Abstract
The functor between operadic algebras given by restriction along an operad map generally has a left adjoint. We give a necessary and sufficient condition for the restriction functor to admit a right adjoint. The condition is a factorization axiom which roughly says that operations in the codomain operad can be written essentially uniquely as operations in arity one followed by operations in the domain operad.
Similar content being viewed by others
Notes
This superscript notation matches with the \(S^m\) and \((S^m)k\) appearing in [17].
In general, a functor is conservative if the only morphisms it sends to isomorphisms are themselves isomorphisms.
References
Batanin, M., Kock, J., Weber, M.: Regular patterns, substitudes, Feynman categories and operads. Theory Appl. Categ. 33(7), 148–192 (2018)
Batanin, M.A., Berger, C.: Homotopy theory for algebras over polynomial monads. Theory Appl. Categ. 32(6), 148–253 (2017)
Bénabou, J.: Introduction to bicategories. In: Reports of the Midwest Category Seminar, Lecture Notes in Math., vol. 47, pp. 1–77. Springer (1967). https://doi.org/10.1007/BFb0074299
Berger, C., Moerdijk, I.: Resolution of coloured operads and rectification of homotopy algebras. In: Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, pp. 31–58. Amer. Math. Soc., Providence (2007). https://doi.org/10.1090/conm/431/08265
Börger, R.: Disjointness and related properties of coproducts. Acta Univ. Carolin. Math. Phys. 35(1), 43–63 (1994)
Cohen, F.: The homology of \(C_{n+1}\)spaces, \(n\ge 0\). In: The Homology of Iterated Loop Spaces, Lecture Notes in Math., vol. 533, pp. 207–351. Springer Berlin Heidelberg, Berlin, Heidelberg (1976). https://doi.org/10.1007/BFb0080467
Dehling, M., Vallette, B.: Symmetric homotopy theory of operads. Algebr. Geom. Topol. 21(4), 1595–1660 (2021). https://doi.org/10.2140/agt.2021.21.1595
DrummondCole, G.C., Hackney, P.: A criterion for existence of rightinduced model structures. Bull. Lond. Math. Soc. 51(2), 309–326 (2019). https://doi.org/10.1112/blms.12232
DrummondCole, G.C., Hackney, P.: Dwyer–Kan homotopy theory for cyclic operads. Proc. Edinb. Math. Soc. 64(1), 29–58 (2021). https://doi.org/10.1017/S0013091520000267
Getzler, E.: Batalin–Vilkovisky algebras and twodimensional topological field theories. Comm. Math. Phys. 159, 265–285 (1994). https://doi.org/10.1007/bf02102639
Getzler, E., Kapranov, M.M.: Cyclic operads and cyclic homology. In: Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, pp. 167–201. Int. Press, Cambridge (1995)
Guillén Santos, F., Navarro, V., Pascual, P., Roig, A.: Moduli spaces and formal operads. Duke Math. J. 129, 291–335 (2005). https://doi.org/10.1215/S0012709405129246
Hackney, P., Robertson, M., Yau, D.: Relative left properness of colored operads. Algebr. Geom. Topol. 16(5), 2691–2714 (2016). https://doi.org/10.2140/agt.2016.16.2691
Hackney, P., Robertson, M., Yau, D.: Modular operads and the nerve theorem. Adv. Math. 370, 107206, 39 (2020). https://doi.org/10.1016/j.aim.2020.107206
Joyal, A., Kock, J.: Feynman graphs, and nerve theorem for compact symmetric multicategories (extended abstract). Electron. Notes Theor. Comput. Sci. 270(2), 105–113 (2011). https://doi.org/10.1016/j.entcs.2011.01.025
Kaufmann, R.M., Ward, B.C.: Feynman categories. Astérisque (387), vii+161 (2017)
Kelly, G.M.: On the operads of J. P. May. Repr. Theory Appl. Categ. 13, 1–13 (2005)
van der Laan, P.: Coloured Koszul duality and strongly homotopy operads (2003). arXiv:math/0312147v2 [math.QA]
Leinster, T.: Higher operads, higher categories, London Mathematical Society Lecture Note Series, vol. 298. Cambridge University Press, Cambridge (2004). https://doi.org/10.1017/CBO9780511525896
Lukács, A.: Cyclic operads, dendroidal structures, higher categories. Ph.D. thesis, Universiteit Utrecht (2010)
Raynor, S.: Compact symmetric multicategories and the problem of loops. Ph.D. thesis, University of Aberdeen (2018)
Rezk, C.: Spaces of algebra structures and cohomology of operads. Ph.D. thesis, Massachusetts Institute of Technology (1996)
Smirnov, V.: On the cochain complex of topological spaces. Math. USSR Sbornik 43, 133–144 (1982). https://doi.org/10.1070/sm1982v043n01abeh002437
Street, R.: The formal theory of monads. J. Pure Appl. Algebra 2, 149–168 (1972). https://doi.org/10.1016/00224049(72)900199
Templeton, J.J.: Selfdualities, graphs and operads. Ph.D. thesis, University of Cambridge (2003)
Ward, B.C.: Six operations formalism for generalized operads. Theory Appl. Categ. 34(6), 121–169 (2019)
Wraith, G.C.: Algebraic theories. Lectures Autumn 1969. Lecture Notes Series, No. 22. Matematisk Institut, Aarhus Universitet, Aarhus (1970)
Yau, D.: Infinity operads and monoidal categories with group equivariance (2019). arXiv:1903.03839v1 [math.CT]
Yau, D., Johnson, M.W.: A Foundation for PROPs, Algebras, and Modules, Mathematical Surveys and Monographs, vol. 203. American Mathematical Society, Providence (2015). https://doi.org/10.1090/surv/203
Acknowledgements
The authors would like to thank Michael Batanin, Rune Haugseng, Robin Koytcheff, Damien Lejay, Marcy Robertson, Ben Ward, and Donald Yau for useful discussions. We would also like to thank Gavin Wraith for pointing out his earlier related work. We are grateful to the referee, whose comments helped us improve the clarity and precision of this paper.
Funding
The first author was supported by IBSR003D1. The second author acknowledges the support of Australian Research Council Discovery Project Grant DP160101519.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Examples of colored operads
Examples of colored operads
Colored operads which describe various types of generalized operads are built on the notion of graphs with loose ends, which we will just call graphs in what follows. In such graphs, edges need not be attached to anything at one or both edges, or may be attached to themselves, forming a circle. A picture is instructive, and we have included one in Fig. 4. Any such graph consists of

