Todd Trimble
Further developments on Trimble n-categories

Calculus of bimodules over operads

Throughout this work, we will be working over a category VV with some baseline assumptions:

(Here “connected” means that the exponential functor () I:VV(-)^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

For all intents and purposes, the reader may assume that VV is a nice category TopTop of topological spaces, such as the category of compactly generated Hausdorff spaces.

The functor hom(1,):VSethom(1, -): V \to Set has a left adjoint 1- \cdot 1 which takes a set SS to the coproduct of an SS-indexed collection of copies of 11, S1S \cdot 1, which may be thought of as a discrete space. The resulting comonad takes an object vv to hom(1,v)1hom(1, v) \cdot 1, and this will be denoted as |v||v| (“|v||v| retopologized with the discrete topology”).

Let

Tx= n0x nT x = \sum_{n \geq 0} x^n

be the free monoid monad on VV, with monad multiplication m:TTTm: T T \to T and monad unit u:1 VTu: 1_V \to T. The monad (T,m,u)(T, m, u) is cartesian, and we define a substitution product (a monoidal product \circ on the slice V/T1V/T 1) on objects f:vT1f: v \to T 1, g:wT1g: w \to T 1 of V/T1V/T 1 to be the pullback

vw Tw Tg TT1 μ1T1 T! v f T1 \array{ v \circ w & \to & T w & \overset{T g}{\to} & T T 1 & \overset{\mu 1}{\to} T1 \\ \downarrow & & \downarrow T ! & & & \\ v & \overset{f}{\to} & T 1 & & & }

where the top horizontal composite exhibits vwv \circ w as an object of V/T1V/T 1. This extends naturally to a monoidal product \circ on V/T1V/ T 1; the monoidal unit 1\mathbf{1} here is provided by u:1T1u: 1 \to T 1, the unit of the monad TT. A (nonpermutative) operad in VV is by definition a monoid in (V/T1,)(V/T 1, \circ): an object MM of V/T1V/T 1 together with a pair of maps

μ:MMM,η:1M\mu: M \circ M \to M, \qquad \eta: \mathbf{1} \to M

satisfying monoid axioms.

An easy but useful result is

Lemma 0: The monoidal product

:V/T1×V/T1V/T1\circ: V/T 1 \times V/T 1 \to V/T 1

preserves pullbacks. \Box

For our next results, recall that a reflexive pair in a category is a pair of parallel maps f,gf, g that have a common right inverse hh:

f,g:XY: h:YX(fh=1 Y=gh)f, g: X \overset{\to}{\to} Y: \exists_{h: Y \to X} (f h = 1_Y = g h)

We call a coequalizer of a reflexive pair a reflexive coequalizer for short.

Lemma 1: T:VVT: V \to V preserves reflexive coequalizers.

Proof: This is generally true for finitary monads on suitable VV (such as the topos of globular sets; cf. Batanin’s theory of nn-categories); a proof for TT is outlined as follows. If products distribute over colimits and over reflexive coequalizers in particular, then by a famous 3×33 \times 3 lemma (see Johnstone’s Topos Theory, page 1)

V×VprodVV \times V \overset{prod}{\to} V

also preserves reflexive coequalizers, and it follows by induction that the n thn^{th} cartesian power xx nx \mapsto x^n preserves reflexive coequalizers. Hence

VV :xx nV \to V^{\mathbb{N}}: x \mapsto \langle x^n \rangle

preserves reflexive coequalizers, as does

:V V\sum: V^{\mathbb{N}} \to V

since left adjoints preserve all colimits, and the result follows. \Box

Theorem 0: The functor w:V/T1V/T1- \circ w: V/T 1 \to V/T 1 preserves colimits.

Proof: Because VV is \infty-extensive, we have an identification

V/T1V V/ T1 \simeq V^{\mathbb{N}}

so that we can think of V/T1V/ T1 as the category of graded VV-objects. Also by \infty-extensivity, the pullback functor

(T!) *:V/T1V/Tw(T !)^*: V/T 1 \to V/T w

is equivalent to a product of functors

n:(×w n): n:V n:V/w n\prod_{n: \mathbb{N}}(- \times w^n): \prod_{n: \mathbb{N}} V \to \prod_{n: \mathbb{N}} V/w^n

each of which is colimit-preserving, so the pullback (T!) *(T !)^* preserves colimits. The pushforward

(m1Tg) != m1Tg:V/TwV/T1,(m1 \circ T g)_!= \sum_{m 1 \circ T g}: V/T w \to V/T 1,

being a left adjoint, also preserves colimits. The result follows. \Box

Lemma 2: The monoidal product

v:V/T1V/T1v \circ -: V/ T 1 \to V/T 1

preserves reflexive coequalizers. Thus the bifunctor (v,w)vw(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/T1V/T 1 as they are in VV. By lemma 1, T:VVT: V \to V preserves reflexive coequalizers. Thus, given a coequalizer diagram (for a reflexive pair)

w 1w 2ww_1 \overset{\to}{\to} w_2 \to w

in V/T1V/T 1, we get a corresponding coequalizer diagram

Tw 1Tw 2TwT1T1T w_1 \overset{\to}{\to} T w_2 \to T w \overset{T 1}{\to} T 1

in V/T1V/T 1. Taking advantage of \infty-extensivity as before, we see that when we pull back this diagram along a map f:vT1f: v \to T 1, we again get a coequalizer diagram

vw 1vw 2vwv \circ w_1 \overset{\to}{\to} v \circ w_2 \to v \circ w

(in the category V/TwV/T w), and pushing forward along

(m1Tg) !:V/TwV/T1(m 1 \circ T g)_!: V/T w \to V/T 1

this coequalizer is again preserved, completing the proof. \Box

Now we come to the key concept of this section. Let MM, NN be operads in VV.

