.