A finite set of vertices V.

A finite set of edges E.

For each vertex v, a set \({{\,\mathrm{nb}\,}}(v)\) of germs of edges at the vertex; there is a function \(\coprod _{v\in V} {{\,\mathrm{nb}\,}}(v) \rightarrow E\) whose fibers have cardinality less than three.

A boundary set B, equipped with a function \(B \rightarrow E\) whose fibers have cardinality less than three.
The only additional condition to be a graph is that each fiber of \(B \amalg \coprod _{v\in V} {{\,\mathrm{nb}\,}}(v) \rightarrow E\) has cardinality either zero or two. If e is an edge such that this fiber is empty, we regard e as being like a circle, while if two elements of B map to e, we regard e as being like an interval. Alternative presentations of such graphs may be found in [2, Definition 13.1], [14, Definition 1.1], and [29, Sect. 1.2]. There is a realization functor from graphs to topological spaces whose details we omit; when we refer to topological properties of a graph we always implicitly use this functor. In what follows we always restrict to connected graphs. An ordered graph is a graph G with the following additional structure:

A total ordering on the set of vertices V.

A total ordering on each \({{\,\mathrm{nb}\,}}(v)\).

A total ordering on the boundary B.
The following is a special case of [21, §4.5], when C is a point. Raynor’s term ‘CSM’ refers to what we call ‘modular operad’ in this paper, whereas what Raynor calls a ‘modular operad’ we would call a ‘nonunital modular operad’.
Example A.1
(Modular operads) Let \(\mathscr {M}\) be the \({\mathbb {N}}\)colored collection whose elements are isomorphism classes of ordered graphs. Specifically, an element of \(\mathscr {M}(k_1, \dots , k_n; p)\) will be represented by an ordered graph G with n vertices \(\{v_1, \dots , v_n\}\) so that \({{\,\mathrm{nb}\,}}(v_j)\) has cardinality \(k_j\) and B(G) has cardinality p. There is a function
which replaces the \(v_j\in V(G)\) with a graph \(H_j\) with the gluing specified by the unique ordered bijection \(B(H_j) \rightarrow {{\,\mathrm{nb}\,}}(v_j)\). This type of graph substitution is both unital (with respect to corollas) by [29, Lemma 5.31] and associative by [29, Theorem 5.32], hence \(\mathscr {M}\) is a colored operad. Algebras over \(\mathscr {M}\) are a kind of modular operad.
The underlying category of \(\mathscr {M}\) consists of ordered graphs with precisely one vertex. Each edge will either be loop at the vertex, or be connected at one end to v. Then an equivalent presentation of this category has objects \({\mathbb {N}}\) and morphisms from k to p consisting of the data:

1.
an involution \(\iota \) on [k] with precisely p fixed points (the boundary edges) and \(\frac{kp}{2}\) free orbits (the loops), and

