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Register a new userPartial Evaluations and the Compositional Structure of the Bar Construction
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The algebraic expression $3 + 2 + 6$ can be evaluated to $11$, but it can also be partially evaluated to $5 + 6$. In categorical algebra, such partial evaluations can be defined in terms of the $1$-skeleton of the bar construction for algebras of a monad. We show that this partial evaluation relation can be seen as the relation internal to the category of algebras generated by relating a formal expression to its total evaluation. The relation is transitive for many monads which describe commonly encountered algebraic structures, and more generally for BC monads on $mathsf{Set}$ (which are those monads for which the underlying functor and the multiplication are weakly cartesian). We find that this is not true for all monads: we describe a finitary monad on $mathsf{Set}$ for which the partial evaluation relation on the terminal algebra is not transitive. With the perspective of higher algebraic rewriting in mind, we then investigate the compositional structure of the bar construction in all dimensions. We show that for algebras of BC monads, the bar construction has fillers for all directed acyclic configurations in $Delta^n$, but generally not all inner horns.
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Monads can be interpreted as encoding formal expressions, or formal operations in the sense of universal algebra. We give a construction which formalizes the idea of evaluating an expression partially: for example, 2+3 can be obtained as a partial evaluation of 2+2+1. This construction can be given for any monad, and it is linked to the famous bar construction, of which it gives an operational interpretation: the bar construction induces a simplicial set, and its 1-cells are partial evaluations. We study the properties of partial evaluations for general monads. We prove that whenever the monad is weakly cartesian, partial evaluations can be composed via the usual Kan filler property of simplicial sets, of which we give an interpretation in terms of substitution of terms. In terms of rewritings, partial evaluations give an abstract reduction system which is reflexive, confluent, and transitive whenever the monad is weakly cartesian. For the case of probability monads, partial evaluations correspond to what probabilists call conditional expectation of random variables. This manuscript is part of a work in progress on a general rewriting interpretation of the bar construction.
In homotopy type theory we can define the join of maps as a binary operation on maps with a common co-domain. This operation is commutative, associative, and the unique map from the empty type into the common codomain is a neutral element. Moreover, we show that the idempotents of the join of maps are precisely the embeddings, and we prove the `join connectivity theorem, which states that the connectivity of the join of maps equals the join of the connectivities of the individual maps. We define the image of a map $f:Ato X$ in $U$ via the join construction, as the colimit of the finite join powers of $f$. The join powers therefore provide approximations of the image inclusion, and the join connectivity theorem implies that the approximating maps into the image increase in connectivity. A modified version of the join construction can be used to show that for any map $f:Ato X$ in which $X$ is only assumed to be locally small, the image is a small type. We use the modified join construction to give an alternative construction of set-quotients, the Rezk completion of a precategory, and we define the $n$-truncation for any $n:mathbb{N}$. Thus we see that each of these are definable operations on a univalent universe for Martin-Lof type theory with a natural numbers object, that is moreover closed under homotopy coequalizers.
Suppose k is a field of characteristic 2, and $n,mgeq 4$ powers of 2. Then the $A_infty$-structure of the group cohomology algebras $H^*(C_n,k)$ and $(H^*(C_m,k)$ are well known. We give results characterizing an $A_infty$-structure on $H^*(C_ntimes C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.
We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modular operads to presheaves is fully faithful, and its essential image can be classified by a Segal condition. This theorem can be used to recover a related statement, due to Andre Joyal and Joachim Kock, concerning a larger category of undirected graphs whose functor to modular operads is not just faithful but also full.
Cheng, Gurski, and Riehl constructed a cyclic double multicategory of multivariable adjunctions. We show that the same information is carried by a double polycategory, in which opposite categories are polycategorical duals. Moreover, this double polycategory is a full substructure of a double Chu construction, whose objects are a sort of polarized category, and which is a natural home for 2-categorical dualities. We obtain the double Chu construction using a general Chu-Dialectica construction on polycategories, which includes both the Chu construction and the categorical Dialectica construction of de Paiva. The Chu and Dialectica constructions each impose additional hypotheses making the resulting polycategory representable (hence *-autonomous), but for different reasons; this leads to their apparent differences.
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