Why is the smash product in a certain slice category associative?

I am trying to show the following. Let $(V, \times , \ast)$ be a cartesian closed symmetric monoidal category (so the monoidal product is the ordinary product). Consider the category $V_\ast$ of based objets, objets under $\ast$, the terminal objet of $V$. One then defines the smash produt in $V_\ast$ of v and w to be the pushout

$$\require{AMScd} \begin{CD} v \amalg w @>>> v \times w ;\\ @VVV @VVV \\ \ast @>>> v \wedge w. \end{CD}$$

Appearently, this is supposed to follow because $V$ is cartesian closed and by this, $v \times -$ preserves colimits. But I don't see how this gives my proposition, unfortunately.

Anyhelp would be appreciated.

I think the best way to proceed is to construct the internal hom for $\mathcal{V}_*$. Writing $[-, -]$ for the internal hom of $\mathcal{V}$, we have the following equaliser diagram in $\mathcal{V}$, $$\mathrm{Hom}_* (V, W) \rightarrow [V, W] \rightrightarrows W$$ where one arrow $[V, W] \to W$ is induced by the distinguished point $1 \to V$, and the other arrow $[V, W] \to W$ is induced by the distinguished point $1 \to W$. The distinguished point of $\mathrm{Hom}_* (V, W)$ corresponds to the zero morphism $V \to W$, of course. One then has to establish the following natural isomorphism: $$\mathrm{Hom}_* (U \wedge V, W) \cong \mathrm{Hom}_* (U, \mathrm{Hom}_* (V, W))$$ Thus, \begin{align} \mathrm{Hom}_* (U \wedge (V \wedge W), X) & \cong \mathrm{Hom}_* (U, \mathrm{Hom}_* (V \wedge W, X)) \\ & \cong \mathrm{Hom}_* (U, \mathrm{Hom}_* (V, \mathrm{Hom}_* (W, X))) \\ & \cong \mathrm{Hom}_* (U \wedge V, \mathrm{Hom}_* (W, X)) \\ & \cong \mathrm{Hom}_* ((U \wedge V) \wedge W, X) \end{align} so we have a natural isomorphism $U \wedge (V \wedge W) \cong (U \wedge V) \wedge W$. One also has to check the various coherence axioms, but this should be straightforward.

It is also possible to bypass the construction of the internal hom and simply work with the natural bijection $$\mathcal{V}_* (U \wedge V, X) \cong \{ \text{bipointed morphisms } U \times V \to X \}$$ where a bipointed morphism is a morphism in $\mathcal{V}$ that preserves the distinguished point in each variable separately. The aim is to show that both $U \wedge (V \wedge W)$ and $(U \wedge V) \wedge W$ represent the functor of tripointed morphisms.

I suppose it may be possible to verify the associativity of $\wedge$ using only the fact that $\times$ preserves colimits in each variable separately, but this looks considerably more tedious than either approach outlined above.

• Thank you for a nice answer! – user101036 Feb 22 '14 at 14:57

Zhen writes: "I suppose it may be possible to verify the associativity of ∧ using only the fact that × preserves colimits in each variable separately, but this looks considerably more tedious than either approach outlined above."

Actually it is straight forward: $U \wedge (V \wedge W)$ is the quotient of $U \times (V \wedge W)$ which identifies $* \times (V \wedge W)$ with $U \times * \times *$. But $U \times (V \wedge W)$ is the quotient of $U \times V \times W$ which identifies $U \times * \times W$ with $U \times V \times *$ (in particular $U \times * \times *$). It follows that $U \wedge (V \wedge W)$ is the quotient of $U \times V \times W$ which identifies $* \times V \times W$ with $U \times V \times *$ and $U \times * \times W$. The reasoning applies to $(U \wedge V) \wedge W$. In fact, this shows that both are isomorphic to the object $U \wedge V \wedge W$ which classifies tripointed morphisms.

By the way, this whole construction can be seen as a special case of a more general theory about monoidal monads (see for example arXiv/1205.0101) applied to the monoidal monad $X \mapsto * + X$.

Finally let me mention that we cannot apply this to $(\mathsf{Top},\times)$, since here $\times$ doesn't preserve colimits in each variable. In fact, the smash product of pointed spaces is not associative in general (proof?).

• This is also nice. I am a bit unsure (although it is surely true) with talking about quotients like this in general categories. How could I, diagrammatially, formulate this? I understand that it is tedious to write out but I would be very grateful for some more details! – user101036 Feb 23 '14 at 12:12
• for example, how $\times$ being colimit preserving is rucial here. – user101036 Feb 23 '14 at 12:12
• It is not tedious at all. I just apply $U \times -$ to the pushout diagram defining $V \wedge W$. It stays a pushout diagram. So we get a pushout description for $U \times (V \wedge W)$. In general, in a category with a zero object $*$, and a morphism $A \to B$, the quotient $B/A$ is defined to be the cokernel of $A \to B$. One says "quotient of $B$ by $A$". If $A$ is a coproduct of $A_1$ and $A_2$, one can also say "quotient of $B$ by $A_1$ and $A_2$". – Martin Brandenburg Feb 23 '14 at 15:09
• OK - just to make a few last things clear : you say that $U \times (V \wedge W)$ is the quotient of $U \times V \times W$ which identifies... For quotient, don't we need a cokernel (i.e pushout along the zero object)? In this case, we pushout along U. I agree that for example in Sets, Top this description works but I am a bit uneasy in general categories like these, so please excuse my ignorance. – user101036 Feb 24 '14 at 13:50
• Like, wy can we say that in a general category, $U \times (V \wedge W)$ is the quotient of $U \times V \times W$ which identifies $U \times \ast \times W$ with $U \times V \times \ast$? – user101036 Feb 24 '14 at 13:54