nLab cocartesian multicategory

Idea

The 2-category $Cat$ of categories embeds fully faithfully into the 2-category $Multicat$ of multicategories in two ways:

1. As the unary multicategories (i.e. in which every multimorphism has unary domain).
2. Via the “discrete cocones” construction, in which we define the multimorphisms of the multicategory $C_\blacktriangleright$ induced by a category $C$ by
   $$C_\blacktriangleright(X_1, \ldots, X_n; Y) := C(X_1, Y) \times \cdots \times C(X_n, Y)$$

A multicategory is isomorphic to one induced by the discrete cocones construction exactly when it is a generalized multicategory relative to the monad on Prof whose algebras are categories with (strict) finite coproducts, i.e. a “virtual cocartesian category”. Thus, such a multicategory is called a cocartesian multicategory.

An older name for a cocartesian multicategory is a sequential multicategory, since its multimorphisms are sequences of unary morphisms.

Equivalence of definitions

Theorem

For a symmetric multicategory $E$ the following are equivalent.

1. $E \cong C_\blacktriangleright$ for some category $C$ (which then must be the category of unary morphisms in $E$).

2. Every object of $E$ is a monoid object, in a way that commutes with all the multimorphisms of $E$.

3. Every object of $E$ is a unital magma object, in a way that commutes with all the multimorphisms of $E$.

4. $E$ is equipped with

   • “expansion” operations (the dual of contraction)
     $$E(\dots, x, \dots) \to E(\dots, x,x,\dots)$$
   • “strengthening” operations (the dual of weakening)
     $$E(\dots, x,\dots) \to E(\dots, \dots)$$

   satisfying certain evident axioms.

5. $E$ has the structure of a virtual $T$-algebra, where $T$ is the monad on Prof whose algebras are cocartesian strict monoidal categories.

The equivalence of (1)–(3) is Proposition 3.6 (3.7 in the arXiv version v1) of Pisani 14.

Proof

In (5), the additional structure in a virtual cocartesian category beyond that of a virtual symmetric monoidal category (that is, a symmetric multicategory) consists of generalized $n$-ary expansion/strengthening operations. These are generated by the binary and nullary operations displayed in (4), and the “evident axioms” in (4) are chosen precisely so as to make it equivalent to (5).

It is easy to check that (1) implies all the other conditions, and of course (2) implies (3).

Now assuming (3), the free symmetric monoidal category $F E$ generated by $E$, whose objects are finite lists of objects of $E$, also inherits the property stated in (3), along with compatibility with its monoidal structure. Therefore, by the dual of Arkor’s version of Fox's theorem, $F E$ is cocartesian monoidal. Since $E$ embeds fully in the underlying multicategory of $F E$, it follows that $E(x_1,\dots, x_n; y) \cong E(x_1;y) \times \cdots \times E(x_n;y)$, which is to say (1) holds.

It therefore suffices to show that (5) implies (3). The magma structure maps $mult_x \in E(x,x;x)$ and $unit_x \in E(;x)$ are obtained by expansion and strengthening from $1_x\in E(x;x)$. To prove the unitality axioms and naturality, we describe a bit more of the structure of $T$.

Write $n = \{1,\dots ,n\}$. For a category $A$, the category $T A$ has as objects finite lists of objects of $A$, and morphisms $(a_1,\dots, a_m) \to (a'_1,\dots ,a'_n)$ given by a function $\phi:m\to n$ and morphisms $a_{i} \to a'_{\phi(i)}$ for all $i\in m$. Similarly, for a profunctor $M:A$ ⇸ $B$, the profunctor $T M : T A$ ⇸ $T B$ has heteromorphisms $(b_1,\dots, b_m)$ ⇸ $(a_1,\dots, a_n)$ given by a function $\phi:m\to n$ and heteromorphisms $b_i$ ⇸ $a_{\phi(i)}$ for all $i\in m$.

Now a virtual $T$-algebra consists, in particular, of a category $A$ and a profunctor $M : A$ ⇸ $T A$. By Cruttwell-Shulman 2010, Theorem 8.6, since $T$ preserves bijective-on-objects functors, we may assume $A$ is a discrete category, so $M$ contains all the morphisms and multimorphisms. The generalized expansion/strengthening operations are then given by the profunctorial action of $T A$ on $M$.

We also have $T M : T A$ ⇸ $T T A$, so we have a covariant action of $T A$ on $T M$. For instance, if $f\in M(a_1,\dots,a_m; c)$ and $g\in M(b_1,\dots ,b_n; c)$, we have

$$(f,g) \in T M ( (a_1,\dots ,a_m), (b_1,\dots,b_n); (c,c)),$$

and we can act on this by the codiagonal $\delta_c : (c,c) \to (c)$ in $T A$ to obtain a morphism $(\delta_c)_*(f,g) \in T M((a_1,\dots ,a_m), (b_1,\dots,b_n); (c))$. Since the magma operation $mult_c : (c,c) \to c$ is by definition the contravariant action $1_c (\delta_c)^*$, by the equivariance of the composition map we have

$$\mult_c \circ (f,g) = \mu (1_c (\delta_c)^*, (f,g)) = \mu(1_c, (\delta_c)_*(f,g)).$$

Here $(\delta_c)_*(f,g) \in T M ((a_1,\dots ,a_m), (b_1,\dots,b_n); (c))$ is determined by the unique function $\phi : 2\to 1$ along with $f$ and $g$.

