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This post is under construction. See the previous post in this series for context and content of the following post, or the bottom of the first in the series for a table of contents.

Symmetric sequences, operads, and free functors

Let \(\Sigma\) denote the groupoid \(\Sigma := \coprod_{n \in \NN} \Sigma_n\); this is equivalent to the core of the category of finite sets.

Definition 6. Fix \(\cC\) a cocomplete symmetric monoidal closed category. The *category of symmetric sequences in \(\cC\) is the functor category \(\cC^\Sigma\). These possess a symmetric monoidal product called the *composition product, computed on objects by

$$\cO \circ \cO'(n) := \coprod_{n_1 + \cdots + n_k = n} \cO(k) \otimes \bigotimes_{i} \cO'(n_i).$$

An operad is a monoid in symmetric sequences.

We could have equivalently replaced \(\Sigma\) with \(\mathbf{Fin}^{\simeq}\); we use the skeleton for the sake of the notation \(\cO(n)\).

I’ll cut to the chase; there is a free forgetful adjunction between pointed symmetric sequences and operads, given by the free monoid construction with respect to \(\circ\). There’s likewise a free-forgetful adjunction between symmetric sequences and left-modules over a given operad, as well as right-modules. Left-modules are a generalization of algebras, which are precisely left modules concentrated in degree 0.

If we view the weight grading of a symmetric sequence \(X_\bullet\) as being the “number of elements,” then we may view right-modules over \(\cO\) as an algebraic structure wherein, given an element of \(X_\bullet\) with \(n\) elements, and \(n\) elements of \(\cO\) with \(n_1,\dots,n_k\) elements, we produce an element of \(X_\bullet\) with \(\sum_i n_i\) elements.

We use this perspective to formalize toroidal composition. But first, we sketch the notion of a operadic congruence generated by a sub-symmetric sequences.

Congruences, quotients, and operads

This section will be far from self contained.

Let \(\cC\) be a finitely complete category. A congruence on $$X$$</a> is an equivalence relation on $$X$$ internal to $$\cC$$; in particular, such a thing is a subobject $$R \hookrightarrow X \times X$$, which yields a parallel pair $$R \rightrightarrows X$$ by composing with the right and left projection. Note that the condition of internal reflexivity is precisely that there exists a common section $$\Delta:X \rightarrow R \rightrightarrows X$$.

The quotient \(X / R\) is, if it exists, the coequalizer of the diagram \(R \rightrightarrows X\).

In particular, any morphism \(S \rightarrow X\) has a kernel pair, defined as the pullback of the cospan \(S \rightarrow X \leftarrow S\), and this supports the structure of a congruence. If all quotients exist for kernel pairs, then every coequalizer is expressed as the quotient of its kernel pair; this is a nonabelian version of the axiom in abelian category that cokernels of kernels agree with kernels of cokernels.

If in fact the morphism property of being a coequalizer of some parallel pair is pullback-stable, then \(\cC\) is called regular; regular categories are the categories with pullback-stable image factorizations, yielding a form of the first isomorphism theorem: every morphism \(f:X \rightarrow Y\) factors canonically as a coequalizer followed by a monomorphism.

Claim. Let \(G \hookrightarrow X \times X\) be a subobject, for \(X\) a monoid. Then, the poset \(P\) of congruences on \(X\) containing \(G\) (ordered by containment) has a unique minimal element.

It suffices to prove that \(P\) is cofiltered and its opposite poset satisfies the conditions of Zorn’s lemma. In fact, we can prove a stronger condition: we show that \(P\) has arbitrary nonempty limits, i.e. it has arbitrary nonempty intersections.

Toroidal composition

There are symmetric sequences \(A^\vee\) of atoroidal toroidal weaves, \(A^{\Sym}\) of atoroidal weaves, \(W^\vee\) of toroidal weaves, and \(W^{\Sym}\) of weaves, in sets. Toroidal composition provides two maps of symmetric sequences:

$$W^\vee \circ W^\vee \xrightarrow{\mu^\vee}W^\vee.$$

$$W^{\Sym} \circ W^\vee \xrightarrow{\mu} W^{\Sym}.$$

We’d like to realize the first as the composition map of an operad, and the second as a structure map of an operadic right module. To do the first, we need to choice a unit. This is not too hard:

Proposition 1. \(\cW^{\vee}(1)\) is contractible; that is, there is a contractible space of one-component toroidal weaves.

