join of categories



The join of two categories CC and CC' is obtained from the disjoint union of CC with CC' by throwing in a unique morphism from every object of CC to every object of CC'.


The join of categories CC and CC' is the category with

  • objects Obj(CC):=Obj(C)⨿Obj(C)Obj(C \star C') := Obj(C) \amalg Obj(C');

  • morphisms given by

    Mor CC(a,b):={Mor C(a,b) if a,bC Mor C(a,b) if a,bC if aC,bC; pt if aC,bC; Mor_{C \star C'}(a,b) := \left\lbrace \array{ Mor_C(a,b) & \text{ if }\;\; a,b \in C \\ Mor_{C'}(a,b) & \text{ if }\;\; a,b \in C'\\ \emptyset & \text{ if }\;\; a \in C', b \in C; \\ \mathrm{pt} & \text{ if } \;\; a \in C, b \in C'; } \right.

In terms of profunctors

The join of categories C,CC,C' can also be described to be the cograph of the unique profunctor W:CCW\colon C ⇸ \; C' sending all objects (c,c)(c,c') to the terminal set (the definition speaks for itself).

In terms of adjoint functors

Consider the inclusion of the boundary of the standard 1-simplex, i:{0,1}[1]i\colon \{0,1\}\to [1] as a functor between the discrete category with two elements and the walking arrow I={01}I=\{0 \leq 1\}. It induces a functor

i *:Cat/ICat×Cat i^*\colon \Cat / I \to \Cat \times \Cat

which admits a right adjoint. This right adjoint is precisely the bifunctor :Cat×CatCat/I\star\colon \Cat \times \Cat \to \Cat / I, once we noticed that the category CCC\star C' comes naturally equipped with an arrow CCI=11C\star C'\to I=1\star 1 induced by (bi)functoriality of \star, starting from the canonical arrows C1,C1C\to 1, C'\to 1 to the terminal category.


It is quite clear that i *i^* is defined by sending CIC\to I to the pair of categories i (0)=C 0,C 1=i (1)i^\leftarrow(0)=C_0, C_1=i^\leftarrow(1). The bijection

Cat 2(i *(CI),(A,B))Cat/I(C,AB) \Cat^{\mathbf{2}}\Big(i^*\big( C\to I\big), (A,B)\Big)\cong \Cat / I \Big( C, A\star B \Big)

is now rather obvious, since any functor i *(C I)(A,B)i^* \Big( \array{ C\\ \downarrow \\ I } \Big) \to (A,B) determines a functor CABC\to A\star B and viceversa.


  • If the small categories C,CC,C' are two posets, their join consists of their ordinal sum;
  • The monoidal structure induced on Cat\Cat by \star is not symmetric (if it was, then the right cone of CC would be equivalent to the left cone, which is blatantly false);
  • The monoidal structure induced on Cat\Cat by \star is not closed, since the functor AA\star - does not preserve colimits.
  • The functor C:CatC/Cat:A(AAC)-\star C'\colon \Cat \to C'\!/\!\Cat\colon A\mapsto (A\to A\star C') is a left adjoint, and similarly is the functor CC\star-; see Joyal, Ch. 3.1.1-2 for a detailed description.


  • The cone below a category CC is the join CptC \star \mathrm{pt}. The cone above CC is the join ptC\mathrm{pt} \star C.


See p. 42 of

See also Ch. 3 of

Last revised on November 20, 2015 at 12:14:27. See the history of this page for a list of all contributions to it.