MacLane Research Articles

String diagrams for Strictification and Coherence

Whereas string diagrams for strict monoidal categories are well understood, and have found application in several fields of Computer Science, graphical formalisms for non-strict monoidal categories are far less studied. In this paper, we provide a presentation by generators and relations of string diagrams for non-strict monoidal categories, and show how this construction can handle applications in domains such as digital circuits and programming languages. We prove the correctness of our construction, which yields a novel proof of Mac Lane's strictness theorem. This in turn leads to an elementary graphical proof of Mac Lane's coherence theorem, and in particular allows for the inductive construction of the canonical isomorphisms in a monoidal category.

Coherence via focusing for symmetric skew monoidal and symmetric skew closed categories

Abstract The symmetric skew monoidal categories of Bourke and Lack are a weakening of Mac Lane’s symmetric monoidal categories where (i) the three structural laws of left and right unitality and associativity are not required to be invertible, they are merely natural transformations with a specific orientation; (ii) the structural law of symmetry is a natural isomorphism involving three objects rather than two. In a similar fashion, symmetric skew closed categories are a weakening of de Schipper’s symmetric closed categories with non-invertible structural laws of left and right unitality. In this paper, we study the structural proof theory of symmetric skew monoidal and symmetric skew closed categories, progressing the project initiated by Uustalu et al. on deductive systems for categories with skew structure. We discuss three equivalent presentations of the free symmetric skew monoidal (resp. closed) category on a set of generating objects: a Hilbert-style categorical calculus; a cut-free sequent calculus; a focused subsystem of derivations, corresponding to a sound and complete goal-directed proof search strategy for the cut-free sequent calculus. Focusing defines an effective normalization procedure for maps in the free symmetric skew monoidal (resp. closed) category, as such solving the coherence problem for symmetric skew monoidal (resp. closed) categories.

On a general notion of a polynomial identity and codimensions

Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category C as well as their codimensions in the case when C is linear over some field. The new cases include coalgebras, bialgebras, Hopf algebras, braided vector spaces, Yetter–Drinfel'd modules, etc. We find bases for polynomial identities and calculate codimensions in some important particular cases.

Matroid Horn functions

Hypergraph Horn functions were introduced as a subclass of Horn functions that can be represented by a collection of circular implication rules. These functions possess distinguished structural and computational properties. In particular, their characterizations in terms of implicate-duality and the closure operator provide extensions of matroid duality and the Mac Lane–Steinitz exchange property of matroid closure, respectively.In the present paper, we introduce a subclass of hypergraph Horn functions that we call matroid Horn functions. We provide multiple characterizations of matroid Horn functions in terms of their canonical and complete CNF representations. We also study the Boolean minimization problem for this class, where the goal is to find a minimum size representation of a matroid Horn function given by a CNF representation. While there are various ways to measure the size of a CNF, we focus on the number of circuits and circuit clauses. We determine the size of an optimal representation for binary matroids, and give lower and upper bounds in the uniform case. For uniform matroids, we show a strong connection between our problem and Turán systems that might be of independent combinatorial interest.

Homotopy cartesian squares in extriangulated categories

Abstract Let ( C , E , s ) \left({\mathcal{C}},{\mathbb{E}},{\mathfrak{s}}) be an extriangulated category. Given a composition of two commutative squares in C {\mathcal{C}} , if two commutative squares are homotopy cartesian, then their composition is also a homotopy cartesian square. This covers the result by Mac Lane [Categories for the Working Mathematician, Second edition, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1998] for abelian categories and by Christensen and Frankland [On good morphisms of exact triangles, J. Pure Appl. Algebra 226 (2022), no. 3, 106846] for triangulated categories.

The étale symmetric Künneth theorem

Let k be an algebraically closed field, lne {{,textrm{char},}}k a prime number, and X a quasi-projective scheme over k. We show that the étale homotopy type of the dth symmetric power of X is textbf{Z}/l-homologically equivalent to the dth strict symmetric power of the étale homotopy type of X. We deduce that the textbf{Z}/l-local étale homotopy type of a motivic Eilenberg–Mac Lane space is an ordinary Eilenberg–Mac Lane space.

