Here is a *tentative* definition based on finitary unbiased monoidal categories – I make no claims of usefulness!

An **infinitary unbiased monoidal category** consists of the following data:

An ordinary category $\mathcal{C}$.

For each small ordinal $\alpha$, a functor $T_\alpha : \mathcal{C}^\alpha \to \mathcal{C}$, which maps a $\alpha$-sequence $(A_0, A_1, \ldots)$ to $A_0 \otimes A_1 \otimes \cdots$.

A natural isomorphism $\textrm{id}_\mathcal{C} \Rightarrow T_1$.

For each partition $\alpha = \sum_{i < \gamma} \beta_i$, a natural isomorphism
$$T_\gamma \circ (T_{\beta_0} \times T_{\beta_1} \times \cdots ) \Rightarrow T_\alpha$$
such that for each double partition $\alpha = \sum_{i < \gamma} \sum_{j < \delta_i} \beta_{i,j}$, where $\beta_i = \sum_{j < \delta_i} \beta_{i,j}$ and $\delta = \sum_{i < \gamma} \delta_i$, the two composites
\begin{multline}
T_\gamma \circ (T_{\delta_0} \circ (T_{\beta_{0,0}} \times T_{\beta_{0,1}} \times \cdots) \times T_{\delta_1} \circ (T_{\beta_{1,0}} \times T_{\beta_{1,1}} \times \cdots) \times \cdots) \newline
\Rightarrow T_\gamma \circ (T_{\beta_0} \times T_{\beta_1} \times \cdots) \Rightarrow T_\alpha
\end{multline}
and
\begin{multline}
T_\gamma \circ (T_{\delta_0} \circ (T_{\beta_{0,0}} \times T_{\beta_{0,1}} \times \cdots) \times T_{\delta_1} \circ (T_{\beta_{1,0}} \times T_{\beta_{1,1}} \times \cdots) \times \cdots) \newline
\Rightarrow T_\delta \circ ((T_{\beta_{0,0}} \times T_{\beta_{0,1}} \times \cdots) \times (T_{\beta_{1,0}} \times T_{\beta_{1,1}} \times \cdots) \times \cdots) \Rightarrow T_\alpha
\end{multline}
are equal, and the two composites
$$T_\alpha \Rightarrow T_\alpha \circ (T_1 \times T_1 \times \cdots ) \Rightarrow T_\alpha$$
$$T_\alpha \Rightarrow T_1 \circ T_\alpha \Rightarrow T_\alpha$$
are the identity natural transformation on $T_\alpha$.

An **infinitary lax monoidal category** is what we get if we replace "natural isomorphism" by "natural transformation" in the above. An **infinitary strict monoidal category** is what we get if we replace "natural isomorphism" by "identity".

This is just a straightforward generalisation of the definition appearing in [Leinster, *Higher operads, higher categories*, §3.1]. The reason why this even makes sense is that ordinal addition is associative – if it weren't, we'd be stuck. This leads us to our first easy example:

**Example.** The category of small ordinals (and monotone maps) is a infinitary strict monoidal category under ordinal addition.

Less tautologically:

**Example.** Any (co)complete category is an infinitary unbiased monoidal category under (co)products.

(Though, in some sense we assumed this in our definition...)

**Conjecture.** Martin's third example probably works if we assume that $X \otimes (-)$ preserves sequential colimits, and that the chosen morphisms $I \to X$ are sufficiently nice. First, by the Mac Lane–Kelly coherence theorem, we can make a finitary biased monoidal category into an unbiased one, and we can use the colimit construction to define the infinitary monoidal products. We use the assumption on $X \otimes (-)$ to prove that, e.g.
$$A_0 \otimes (A_1 \otimes A_2 \otimes \cdots) \cong A_0 \otimes A_1 \otimes A_2 \otimes \cdots$$
The universal property of colimits *should* be enough to ensure the coherence of the infinitary associators – but I haven't checked. We need to assume something about the choice of morphisms $I \to X$ so that
$$I \otimes I \otimes I \otimes \cdots \cong I$$
holds. (For example, choosing a non-isomorphism for $I \to I$ is a bad idea!)