2.
a bijection between the fixed points of \(\iota \) with [p].
In particular the set of morphisms is nonempty if and only if \(k\ge p\) and \(k\equiv p\pmod 2\).
The following example can be recovered from the Feynman category of [16, §2.3.3] using the biequivalence of [1, Theorem 5.16].
Example A.2
(A genusaware version) We also have \(\mathscr {M}^{\mathrm {g}}\), which is a \({\mathbb {N}}^2\)colored collection defined as follows. Let \(\mathscr {M}^{\mathrm {g}}((k_1,g_1),\ldots , (k_n,g_n);(p,g))\) be the set of isomorphism classes of ordered graphs as before but where the graph G is restricted to have first betti number \(g\sum g_j\). That is, operations of \(\mathscr {M}^{\mathrm {g}}\) are genusdecorated graphs. The composition map defined for \(\mathscr {M}\) respects genus appropriately, making \(\mathscr {M}^{\mathrm {g}}\) an operad. There is a map of operads \(\mathscr {M}^{\mathrm {g}}\) to \(\mathscr {M}\) which on color sets is projection on the first factor, \((k,g)\mapsto k\). In the underlying category of \(\mathscr {M}^{\mathrm {g}}\), the set of morphisms from (k, g) to (p, h) is given by \(\mathscr {M}(k;p)\) when \(kp = 2(hg)\), while in other cases it is empty.
Example A.3
(Cyclic operads) Let \(\mathscr {C}\) be the suboperad of \(\mathscr {M}\) with the same color set consisting of ordered graphs which are simplyconnected. Algebras over \(\mathscr {C}\) are a variant of cyclic operads. They are slightly more general than the cyclic operads of [11] because they contain “constants” in level 0 and two “elements” in level 1 can be paired to give a constant.
We can define a further suboperad \(\mathscr {C}_{\mathrm {GK}}\) recovering Getzler and Kapranov’s cyclic operads precisely. This suboperad has colors the positive integers, and its elements have the additional restriction that the boundary set B is nonempty. A version of \(\mathscr {C}_{\mathrm {GK}}\) appeared in [20, §1.6.4].
In both cases, the underlying category is a disjoint union of symmetric groups. That is, the morphisms between different colors are empty, while the endomorphisms of n are the symmetric group \(\varSigma _n\).
Example A.4
(Operads) We further restrict \(\mathscr {C}_{\mathrm {GK}}\subseteq \mathscr {M}\) to give an operad \(\mathscr {O}\) governing monochrome operads. This is a variant of the description of [4, §1.5.6]. Our presentation is slightly more complicated but has the virtue of having a direct relationship with \(\mathscr {C}\) and \(\mathscr {M}\). See also [7, §1.2] and [29, §14.1].
Suppose that G is an ordered graph in \(\mathscr {C}_{\mathrm {GK}}\), that is, suppose that G is a tree with at least one boundary element. There is a unique edge flow in the direction of the first element of B(G), which we call the root. That is, we have a partial order with the root as the minimal element. This allows us to declare that the root of a vertex v is the element of \({{\,\mathrm{nb}\,}}(v)\) that is nearest to the global root. We call G a rooted tree just when, for each v, the root of v is also the minimal element of \({{\,\mathrm{nb}\,}}(v)\). We declare that \(\mathscr {O} \subseteq \mathscr {C}_{\mathrm {GK}}\) is the collection of all rooted trees. Algebras over \(\mathscr {O}\) are operads.