Eilenberg Mac Lane spectra as p$p$‐cyclonic Thom spectra

Hopkins and Mahowald gave a simple description of the mod p $p$ Eilenberg Mac Lane spectrum F p ${\mathbb {F}}_p$ as the free E 2 $\mathbb {E}_2$ -algebra with an equivalence of p $p$ and 0. We show for each faithful 2-dimensional representation λ $\lambda$ of a p $p$ -group G $G$ that the G $G$ -equivariant Eilenberg Mac Lane spectrum F ̲ p $\underline{\mathbb {F}}_p$ is the free E λ $\mathbb {E}_{\lambda }$ -algebra with an equivalence of p $p$ and 0. This unifies and simplifies recent work of Behrens, Hahn, and Wilson, and extends it to include the dihedral 2-subgroups of O ( 2 ) $\textnormal {O}(2)$ . The main new idea is that F ̲ p $\underline{\mathbb {F}}_p$ has a simple description as a p $p$ -cyclonic module over THH ( F p ) $\mathop {\rm THH}\nolimits (\mathbb {F}_p)$ . We show that our result is the best possible one in that it gives all groups G $G$ and representations V $V$ such that F ̲ p $\underline{\mathbb {F}}_p$ is the free E V $\mathbb {E}_V$ -algebra with an equivalence of p $p$ and 0.

Explicit minimal embedded resolutions of divisors on models of the projective line

Let K be a discretely valued field with ring of integers \(\mathcal {O}_K\) with perfect residue field. Let K(x) be the rational function field in one variable. Let \({\mathbb {P}}^1_{\mathcal {O}_K}\) be the standard smooth model of \({\mathbb {P}}^1_K\) with coordinate x. Let \(f(x) \in \mathcal {O}_K[x]\) be a squarefree polynomial with corresponding divisor of zeroes \({{\,\mathrm{div}\,}}_0(f)\) on \({\mathbb {P}}^1_{\mathcal {O}_K}\). We give an explicit description of the minimal embedded resolution \(\mathcal {Y}\) of the pair \(({\mathbb {P}}^1_{\mathcal {O}_K}, {{\,\mathrm{div}\,}}_0(f))\) by using Mac Lane’s theory to write down the discrete valuations on K(x) corresponding to the irreducible components of the special fiber of \(\mathcal {Y}\).

Non-Trivial Universes and Sequences of Universes

Summary Universe is a concept which is present from the beginning of the creation of the Mizar Mathematical Library (MML) in several forms (Universe, Universe_closure, UNIVERSE) [25], then later as the_universe_of, [33], and recently with the definition GrothendieckUniverse [26], [11], [11]. These definitions are useful in many articles [28, 33, 8, 35], [19, 32, 31, 15, 6], but also [34, 12, 20, 22, 21], [27, 2, 3, 23, 16, 7, 4, 5]. In this paper, using the Mizar system [9] [10], we trivially show that Grothendieck’s definition of Universe as defined in [26], coincides with the original definition of Universe defined by Artin, Grothendieck, and Verdier (Chapitre 0 Univers et Appendice “Univers” (par N. Bourbaki) de l’Exposé I. “PREFAISCE-AUX”) [1], and how the different definitions of MML concerning universes are related. We also show that the definition of Universe introduced by Mac Lane ([18]) is compatible with the MML’s definition. Although a universe may be empty, we consider the properties of non-empty universes, completing the properties proved in [25]. We introduce the notion of “trivial” and “non-trivial” Universes, depending on whether or not they contain the set ω (NAT), following the notion of Robert M. Solovay2. The following result links the universes U 0 (FinSETS) and U 1 (SETS): Grothendieck Universe ω = Grothendieck Universe U 0 = U 1 {\rm{Grothendieck}}\,{\rm{Universe}}\,\omega = {\rm{Grothendieck}}\,{\rm{Universe}}\,{{\bf{U}}_0} = {{\bf{U}}_1} Before turning to the last section, we establish some trivial propositions allowing the construction of sets outside the considered universe. The last section is devoted to the construction, in Tarski-Grothendieck, of a tower of universes indexed by the ordinal numbers (See 8. Examples, Grothendieck universe, ncatlab.org [24]). Grothendieck’s universe is referenced in current works: “Assuming the existence of a sufficient supply of (Grothendieck) univers”, Jacob Lurie in “Higher Topos Theory” [17], “Annexe B – Some results on Grothendieck universes”, Olivia Caramello and Riccardo Zanfa in “Relative topos theory via stacks” [13], “Remark 1.1.5 (quoting Michael Shulman [30])”, Emily Riehl in “Category theory in Context” [29], and more specifically “Strict Universes for Grothendieck Topoi” [14].