**Non-example.** Take $\mathcal{C}$ to be the category of small ordinals, and take $T_\alpha$ to be ordinal addition for all finite ordinals $\alpha$, and $T_\alpha = 0$ for all infinite ordinals $\alpha$. This is not an infinitary unbiased monoidal category, because
$$T_\omega \circ (T_1 \times T_1 \times T_1 \times T_0 \times T_0 \times \cdots ) \ncong T_3$$
and so we see that there is *some* form of continuity required.

Also, a reminder about a famous trick:

**Remark.** Suppose we define infinitary monoids to be *discrete* infinitary strict monoidal categories. There are non-trivial *small* infinitary monoids: for example, take any complete small semilattice with $\sup$ as the monoid operation. However, any infinitary monoid that is also a group must be the trivial monoid: after all, for any element $x$,
$$x^{-1} \cdot (x \cdot x \cdot x \cdot \cdots) = (x^{-1} \cdot x) \cdot (x \cdot x \cdot x \cdot \cdots) = x \cdot x \cdot x \cdot \cdots$$
and then cancel $x \cdot x \cdot x \cdot \cdots$ on both sides of the equation to obtain $x^{-1} = \textrm{id}$.

**(EDIT) Non-example.** The tensor product of modules doesn't give an infinitary monoidal product. To be precise, if we define
$$\textrm{Hom}_R (A_0 \otimes_R A_1 \otimes_R A_2 \otimes_R \cdots, B) \cong \textrm{Multi}_R(A_0, A_1, A_2, \ldots ; B)$$
where by $R$-multilinear we mean that $f(\ldots, r a, \ldots) = r f (\ldots, a, \ldots)$, then I don't see how $R \otimes_R R \otimes_R R \otimes_R \cdots$ can be isomorphic to $R$. (A multilinear map $R \times R \times R \times \cdots \to B$ is not necessarily determined by just what it does to $(1, 1, 1, \ldots)$, unlike the finite case.)

Now, some closing remarks:

How do we define an "infinitary braided/symmetric monoidal category" as extra structure on top of an infinitary unbiased monoidal category? The trouble is that a permutation of an infinite ordinal can change its order type (e.g. $\omega + \omega_1 = \omega_1 \ne \omega_1 + \omega$)... but this probably isn't a big problem. It seems to me that the morally correct way of defining an "infinitary unbiased symmetric monoidal category" would define a functor $S_\kappa : \mathcal{C}^\kappa / \textrm{Sym}(\kappa) \to \mathcal{C}$ for each cardinal $\kappa$ and some natural isomorphisms to the various $T_\alpha$.

Is there a sensible notion of an "infinitary biased monoidal category"? We would need to define $T_\alpha$ for at least $\alpha = 0$, $\alpha = 2$, and every infinite regular ordinal $\alpha$. This is still a large amount of data! For the finitary fragment we can adopt the same coherence axioms, but I don't know what the coherence axioms for the infinitary fragment ought to be. There will have to be some new ones that don't show up in the finitary fragment: For example, we would need an axiom like
$$I \otimes I \otimes I \otimes \cdots \cong I$$
for the monoidal unit $I$ – because inductively applying the left unitor can never delete infinitely many copies of $I$; or we could instead make
$$A_0 \otimes A_1 \otimes \cdots \otimes A_n \otimes I \otimes I \otimes I \otimes \cdots \cong A_0 \otimes A_1 \otimes \cdots \otimes A_n$$
into an axiom, because we can't even apply the right unitor here!

Higher operads, higher categories. This seems easy to generalise in comparison to the traditional one involving only a monoidal unit and a binary monoidal product. The coherence axioms are based on partitions, as Buschi Sergio indicated. $\endgroup$ – Zhen Lin Sep 8 '12 at 15:40usesa transfinite composition, and so do the largest Hausdorff quotient, the associated sheaf, and colimits of algebras. Transfinite composition is of course a widely applicable technique. But are you claiming that those three other constructions are actually infinitary tensor products? $\endgroup$ – Mike Shulman Sep 8 '12 at 19:51