Throughout this work, we will be working over a category $V$ with some baseline assumptions:
$V$ is complete, cocomplete, and cartesian closed, and $hom(1, -): V \to Set$ preserves coproducts;
There is given a connected object $I$ of $V$ equipped with an internal meet-semilattice structure and a bottom element $0$. $I$ will be called “the interval” of $V$.
(Here “connected” means that the exponential functor $(-)^I: V \to V$ preserves coproducts.) We also assume, although it is not critical to the development of the theory and is included just for the sake of convenience, that
is an equivalence for any small collection of objects $\langle v_a\rangle_{a \in A}$.
For all intents and purposes, the reader may assume that $V$ is a nice category $Top$ of topological spaces, such as the category of compactly generated Hausdorff spaces.
The functor $hom(1, -): V \to Set$ has a left adjoint $- \cdot 1$ which takes a set $S$ to the coproduct of an $S$-indexed collection of copies of $1$, $S \cdot 1$, which may be thought of as a discrete space. The resulting comonad takes an object $v$ to $hom(1, v) \cdot 1$, and this will be denoted as $|v|$ (“$|v|$ retopologized with the discrete topology”).
Let
be the free monoid monad on $V$, with monad multiplication $m: T T \to T$ and monad unit $u: 1_V \to T$. The monad $(T, m, u)$ is cartesian, and we define a substitution product (a monoidal product $\circ$ on the slice $V/T 1$) on objects $f: v \to T 1$, $g: w \to T 1$ of $V/T 1$ to be the pullback
where the top horizontal composite exhibits $v \circ w$ as an object of $V/T 1$. This extends naturally to a monoidal product $\circ$ on $V/ T 1$; the monoidal unit $\mathbf{1}$ here is provided by $u: 1 \to T 1$, the unit of the monad $T$. A (nonpermutative) operad in $V$ is by definition a monoid in $(V/T 1, \circ)$: an object $M$ of $V/T 1$ together with a pair of maps
satisfying monoid axioms.
An easy but useful result is
Lemma 0: The monoidal product
preserves pullbacks. $\Box$
For our next results, recall that a reflexive pair in a category is a pair of parallel maps $f, g$ that have a common right inverse $h$:
We call a coequalizer of a reflexive pair a reflexive coequalizer for short.
Lemma 1: $T: V \to V$ preserves reflexive coequalizers.
Proof: This is generally true for finitary monads on suitable $V$ (such as the topos of globular sets; cf. Batanin’s theory of $n$-categories); a proof for $T$ is outlined as follows. If products distribute over colimits and over reflexive coequalizers in particular, then by a famous $3 \times 3$ lemma (see Johnstone’s Topos Theory, page 1)
also preserves reflexive coequalizers, and it follows by induction that the $n^{th}$ cartesian power $x \mapsto x^n$ preserves reflexive coequalizers. Hence
preserves reflexive coequalizers, as does
since left adjoints preserve all colimits, and the result follows. $\Box$
Theorem 0: The functor $- \circ w: V/T 1 \to V/T 1$ preserves colimits.
Proof: Because $V$ is $\infty$-extensive, we have an identification
so that we can think of $V/ T1$ as the category of graded $V$-objects. Also by $\infty$-extensivity, the pullback functor
is equivalent to a product of functors
each of which is colimit-preserving, so the pullback $(T !)^*$ preserves colimits. The pushforward
being a left adjoint, also preserves colimits. The result follows. $\Box$
Lemma 2: The monoidal product
preserves reflexive coequalizers. Thus the bifunctor $(v, w) \mapsto v \circ w$ preserves reflexive coequalizers in each of its two separate arguments.
Proof: As is the case for general colimits, reflexive coequalizers are computed in $V/T 1$ as they are in $V$. By lemma 1, $T: V \to V$ preserves reflexive coequalizers. Thus, given a coequalizer diagram (for a reflexive pair)
in $V/T 1$, we get a corresponding coequalizer diagram
in $V/T 1$. Taking advantage of $\infty$-extensivity as before, we see that when we pull back this diagram along a map $f: v \to T 1$, we again get a coequalizer diagram
(in the category $V/T w$), and pushing forward along
this coequalizer is again preserved, completing the proof. $\Box$
Now we come to the key concept of this section. Let $M$, $N$ be operads in $V$.
Definition: A left $M$ right $N$ bimodule is an object $X$ of $V/T 1$ together with
We write $X: M \to N$ to indicate such a bimodule; this should suggest that a bicategory whose objects are operads and whose 1-cells are bimodules is in the offing.
Lemma 2 makes this a routine matter. If $(X, \beta)$ is a right $M$-module and $(Y, \alpha)$ is a left $M$-module, then we can form a “tensor product” $X \circ_M Y$ as the coequalizer of the diagram
The parallel pair here is reflexive (consider $X \circ \eta \circ Y: X \circ \mathbf{1} \circ Y \to X \circ M \circ Y$). Thus, by lemma 2, for any object $L$ of $V/T 1$, the functor $L \circ -$ preserves tensor products: the canonical map
is an isomorphism. In particular, if $L$ carries an operad structure, then by the usual universality arguments, it follows that for $X$ a left $L$ right $M$ bimodule and $Y$ a left $M$-module, the object $X \circ_M Y$ carries a canonical left $L$-module structure.