The underlying category is again a disjoint union of symmetric groups. But in the underlying category of \(\mathscr {O}\), the endomorphisms of n are the symmetric group \(\varSigma _{n1}\) (the root remains fixed).
Let us give a derived example. There is a suboperad \(\mathscr {O}_{\mathrm {ns}}\) with colors again the positive integers, but fewer operations in most profiles. Namely, for a rooted tree to be in \(\mathscr {O}_{\mathrm {ns}}(k_1,\dots , k_n; p)\), the orderings on B must be compatible with the orderings on each \({{\,\mathrm{nb}\,}}(v)\). Precisely, suppose that \(a_1\) and \(a_2\) are elements of \({{\,\mathrm{nb}\,}}(v)\) and \(b_1\) and \(b_2\) are elements of B so that the image of \(b_i\) in E is greater than or equal to the image of \(a_i\) in the partial order on E. Compatibility means that if \(a_1 < a_2\) in the total ordering on \({{\,\mathrm{nb}\,}}(v)\), then \(b_1 < b_2\) in the total ordering on B. Algebras over \(\mathscr {O}_{\mathrm {ns}}\) are nonsymmetric operads.
The underlying category of \(\mathscr {O}_{\mathrm {ns}}\) has only the identity in each color because, for a graph with one vertex, the compatibility condition forces the orders on \(B \cong E\) and \({{\,\mathrm{nb}\,}}(v) \cong E\) to coincide. This version was studied by van der Laan [18].
Our convention for the colors of \(\mathscr {O}\) are shifted by one from all conventions in the literature. We make this nonstandard choice because we are interested in the comparison with \(\mathscr {C}\) and \(\mathscr {M}\) where this shift is natural.
The operad \(\mathscr {O}\) has operations given by rooted trees. One can imagine analogous operads whose operations are other kinds of directed graphs and whose algebras are dioperads, properads, wheeled operads, and so on. A general construction of such operads is included in Sect. 14.1 of [29], so we will omit further details here.
Example A.5
(Colored variants) Given a set \({\textsf {A}}\) of colors, one can form an operad \(\mathscr {O}^{\textsf {A}}\) whose set of colors is \(\coprod _{n \ge 1} {\textsf {A}}^{\times n}\) and whose operations are isomorphism classes of ordered rooted trees equipped with a function from the set of edges to the set \({\textsf {A}}\). Algebras over this \(\coprod _{n \ge 1} {\textsf {A}}^{\times n}\)colored operad are precisely \({\textsf {A}}\)colored operads. When \({\textsf {A}}\) is a point, one recovers the operad \(\mathscr {O}\). The underlying category of \(\mathscr {O}^{\textsf {A}}\) is a groupoid of positive length lists of elements of \({\textsf {A}}\).
This same pattern extends in a straightforward way to other types of directed graphs, and actually falls under the general construction of [29, §14.1]. For operadic structures built on undirected graphs, like cyclic operads and modular operads, one has the flexibility to work with an involutive set of colors \({\textsf {A}}\). The main difference is that the coloring function \(E\rightarrow {\textsf {A}}\) should be replaced with an involutive function from the involutive set of oriented edges to \({\textsf {A}}\). See, e.g., [9, §2], [15, 21, §4.5], and [14, §2] for implementations of this involutive perspective.
Rights and permissions
About this article
Cite this article
DrummondCole, G.C., Hackney, P. Coextension of scalars in operad theory. Math. Z. 301, 275–314 (2022). https://doi.org/10.1007/s00209021028405
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209021028405