Ideal ring extensions and trusses

It is shown that there is a close relationship between ideal extensions of rings and trusses, that is, sets with a semigroup operation distributing over a ternary abelian heap operation. Specifically, a truss can be associated to every element of an extension ring that projects down to an idempotent in the extending ring; every weak equivalence of extensions yields an isomorphism of corresponding trusses. Furthermore, equivalence classes of ideal extensions of rings by integers are in one-to-one correspondence with associated trusses up to isomorphism given by a translation. Conversely, to any truss T and an element of this truss one can associate a ring and its extension by integers in which T is embedded as a truss. Consequently any truss can be understood as arising from an ideal extension by integers. The key role is played by interpretation of ideal extensions by integers as extensions defined by double homothetisms of Redei (L. Redei (1952) [24]) or by self-permutable bimultiplications of Mac Lane (S. Mac Lane (1958) [18]), that is, as integral homothetic extensions. The correspondence between homothetic ring extensions and trusses is used to classify fully up to isomorphism trusses arising from rings with zero multiplication and rings with trivial annihilators.

Localization and nilpotent spaces in -homotopy theory

AbstractFor a subring$R$of the rational numbers, we study$R$-localization functors in the local homotopy theory of simplicial presheaves on a small site and then in${\mathbb {A}}^1$-homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in${\mathbb {A}}^1$-homotopy theory, paying attention to future applications for vector bundles. We show that$R$-localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space$BGL_n$is${\mathbb {A}}^1$-nilpotent when$n$is odd, and analyze the (more complicated) situation where$n$is even as well. We establish analogs of various classical results about rationalization in the context of${\mathbb {A}}^1$-homotopy theory: if$-1$is a sum of squares in the base field,${\mathbb {A}}^n \,{\setminus}\, 0$is rationally equivalent to a suitable motivic Eilenberg–Mac Lane space, and the special linear group decomposes as a product of motivic spheres.

What do Abelian categories form?

Abstract Given two finitely presentable Abelian categories and , we outline a construction of an Abelian category of functors from to , which has nice 2-categorical properties and provides an explicit model for a stable category of stable functors between the derived categories of and . The construction is absolute, so it makes it possible to recover not only Hochschild cohomology but also Mac Lane cohomology. Bibliography: 29 titles.

A multiplicative comparison of Mac Lane homology and topological Hochschild homology

Let $Q$ denote MacLane's $Q$-construction, and $\otimes$ denote the smash product of spectra. In this paper we construct an equivalence $Q(R)\simeq \mathbb Z\otimes R$ in the category of $A_\infty$ ring spectra for any ring $R$, thus proving a conjecture made by Fiedorowicz, Schw\"anzl, Vogt and Waldhausen in "MacLane homology and topological Hochschild homology". More precisely, we construct is a symmetric monoidal structure on $Q$ (in the $\infty$-categorical sense) extending the usual monoidal structure, for which we prove an equivalence $Q(-)\simeq \mathbb Z\otimes -$ as symmetric monoidal functors, from which the conjecture follows immediately. From this result, we obtain a new proof of the equivalence $\mathrm{HML}(R,M)\simeq \mathrm{THH}(R,M)$ originally proved by Pirashvili and Waldaushen in "MacLane homology and topological Hochschild homology" (a different paper from the one cited above). This equivalence is in fact made symmetric monoidal, and so it also provides a proof of the equivalence $\mathrm{HML}(R)\simeq \mathrm{THH}(R)$ as $E_\infty$ ring spectra, when $R$ is a commutative ring.

