Applied Categorical Structures

, Volume 23, Issue 4, pp 543–573 | Cite as

On the Category of Props

  • Philip Hackney
  • Marcy RobertsonEmail author


The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. The titular category has nice formal properties: it is bicomplete and is a symmetric monoidal category, with monoidal product closely related to the Boardman-Vogt tensor product of operads. Tools developed in this article, which is the first part of a larger work, include a generalized version of multilinearity of functors, a free prop construction defined on certain “generalized” graphs, and the relationship between the category of props and the categories of permutative categories and of operads.


Colored operad Colored prop Multicategory Permutative category 


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  1. 1.
    Barr, M., Wells, C.: Toposes, triples and theories. Repr. Theory Appl. Categ. 12, x+288 (2005). Corrected reprint of the 1985 original [MR0771116]. MR 2178101MathSciNetGoogle Scholar
  2. 2.
    Berger, C., Moerdijk, I.: Resolution of coloured operads and rectification of homotopy algebras, categories in algebra, geometry and mathematical physics. Contemp. Math. Amer. Math. Soc. Providence RI 431, 31–58 (2007). MR 2342815 (2008k:18008)MathSciNetGoogle Scholar
  3. 3.
    Boardman, J.M., Vogt, R.M.: Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Mathematics, vol. 347. Springer-Verlag, Berlin (1973). MR 0420609 (54 #8623a)Google Scholar
  4. 4.
    Cisinski, D.-C., Moerdijk, I.: Dendroidal sets as models for homotopy operads. J. Topol. 4(2), 257–299 (2011). MR 2805991. doi: 10.1112/jtopol/jtq039 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cisinski, D.-C., Moerdijk, I.: Dendroidal Segal spaces and ∞-operads. J. Topol. 6(3), 675–704 (2013). doi: 10.1112/jtopol/jtt004 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cisinski, D.-C., Moerdijk, I.: Dendroidal sets and simplicial operads. J. Topol. 6(3), 705–756 (2013). doi: 10.1112/jtopol/jtt006 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hess, K., Voronov, A., Cohen, R.L.: String Topology and Cyclic Homology. Adv. Courses in Math CRM Barcelona. Birkhäuser, Basel (2006). Translated from the Japanese by M. ReidGoogle Scholar
  8. 8.
    Elmendorf, A.D., Mandell, M.A.: Rings, modules, and algebras in infinite loop space theory. Adv. Math. 205(1), 163–228 (2006). MR 2254311 (2007g:19001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Elmendorf, A.D., Mandell, M.A.: Permutative categories, multicategories and algebraic K-theory. Algebr. Geom. Topol. 9(4), 2391–2441 (2009). MR 2558315 (2011a:19002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Frégier, Y., Markl, M., Yau, D.: The \(L_\infty \)-deformation complex of diagrams of algebras, New York. J. Math. 15, 353–392 (2009). MR 2530153 (2011b:16039)zbMATHGoogle Scholar
  11. 11.
    Fresse, B.: Props in model categories and homotopy invariance of structures. GeorgianMath. J. 17(1), 79–160 (2010). MR 2640648 (2011h:18011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gallo, G., Longo, G., Pallottino, S., Nguyen, S.: Directed hypergraphs and applications. Discret. Appl. Math. 42(2–3), 177–201 (1993). Combinatorial structures and algorithms. MR 1217096 (94e:05187)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ginzburg, V., Kapranov, M.: Koszul duality for operads. Duke Math. J. 76(1), 203–272 (1994). MR 1301191 (96a:18004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hackney, P., Robertson, M.: The homotopy theory of propsGoogle Scholar
  15. 15.
    Johnson, M.W., Yau, D.: On homotopy invariance for algebras over colored PROPs. J. Homotopy Relat. Struct. 4(1), 275–315 (2009). MR 2559644 (2010j:18014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Leinster, T.: Higher Operads, Higher Categories. London Mathematical Society Lecture Note Series, vol. 298. Cambridge University Press, Cambridge (2004). MR 2094071 (2005h:18030)CrossRefGoogle Scholar
  17. 17.
    Mac Lane, S.: Categorical algebra. Bull. Am. Math. Soc. 71, 40–106 (1965). MR 0171826 (30 #2053)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mac Lane, S.: Categories for the working mathematician, 2nd edn.Graduate Texts in Mathematics, vol. 5. Springer-Verlag, New York (1998). MR 1712872 (2001j:18001)Google Scholar
  19. 19.
    Markl, M.: Operads and PROPs. Handbook of Algebra, vol. 5, pp. 87–140. Elsevier, North-Holland, Amsterdam (2008). MR 2523450 (2010j:18015)Google Scholar
  20. 20.
    Markl, M., Shnider, S., Stasheff, J.: Operads in Algebra, Topology and Physics. Mathematical Surveys and Monographs, vol. 96. American Mathematical Society, Providence, RI (2002). MR 1898414 (2003f:18011)Google Scholar
  21. 21.
    Peter May, J.: May, The Geometry of Iterated Loop Spaces. Lectures Notes in Mathematics, vol, 271. Springer-Verlag, Berlin (1972). MR 0420610 (54 #8623b)Google Scholar
  22. 22.
    Moerdijk, I.: Lectures on dendroidal sets. Simplicial methods for operads and algebraic geometry. Adv. Courses Math, pp. 1–118. CRM Barcelona, Birkhauser/Springer Basel AG, Basel (2010). Notes written by Javier J. Gutierrez. MR 2778589Google Scholar
  23. 23.
    Moerdijk, I., Weiss, I.: Dendroidal sets. Algebr. Geom. Topol. 7, 1441–1470 (2007). MR 2366165 (2009d:55014). doi: 10.2140/agt.2007.7.1441 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Moerdijk, I., Weiss, I.: On inner Kan complexes in the category of dendroidal sets. Adv. Math. 221(2), 343–389 (2009). MR 2508925 (2010a:55021). doi: 10.1016/j.aim.2008.12.015 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pirashvili, T.: On the PROP corresponding to bialgebras. Cah. Topol. Géom. Différ. Catég. 43(3), 221–239 (2002). MR 1928233 (2003i:18012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Segal, G.B.: The definition of conformal field theory. Differential geometrical methods in theoretical physics (Como, 1987). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Kluwer Acad. Publ. Dordrecht 250, 165–171 (1988). MR 981378 (90d:58026)Google Scholar
  27. 27.
    Torres, A.F., Aráoz, J.D.: Combinatorial models for searching in knowledge bases. Acta Cient. Venezolana 39(5–6), 387–394 (1988). MR 1005374 (90k:68162)MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California-RiversideRiversideUSA
  2. 2.Department of MathematicsUniversity of Western OntarioLondonCanada

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