Definition: A left MM right NN bimodule is an object XX of V/T1V/T 1 together with

We write X:MNX: 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,β)(X, \beta) is a right MM-module and (Y,α)(Y, \alpha) is a left MM-module, then we can form a “tensor product” X MYX \circ_M Y as the coequalizer of the diagram

(βY,Xα):XMYXYX MY(\beta \circ Y, X \circ \alpha): X \circ M \circ Y \overset{\to}{\to} X \circ Y \to X \circ_M Y

The parallel pair here is reflexive (consider XηY:X1YXMYX \circ \eta \circ Y: X \circ \mathbf{1} \circ Y \to X \circ M \circ Y). Thus, by lemma 2, for any object LL of V/T1V/T 1, the functor LL \circ - preserves tensor products: the canonical map

(LX) MYL(X MY)(L \circ X) \circ_M Y \to L \circ (X \circ_M Y)

is an isomorphism. In particular, if LL carries an operad structure, then by the usual universality arguments, it follows that for XX a left LL right MM bimodule and YY a left MM-module, the object X MYX \circ_M Y carries a canonical left LL-module structure.

Similarly, given bimodules X:LMX: L \to M and Y:MPY: M \to P, the object X MYX \circ_M Y becomes a bimodule LPL \to P, and this defines bimodule composition. The identity MMM \to M is MM as a bimodule over itself. With the structure thus sketched, we have in summary

Theorem 1: Operads in VV, bimodules, and bimodule homomorphisms form a bicategory. \Box

Notice incidentally that the monoidal unit 1\mathbf{1} for the monoidal product \circ is an operad, and that a left MM-module is the same thing as a bimodule M1M \to \mathbf{1}, and similarly that a right MM-module is a bimodule 1M\mathbf{1} \to M.

A left MM right MM bimodule is called simply an MM-bimodule. Thus MM-bimodules form a monoidal category (or a bicategory with just one object MM).

Acyclic structures on bimodules

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 VV has a model category structure, as for example on V=Set Δ opV = Set^{\Delta^{op}} (simplicial sets) or on V=TopV = 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 II with bottom element 00, which we regard as a monoid whose multiplication is the meet operation. In V/T1V/T 1 there is a corresponding constant interval π 2:I×T1T1\pi_2: I \times T 1 \to T 1, denoted I *I^*.

Because II is a connected monoid, there is a structure of limit-preserving and coproduct-preserving comonad on the functor () I:VV(-)^I: V \to V. The discrete space functor

|()|:VV|(-)|: V \to V

also carries a structure of limit-preserving and coproduct-preserving comonad, as does the identity functor 1 V:VV1_V: V \to V. The functor PP defined by pullback

PX X I X 1=eval 1X eval 0 |X| counit X\array{ P X & \to & X^I & \overset{X^1 = eval_1}{\to} X\\ \downarrow & & \downarrow eval_0 & \\ |X| & \overset{counit}{\to} & X }

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(-)^I. We call PP the path comonad. It plays a role analogous to the decalage functor on simplicial objects.

Definition: An acyclic structure on an object XX is a PP-coalgebra structure on XX.

We also have a path comonad P *P^* defined componentwise on the category V/T1V V/T 1 \simeq V^{\mathbb{N}}, also limit-preserving and coproduct preserving, and a corresponding notion of acyclic structure on V/T1V/T 1.

In some cases, the comonad PP admits a left adjoint CC, which then automatically carries a monad structure such that the category of CC-algebras is equivalent to the category of PP-coalgebras (by a classical result of Eilenberg-Moore), and in that case it may be easier to comprehend acyclic structures in terms of CC-algebras. Thus, suppose that the discretization functor

1:SetV- \cdot 1: Set \to V

has a left adjoint π 0:VSet\pi_0: V \to Set; this occurs for instance in the case where VV is the category of simplicial sets. Then the left adjoint CC of PP is constructed objectwise as the pushout

X {0}×1 X I×X π π 0(X) CX\array{ & X & \overset{\{0\} \times 1_X}{\to} & I \times X \\ \pi & \downarrow & & \downarrow \\ & \pi_0(X) & \to & C X }

so that CXC X is, intuitively, the coproduct of all the cones of path components of XX. A CC-algebra structure α:CXX\alpha: C X \to X would pick out a basepoint in each path component of XX, and give an action of the monoid II,

α:I×XX\alpha: I \times X \to X

such that α(0,):XX\alpha(0, -): X \to X contracts each path component of XX down to its basepoint. In this way a CC-algebra structure on XX witnesses the acyclicity of XX. But we don’t need the monad CC: more generally, a PP-coalgebra structure on XX does exactly the same thing.

Theorem 2: The comonad P *:V/T1V/T1P^*: 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/T1V/T 1 as graded objects V V^{\mathbb{N}}, where the substitution product is given explicitly by the formula

(vw)(n)= n 1++n k=nv(k)×w(n 1)××w(n k)(v \circ w)(n) = \sum_{n_1 + \ldots + n_k = n} v(k) \times w(n_1) \times \ldots \times w(n_k)

The exponential functor () I(-)^I preserves both sums (because II is connected) and products, so it preserves the formula above, i.e., there is a canonical isomorphism

(vw) Iv Iw I(v \circ w)^I \cong v^I \circ w^I

Similar observations hold for the identity functor and the functor |()|:VV|(-)|: V \to V. Then, applying lemma 0, we conclude that there is a canonical isomorphism

P(vw)PvPwP(v \circ w) \cong P v \circ P w

exhibiting PP 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 vv and ww are PP-coalgebras, then vwv \circ w carries a natural PP-coalgebra structure.

Revised on July 5, 2009 at 13:11:55 by Todd Trimble