A metric measure space $(X,d,\mu)$ consists of a metric space $(X,d)$ together with a Borel measure $\mu$. A coupling between metric measure spaces $(X_1,d_1,\mu_1)$ and $(X_2,d_2,\mu_2)$ consists of a Borel measure $\nu$ on the product metric space $X_1\times X_2$ such that $(\pi_1)_*(\nu) = \mu_1$ and $(\pi_2)_*(\nu) = \mu_2$.

I would like to see composition of couplings between metric measure spaces (explained below) as a pullback construction, in the same way that composition of correspondences between sets can be seen as a pullback.

Classical set up for composition of couplings. I follow Section 6.2 of [1]. There, it is stated that a coupling $\nu$ between measures $\mu,\mu'$ on Polish metric spaces $X$ and $X'$ can be written in the form $\mu(\mathbb{d} x)Q_\nu(x,\mathbb{d}x)$, where $Q_\nu$ is a Markov kernel from $X$ to $X'$. In particular, if $\mu_1,\mu_2,\mu_3$ are probability measures on compact spaces $X_1$, $X_2$, and $X_3$, and $\nu_1$ and $\nu_2$ are couplings between $\mu_1,\mu_2$ and $\mu_2,\mu_3$ respectively, one can define a Markov chain $(A_1,A_2,A_3)$ where $A_i$ is a random variable with distribution $\mu_i$ on $X_i$, with initial distribution $\mu_1$, and with transition kernel $Q_{\nu_i}$ at step $i$. Then, the joint law of $(A_1,A_3)$ gives a coupling between $\mu_1$ and $\mu_3$, denoted by $\nu_1\nu_2$.

My hope is that the above situation gives a diagram (in some suitable category of metric measure spaces) \begin{array}{ccccccccc} & & (X_1 \times X_2,\nu_1) & & & & (X_2 \times X_3,\nu_2) \\ & \stackrel{\pi_1}{\swarrow} & & \stackrel{\pi_2}{\searrow} & & \stackrel{\pi_2}{\swarrow} & & \stackrel{\pi_3}{\searrow} \\ (X_1,\mu_1) & & & & (X_2,\mu_2) & & & & (X_3,\mu_3) \end{array} such that the pullback of the middle cospan exist, and is given by say $(X_1\times X_2\times X_3, \nu')$, which in turn induces a coupling $(X_1\times X_3, (\pi_1,\pi_3)_*(\nu'))$ between $\mu_1$ and $\mu_3$, that coincides with the composite coupling $\nu_1\nu_2$.

Question. Is there a reasonable category of metric measure spaces that admits at least certain kinds of pullbacks and realizes my hope?

[1] Tessellations of random maps of arbitrary genus, Grégory Miermont.

A metric measure space $(X,d,\mu)$ consists of a metric space $(X,d)$ together with a Borel measure $\mu$. A coupling between metric measure spaces $(X_1,d_1,\mu_1)$ and $(X_2,d_2,\mu_2)$ consists of a Borel measure $\nu$ on the product metric space $X_1\times X_2$ such that $(\pi_1)_*(\nu) = \mu_1$ and $(\pi_2)_*(\nu) = \mu_2$.

I would like to see composition of couplings between metric measure spaces (explained below) as a pullback construction, in the same way that composition of correspondences between sets can be seen as a pullback.

Classical set up for composition of couplings. I follow Section 6.2 of [1]. There, it is stated that a coupling $\nu$ between measures $\mu,\mu'$ on Polish metric spaces $X$ and $X'$ can be written in the form $\mu(\mathbb{d} x)Q_\nu(x,\mathbb{d}x)$, where $Q_\nu$ is a Markov kernel from $X$ to $X'$. In particular, if $\mu_1,\mu_2,\mu_3$ are probability measures on compact spaces $X_1$, $X_2$, and $X_3$, and $\nu_1$ and $\nu_2$ are couplings between $\mu_1,\mu_2$ and $\mu_2,\mu_3$ respectively, one can define a Markov chain $(A_1,A_2,A_3)$ where $A_i$ is a random variable with distribution $\mu_i$ on $X_i$, with initial distribution $\mu_1$, and with transition kernel $Q_{\nu_i}$ at step $i$. Then, the joint law of $(A_1,A_3)$ gives a coupling between $\mu_1$ and $\mu_3$, denoted by $\nu_1\nu_2$.

My hope is that the above situation gives a diagram (in some suitable category of metric measure spaces) \begin{array}{ccccccccc} & & (X_1 \times X_2,\nu_1) & & & & (X_2 \times X_3,\nu_2) \\ & \stackrel{\pi_1}{\swarrow} & & \stackrel{\pi_2}{\searrow} & & \stackrel{\pi_2}{\swarrow} & & \stackrel{\pi_3}{\searrow} \\ (X_1,\mu_1) & & & & (X_2,\mu_2) & & & & (X_3,\mu_3) \end{array} such that the pullback of the middle cospan exist, and is given by say $(X_1\times X_2\times X_3, \nu')$, which in turn induces a coupling $(X_1\times X_3, (\pi_1,\pi_3)_*(\nu'))$ between $\mu_1$ and $\mu_3$, that coincides with the composite coupling $\nu_1\nu_2$.

Question. Is there a reasonable category of metric measure spaces that admits at least certain kinds of pullbacks and realizes my hope?

The problems that I have encountered are similar to the problems pointed out in Is there a category structure one can place on measure spaces so that category-theoretic products exist?. I would like the morphisms to be measure-preserving, or measure non-compressing (and distance non-increasing). The projection from the product measure is of those forms, if we restrict attention to metric probability spaces, which I am willing to do. But it is not clear to me what the measure on an arbitrary pullback should be. There is a measure on an arbitrary pullback, by seeing it as a subspace of the product. But it is not clear to me that the projections are measure preserving (or non-compressing) in that case.

[1] Tessellations of random maps of arbitrary genus, Grégory Miermont.