symmetric monoidal (∞,1)-category of spectra
symmetric monoidal (∞,1)-category of spectra
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
A partial evaluation is an instruction to evaluate a formal expression? piecewise, without necessarily obtaining the final result.
For a simple example, the sum “$3+4+5$” can be evaluated to “$12$”, but also partially evaluated to “$7+5$”.
Let $(T,\mu,\eta)$ be a monad on Set, and let $(A,a)$ be an algebra over $T$. Given two elements $p,q\in T A$, a partial evaluation from $p$ to $q$ is an element $r\in T T A$ such that $\mu(r)=p$, and $T a(r)=q$.
Equivalently, partial evaluations are the 1-simplices of the bar construction (considered as a simplicial set) of the algebra $(A,a)$.
The partial evaluation relation on $T A$ is defined by: $p\to q$ if and only if there exists a partial evaluation from $p$ to $q$. In other words, it is the (0,1)-truncation of the bar construction.
Similar definitions can be given internally to any category: instead of elements $p,q\in T A$, $r\in T T A$ one can take generalized elements (i.e. arrows) $p,q:X\to T A$, $r:X\to T T A$.
Every expression $p\in T A$ admits a partial evaluation to itself (given by $T\eta(p)\in T T A$). This makes the partial evaluation relation reflexive.
Every expression $p\in T A$ admits a partial evaluation to its total evaluation $\eta(a(p))\in T A$, given by $\eta(p)\in T T A$.
The partial evaluation relation is the smallest internal relation (in the category of $T$-algebras) which relates a formal expression? to its result.
If the multiplication $\mu$ of the monad is a weakly cartesian natural transformation (i.e. its naturality square is a weak pullback), the partial evaluation relation is transitive, and hence a preorder.
Tobias Fritz and Paolo Perrone, Monads, partial evaluations, and rewriting, MFPS 2020. (arXiv)
Carmen Constantin, Tobias Fritz, Paolo Perrone and Brandon Shapiro, Partial evaluations and the compositional structure of the bar construction, Theory and Applications of Categories 39, 2023. (arXiv:2009.07302)
Carmen Constantin, Tobias Fritz, Paolo Perrone and Brandon Shapiro, Weak cartesian properties of simplicial sets, Journal of Homotopy and Related Structures, 2023. (arXiv:2105.04775)
Paolo Perrone, Walter Tholen, Kan extensions are partial colimits, Applied Categorical Structures, 2022. (arXiv:2101.04531)
Last revised on July 21, 2024 at 16:09:08. See the history of this page for a list of all contributions to it.