arXiv subjects

Andrew W. Macpherson

Publications and source records attributed to Andrew W. Macpherson.

13 recordsLinked to original sources

arXiv2024

Do backrun auctions protect traders?

We study a new "laminated" queueing model for orders on batched trading venues such as decentralised exchanges. The model aims to capture and generalise transaction queueing infrastructure that has arisen to organise MEV activity on public blockchains such as Ethereum, providing convenient channels for sophisticated agents to extract value by acting on end-user order flow by performing arbitrage and related HFT activities. In our model, market orders are interspersed with orders created by arbitrageurs that under idealised conditions reset the marginal price to a global equilibrium between each trade, improving predictability of execution for liquidity traders. If an arbitrageur has a chance to land multiple opportunities in a row, he may attempt to manipulate the execution price of the intervening market order by a probabilistic blind sandwiching strategy. To study how bad this manipulation can get, we introduce and bound a price manipulation coefficient that measures the deviation from global equilibrium of local pricing quoted by a rational arbitrageur. We exhibit cases in which this coefficient is well approximated by a "zeta value' with interpretable and empirically measurable parameters.

q-fin.TR↗

arXiv2023

Adversarial blockchain queues and trading on a CFMM

We describe a plausible probabilistic model for a blockchain queueing environment in which rational, profit-maximising schedulers impose adversarial disciplines on incoming messages containing a payload that encodes a state transition in a machine. The model can be specialised to apply to chains with fixed or variable block times, traditional priority queue disciplines with `honest' schedulers, or adversarial public mempools. We find conditions under which the model behaves as a bulk-service queue with priority discipline and derive practical expressions for the relative block and message number of a transaction. We study this setup in the context of orders to a CFMM DEX where the execution price a user receives may be quite sensitive to its positioning in the chain -- in particular, to a string of transactions scheduled for prior execution which is not knowable at the time of order creation. We derive statistical models for the price impact of this order flow both in the presence and absence of MEV extraction activity.

math.PR↗

arXiv2021

Adjoining colimits

This paper develops a theory of colimit sketches "with constructions" in higher category theory, formalising the input to the ubiquitous procedure of adjoining specified "constructible" colimits to a category such that specified "relation" colimits are enforced (or preserved). From a more technical standpoint, sketches are a way to describe dense functors using techniques from the homotopy theory of diagrams. We establish basic properties of diagrams in an infinity-category C as a model for presheaves on C and Bousfield localisations thereof, discuss extensions of functors and adjunctions, and equivalences of sets of diagrams. We introduce categories of presheaves which are "constructible in one step" by a set of diagrams and explore, via well-known examples, when constructible cocompletion is idempotent, i.e. when any iterated construction can be completed in one step.

math.CT↗

arXiv2021

Fibrations and Koszul duality in locally Cartesian localisations

I show that any locally Cartesian left localisation of a presentable infinity-category admits a right proper model structure in which all morphisms are cofibrations, and obtain a Koszul duality classification of its fibrations. By a simple criterion in terms of generators for a localisation to be locally Cartesian, this applies to any nullification functor. In particular, it includes examples with non-trivial "homotopical content." I further describe, and provide examples from, the set of fibrations in three contexts: the higher categorical Thomason model structure of Mazel-Gee, where fibrations are local systems; Morel-Voevodsky A1-localisation, where they are a higher analogue of A1-covering spaces; and the Quillen plus construction, where they are related to loop space modules trivialised over the universal acyclic extension.

math.CT↗

arXiv2021

A bivariant Yoneda lemma and $(\infty,2)$-categories of correspondences

Everyone knows that if you have a bivariant homology theory satisfying a base change formula, you get an representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual adjunction, these data are actually equivalent. In other words, a 2-category of correspondences is the universal way to attach to a given 1-category a set of right adjoints that satisfy a base change formula. Through a bivariant version of the Yoneda paradigm, I give a definition of correspondences in higher category theory and prove an extension theorem for bivariant functors. Moreover, conditioned on the existence of a 2-dimensional Grothendieck construction, I provide a proof of the aforementioned universal property. The methods, morally speaking, employ the `internal logic' of higher category theory: they make no explicit use of any particular model.

math.CT↗

arXiv2021

The universal property of derived geometry

Derived geometry can be defined as the universal way to adjoin finite homotopical limits to a given category of manifolds compatibly with products and glueing. The point of this paper is to show that a construction closely resembling existing approaches to derived geometry in fact produces a geometry with this universal property. I also investigate consequences of this definition in particular in the differentiable setting, and compare the theory so obtained to D. Spivak's axioms for derived C-infinity geometry.

math.CT↗

arXiv2020

An alternative to hypercovers

I introduce a class of diagrams in a Grothendieck site called "atlases" which can be used to study hyperdescent, and show that hypersheaves take atlases to limits using an indexed `nerve' construction that produces hypercovers from atlases. Atlases have the flexibility to be at the same time more explicit and more universal than hypercovers.

math.CT↗

arXiv2019

The operad that corepresents enrichment

I show that the theories of enrichment in a monoidal infinity-category defined by Hinich and by Gepner-Haugseng agree, and that the identification is unique. Among other things, this makes the Yoneda lemma available in the former model.

math.CT↗

arXiv2017

Skeleta in non-Archimedean and tropical geometry

I describe an algebro-geometric theory of skeleta, which provides a unified setting for the study of tropical varieties, skeleta of non-Archimedean analytic spaces, and affine manifolds with singularities. Skeleta are spaces equipped with a structure sheaf of topological semirings, and are locally modelled on the spectra of the same. The primary result of this paper is that the topological space underlying a non-Archimedean analytic space may locally be recovered from the sheaf of `pointwise valuations' of its analytic functions.

math.AG↗

arXiv2016

On the difference between `tropical functions' and real-valued functions

I introduce the concept of integral closure for elements and ideals in idempotent semirings, and establish how it corresponds to its namesake in commutative algebra. In the case of free semirings, integral closure can be understood in terms of a certain monoid of convex bodies under Minkowski sum. I argue that integral closure therefore accounts for the difference between `tropical functions' and real functions.

math.AC↗

arXiv2015

Projective modules over polyhedral semirings

I classify projective modules over idempotent semirings that are free on a monoid. The analysis extends to the case of the semiring of convex, piecewise-affine functions on a polyhedron, for which projective modules correspond to convex families of weight polyhedra for the general linear group.

math.AC↗

arXiv2015

Affine manifolds are rigid analytic spaces in characteristic one, I

I extend the definitions of schemes relative to monoids with zero - and therefore, toric geometry - to the world of formal schemes. This expands the usual framework to include, for instance, models for Mumford's degenerating Abelian varieties. Following the usual toric paradigm, normal formal monoid schemes can be classified in terms of certain cone complexes, and their properties understood in combinatorial terms. I use this to give a simple algebraisation criterion. I also reformulate the traditional notions of separated and proper morphism in a manner amenable to the context of relative formal geometry, and give characterisations in terms of the topology of cone complexes.

math.AG↗

arXiv2015

Affine manifolds are rigid analytic spaces in characteristic one, II

I extend the framework of rigid analytic geometry to the setting of algebraic geometry relative to monoids, and study the associated notions of separated, proper, and overconvergent morphisms. The category of affine manifolds embeds as a subcategory defined by simple algebraic (normal) and topological (overconvergent) criteria. The affine manifold of a rigid space can be recovered either as a set of `Novikov field' points or as a universal Hausdorff quotient. After base change to any topological field, one obtains a `toric' analytic space that fibres over the affine manifold.

math.AG↗