To prove this, note that there is a deformation retract of \(\cW^{\vee}(1)\) onto the realization given by the core curve of the solid torus; this is a translation followed by a rotation.

Let \(\eta \in \Hom(\cO_{\operatorname{triv}}, W^\vee) \simeq W^{\vee}(1) \simeq *\) be the for \(W^\vee\).

Proposition 2. The maps \(\mu^\vee, \eta\) endow on \(W^{\vee}\) the structure of an operad.

This is more-or-less obvious, so I’ll skip it. The following proposition is also obvious, and I’ll also skip it

Proposition 3. The map \(\mu\) endows \(W^{\Sym}\) with the structure of a right module over \(W^\vee\).

These summarize the existence of algebraic structure on weaves; there’s both an operadic composition within toroidal weaves, and an operadic right-action of toroidal weaves on weaves. We next describe toroidal decomposition, which amounts to giving presentations of \(W\) and \(W^\vee\) as quotients of free objects.

Toroidal decomposition

Note that \(\cA^\vee\) is contractible by Proposition 1, so \(A^\vee\) is uniquely pointed. There is a free operad \(\Fr_{\Op}\prn{A^\vee}\) on \(A^\vee\) as a pointed object; this is the operad parameterizing trees labelled with atoroidal toroidal weaves, modulo the one-component weave \(O\) acting as unit. The operad structure on \(W^\vee\) is given by a structure map \(\Fr_{\Op}\prn{W^\vee} \rightarrow W^\vee\). This induces a map

$$\Fr_{\Op}\prn{A^\vee} \rightarrow \Fr_{\Op}\prn{W^\vee} \rightarrow W^\vee.$$

In the other direction, the previous discussion of toroidal decomposition yields a map of operads \(W^\vee \rightarrow \Fr_{\Op}\prn{A^\vee}\), after choosing a (non-canonical) decomposition of each fully toroidal weave into a tree of 2-component weaves. We’d like to realize these as maps within a split coequalizer diagram in operads; to do so, we need the technology of operadic congruence envelopes.

TODO: put discussion of operadic congruence envelopes here.

Now, let \(r^\vee_{M} \in \Fr_{\Op}\prn{A^\vee}^{\times 2}\) be the relation between the two realizations of \(M_3\) within \(\Fr_{\Op}\prn{A^\vee}\); these are both trees of height 2 with one leaf node \(M_2\) and the other \(O\), with the first making one choice of position for the \(M_2\) and the other making the other. Similarly, let \(r^\vee_O \in \Fr{|Op}\prn{A^\vee}^{\times 2}\) be the relation between the two realizations of \(3 \cdot O\).

Toroidal decomposition for toroidal weaves can be phrased as follows:

Theorem 4. The operadic cofork diagram

$$\langle r_M, r_O \rangle \rightrightarrows \Fr_{\Op}(A^\vee) \rightarrow \cW^\vee$$

is split by toroidal decomposition; hence \(\cW^\vee\) is the quotient of the operad of trees labelled by \(A^\vee\) by the realizations of \(M_3\) and \(3 \cdot O\).

Similarly, note that \(A \circ W^\vee\) is the free right \(W^\vee\)-module on \(A\). Let \(r_M\) and \(r_0\) be the relations in \(A \circ W^\vee\) corresponding with the realizations of \(M_3\) and \(3 \cdot O\). Toroidal decomposition for weaves can be phrased as follows:

Theorem 5. The right \(W^\vee\)-module cofork diagram

$$\langle r_M, r_O \rangle \rightrightarrows A \circ W^\vee \rightarrow W^\Sym$$

is split by toroidal decomposition; hence \(W^{\Sym}\) arises by affixing a single atoroidal weave to a toroidal weave, modulo the pairs of such decompositions for \(M_3\) and \(3 \cdot O\).

We then may pass from \(W^{\Sym}\) to \(W\) by taking \(\Sigma\)-orbits.

The various functors employed in this construction are summarized in the following chart:

This illustration is color coded for clarity; red describes the free operad construction, green the free module construction, orange the structure of toroidal weaves, and blue the structure of weaves. Each cofork diagram is a (split) coequalizer.