Now suppose $f=g$. Then we have $(f)(\delta_a)^* \in T M((a_1,\dots ,a_m),(a_1,\dots ,a_m); (c)$, which is also determined by the unique function $\phi : 2\to 1$ along with $f$ and $f$, and therefore is equal to $(\delta_c)_*(f,f)$. So we have

$$\mult_c \circ (f,f) = \mu(1_c, (\delta_c)_*(f,f)) = \mu(1_c, (f) (\delta_a)^*) = \mu(1_c, (f))(\delta_a')^* = f (\delta'_a)^*.$$

Here $\delta_a' \in T A( (a_1,\dots,a_m,a_1,\dots,a_m);(a_1,\dots, a_m)$ (as opposed to $\delta_a \in T T A ((a_1,\dots,a_m),(a_1,\dots,a_m)), ((a_1,\dots,a_m)))$), and the equation relating the actions by the two is because $\mathrm{mult}_T(\delta_a) = \delta'_a$. But we also have

$$f (\delta'_a)^* = \mu(f, (1_{a_1},\dots 1_{a_m}))(\delta_a')^* = \mu(f, (1_{a_1}\delta_{a_1}^*,\dots 1_{a_m}\delta_{a_m}^*)\sigma^* = \mu(f, (mult_{a_1},\dots ,mult_{a_m})\sigma^*$$

where $\sigma$ is an appropriate permutation. This shows that the magma multiplications are natural with respect to $f$. A similar argument applies to the units.

The proof of the unitality laws is less complicated. If $\epsilon_c : () \to c$ in $T A$, so that $\unit_c = 1_c (\epsilon_c)^*$, then

$$\mult_c \circ (\unit_c, 1_c) = \mu(1_c (\delta_c)^*, (1_c (\epsilon_c)^*, 1_c)) = \mu(1_c (\delta_c)^*, (1_c,1_c))(\epsilon'_c)^* = 1_c (\delta_c)^* (\epsilon'_c)^* = 1_c.$$

Here $\epsilon'_c : (c) \to (c,c)$ in $T A$ inserts into the right-side $c$, and the final equation is by functoriality.

Properties

• Being cocartesian is a property of a multicategory, in contrast to being a cartesian multicategory, which is structure.

• Every cocartesian multicategory has the structure of a symmetric multicategory.

• A cocartesian multicategory $C_\blacktriangleright$ is representable if and only if $C$ has coproducts (Example 8.8(2) of Hermida 2000). This is another motivation for the terminology.

• A category $C$ is preadditive if and only if its corresponding cocartesian multicategory $C_\blacktriangleright$ is cartesian (see Pisani 2013 and Pisani 2014); and is hence additive if and only if its corresponding multicategory is cartesian and representable.

• The discrete cocones construction $(-)_\blacktriangleright : Cat \to Multicat$ is left adjoint to the category of monoids construction $Mon : Multicat \to Cat$, and this restricts to cartesian multicategories. In fact, this is a consequence of the more general fact that $Multicat$ is copowered over $Cat$: see Pisani 2013 and Pisani 2014.

References

The discrete cocones construction was introduced in Example 2.2(2) of:

• Claudio Hermida, Representable multicategories, Adv. Math. 151 (2000), no. 2, 164-225 (pdf)

and studied extensively in the following papers, where the terminology sequential multicategory is introduced:

• Claudio Pisani, Some remarks on multicategories and additive categories, arXiv:1304.3033 (2013).

• Claudio Pisani, Sequential multicategories, Theory and Applications of Categories 29.19 (2014), arXiv:1402.0253

It is further studied, under the name cocartesian multicategory in:

• Claudio Pisani, Unbiased multicategory theory, Theory and Applications of Categories, Vol. 44, 2025, No. 28, pp 826-868. web

The discrete cocones construction is also discussed in Example 2.1.6 of Higher Operads, Higher Categories.

An analogue of sequential multicategories for (infinity, 1)-categories is contained in §2.4.3 of Higher Algebra.

The sequential multicategory structure was rediscovered in the one-object setting as the “T construction” of:

• Samuele Giraudo, Combinatorial operads from monoids, Journal of Algebraic Combinatorics 41 (2015): 493-538.

A generalisation of the concept to generalised multicategories is given in:

• Alex Cebrian, Plethysms and operads, Collectanea Mathematica 75.1 (2024): 247-303.

References for generalized multicategories:

• Geoff Cruttwell and Mike Shulman, A unified framework for generalized multicategories, Theory Appl. Categ. 24 21 (2010) 580-655 [arxiv/0907.2460, TAC:24-21]