symmetric monoidal (∞,1)-category of spectra
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
An observational monad is a monad where, intuitively, one can distinguish two monadic (for example probabilistic) values by means of taking repeated independent observations.
It is of particular importance in categorical probability, where these observations correspond to iid random variables.
Let $C$ be a category with products, and let $P$ be a symmetric monoidal monad (or commutative monad) on $C$.
Denote also by $C_T$ its Kleisli category, which is then a copy-discard category. As usual, we have an adjunction $L:C\to C_T, R:C_T\to C$ such that $P=R\circ L$.
Denote the comonad $L\circ R$ on $C_T$ again by $P$. (Note that this equips $C_T$ with a thunk-force structure.) Finally, denote $\varepsilon_X:P X\to X$ the components of the counit of the adjunction. (In categorical probability, this is known as the sampling map.)
The monad $P$ is called observational if and only if for every object $X$, the family of morphisms of $C_T$
is jointly monic.
The intuition is that one can distinguish two morphisms (for example, states) into $P X$ by sampling independently as many times as needed.
A representable Markov category whose probability monad is observational is called an observationally representable Markov category.
The Giry monad is observational thanks to the pi-lambda theorem? or, equivalently, the monotone class theorem?. (Moss-Perrone’22)
The Kleisli category of an observational monad, with its canonical copy-discard and thunk-force structures, is such that every deterministic morphism is thunkable (the converse is always true). (Moss-Perrone’22)
Every observationally representable Markov category is necessarily almost-surely-compatibly representable. That is, in the bijection
we have that for every state (probability measure) $p$ on $X$, two morphisms $f,g:X\to Y$ are $p$-almost surely equal? if and only if $f^\sharp$ and $g^\sharp$ are. (FGLPS)
Observationality is related to de Finetti's theorem: stating the theorem as a limit, observationality is closely related to the uniqueness part in the universal property. It coincides exactly with this uniqueness whenever the necessary Kolmogorov products exist. (Moss-Perrone’22)
The concept, in the form presented here, was first introduced in:
It also appears in:
Tobias Fritz, Tomáš Gonda, Antonio Lorenzin, Paolo Perrone and Dario Stein, Absolute continuity, supports and idempotent splitting in categorical probability, 2023. (arXiv)
Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267
Tobias Fritz and Antonio Lorenzin, Involutive Markov categories and the quantum de Finetti theorem, 2023. (arXiv)
Related papers:
Tobias Fritz, Tomáš Gonda, Paolo Perrone, Eigil Fjeldgren Rischel, Representable Markov categories and comparison of statistical experiments in categorical probability, Theoretical Computer Science 961, 2023. (arXiv:2010.07416)
Tobias Fritz, Tomáš Gonda, Paolo Perrone, De Finetti’s theorem in categorical probability. Journal of Stochastic Analysis, 2021. (arXiv:2105.02639)
Created on September 28, 2024 at 10:15:55. See the history of this page for a list of all contributions to it.