Similarly, given bimodules $X: L \to M$ and $Y: M \to P$, the object $X \circ_M Y$ becomes a bimodule $L \to P$, and this defines bimodule composition. The identity $M \to M$ is $M$ as a bimodule over itself. With the structure thus sketched, we have in summary
Theorem 1: Operads in $V$, bimodules, and bimodule homomorphisms form a bicategory. $\Box$
Notice incidentally that the monoidal unit $\mathbf{1}$ for the monoidal product $\circ$ is an operad, and that a left $M$-module is the same thing as a bimodule $M \to \mathbf{1}$, and similarly that a right $M$-module is a bimodule $\mathbf{1} \to M$.
A left $M$ right $M$ bimodule is called simply an $M$-bimodule. Thus $M$-bimodules form a monoidal category (or a bicategory with just one object $M$).
Remark: Since the category $V$ is complete, $- \circ X$ has a right adjoint $(-) \Leftarrow X$ given by the formula
However, $X \circ -$ is very far from colimit-preserving, so $(V/T 1, \circ)$ becomes in this way closed monoidal only on one side. Similarly, at the level of bimodules, functors of the form $- \circ_M X$ have right adjoints (in the parlance, right Kan lifts through $X$ exist in the bicategory of bimodules), but there will be no corresponding right adjoints to composing on the other side, or right Kan extensions, in general.
Remark: An operad morphism $f: M \to N$ does give rise to a bimodule $X_f: M \to N$ whose underlying object is $N$, with left $M$-action given by the composite
and right $N$-action given by $\mu: N \circ N \to N$. (Question of whether $X_f$ has a right adjoint given by the right Kan lift $Ran_f(1_N)$, and whether all left adjoints come from operad maps, etc. Generalizations to multicategories, etc.)
We next introduce objects which carry structures bearing witness to their “contractibility” in some sense of that word, based on the presence of an interval object. At some ultimate stage of the theory, it should be sufficient to assume $V$ has a model category structure, as for example on $V = Set^{\Delta^{op}}$ (simplicial sets) or on $V = Top$, and work with an interval object in the sense of model categories. However, in this work we take a thoroughly algebraic (or equational) approach, for which model categories will not be precisely appropriate. So instead we work with a relatively strong algebraic notion of contractibility, or more to the point algebraic notions of contraction and of acyclic models.
Recall our baseline assumption that our interval object is a connected meet-semilattice $I$ with bottom element $0$, which we regard as a monoid whose multiplication is the meet operation. In $V/T 1$ there is a corresponding constant interval $\pi_2: I \times T 1 \to T 1$, denoted $I^*$.
Because $I$ is a connected monoid, there is a structure of limit-preserving and coproduct-preserving comonad on the functor $(-)^I: V \to V$. The discrete space functor
also carries a structure of limit-preserving and coproduct-preserving comonad, as does the identity functor $1_V: V \to V$. The functor $P$ defined by pullback
is also limit-preserving and coproduct-preserving. It has a comonad structure whose counit is exhibited as the top horizontal composite, and whose comultiplication comes via restriction of the comultiplication on the comonad $(-)^I$. We call $P$ the path comonad. It plays a role analogous to the decalage functor on simplicial objects.
Definition: An acyclic structure on an object $X$ is a $P$-coalgebra structure on $X$.
We also have a path comonad $P^*$ defined componentwise on the category $V/T 1 \simeq V^{\mathbb{N}}$, also limit-preserving and coproduct preserving, and a corresponding notion of acyclic structure on $V/T 1$.
In some cases, the comonad $P$ admits a left adjoint $C$, which then automatically carries a monad structure such that the category of $C$-algebras is equivalent to the category of $P$-coalgebras (by a classical result of Eilenberg-Moore), and in that case it may be easier to comprehend acyclic structures in terms of $C$-algebras. Thus, suppose that the discretization functor
has a left adjoint $\pi_0: V \to Set$; this occurs for instance in the case where $V$ is the category of simplicial sets. Then the left adjoint $C$ of $P$ is constructed objectwise as the pushout
so that $C X$ is, intuitively, the coproduct of all the cones of path components of $X$. A $C$-algebra structure $\alpha: C X \to X$ would pick out a basepoint in each path component of $X$, and give an action of the monoid $I$,
such that $\alpha(0, -): X \to X$ contracts each path component of $X$ down to its basepoint. In this way a $C$-algebra structure on $X$ witnesses the acyclicity of $X$. But we don’t need the monad $C$: more generally, a $P$-coalgebra structure on $X$ does exactly the same thing.
Theorem 2: The comonad $P^*: V/T 1 \to V/T 1$ is strong monoidal with respect to the monoidal product $\circ$. That is, the underlying functor is strong monoidal, and the counit and comultiplication are monoidal transformations.
Proof: It may help to represent $V/T 1$ as graded objects $V^{\mathbb{N}}$, where the substitution product is given explicitly by the formula
The exponential functor $(-)^I$ preserves both sums (because $I$ is connected) and products, so it preserves the formula above, i.e., there is a canonical isomorphism
Similar observations hold for the identity functor and the functor $|(-)|: V \to V$. Then, applying lemma 0, we conclude that there is a canonical isomorphism
exhibiting $P$ as a strong monoidal functor. The fact that the counit and comultiplication are monoidal natural transformations is routine and left to the reader. $\Box$
Corollary: If $v$ and $w$ are $P$-coalgebras, then $v \circ w$ carries a natural $P$-coalgebra structure.