Right-angled Artin groups, polyhedral products and the -generating function

For a graph $\Gamma$, let $K(H_{\Gamma },\,1)$ denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group $H_{\Gamma }$ defined by $\Gamma$. We use the relationship between the combinatorics of $\Gamma$ and the topological complexity of $K(H_{\Gamma },\,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$, we construct a graph $\mathcal {O}_n$ whose TC-generating function has polynomial numerator of degree $n$. Additionally, motivated by the fact that $K(H_{\Gamma },\,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

Eilenberg–Mac Lane spectra as equivariant Thom spectra

We prove that the $G$-equivariant mod $p$ Eilenberg--MacLane spectrum arises as an equivariant Thom spectrum for any finite, $p$-power cyclic group $G$, generalizing a result of Behrens and the second author in the case of the group $C_2$. We also establish a construction of $\mathrm{H}\underline{\mathbb{Z}}_{(p)}$, and prove intermediate results that may be of independent interest. Highlights include constraints on the Hurewicz images of equivariant spectra that admit norms, and an analysis of the extent to which the non-equivariant $\mathrm{H}\mathbb{F}_p$ arises as the Thom spectrum of a more than double loop map.

The groups (2, 𝑚 | 𝑛, 𝑘 | 1, 𝑞): Finiteness and homotopy

Abstract We initiate the study of the groups ( l , m ∣ n , k ∣ p , q ) (l,m\mid n,k\mid p,q) defined by the presentation ⟨ a , b ∣ a l , b m , ( a ⁢ b ) n , ( a p ⁢ b q ) k ⟩ \langle a,b\mid a^{l},b^{m},(ab)^{n},(a^{p}b^{q})^{k}\rangle . When p = 1 p=1 and q = m - 1 q=m-1 , we obtain the group ( l , m ∣ n , k ) (l,m\mid n,k) , first systematically studied by Coxeter in 1939. In this paper, we restrict ourselves to the case l = 2 l=2 and 1 n + 1 k ≤ 1 2 \frac{1}{n}+\frac{1}{k}\leq\frac{1}{2} and give a complete determination as to which of the resulting groups are finite. We also, under certain broadly defined conditions, calculate generating sets for the second homotopy group π 2 ⁢ ( Z ) \pi_{2}(Z) , where 𝑍 is the space formed by attaching 2-cells corresponding to ( a ⁢ b ) n (ab)^{n} and ( a ⁢ b q ) k (ab^{q})^{k} to the wedge sum of the Eilenberg–MacLane spaces 𝑋 and 𝑌, where π 1 ⁢ ( X ) ≅ C 2 \pi_{1}(X)\cong C_{2} and π 1 ⁢ ( Y ) ≅ C m \pi_{1}(Y)\cong C_{m} ; in particular, π 1 ( Z ) ≅ ( 2 , m ∣ n , k ∣ 1 , q ) \pi_{1}(Z)\cong(2,m\mid n,k\mid 1,q) .

Monads, Partial Evaluations, and Rewriting

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 [Saunders Mac Lane, Categories for the Working Mathematician, Springer, 2000, VII.6], of which it gives an operational interpretation: the bar construction is 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.For the case of probability monads, partial evaluations correspond to what probabilists call conditional expectation of random variables, and partial evaluation relation is known as second-order stochastic dominance.In terms of rewritings, partial evaluations give an abstract reduction system which is reflexive, confluent, and transitive whenever the monad is weakly cartesian. This manuscript is part of a work in progress on a general rewriting interpretation of the bar construction.

Partially ordered objects in a topos

In this paper we study the category of partially ordered objects in a topos E . The definition of a partially ordered object or internal poset is taken from [10], ϵ. Mac Lane and I. Moerdijk, Sheaves in geometry and logic, 1992. We will define the concept of monotone morphism between internal posets and then study the resulting category. We will show that the category of partially ordered objects in a topos, denoted PO ϵ, is finitely complete and has exponential objects. This paper would be our first step in investigating partially ordered objects within a topos.

Nilpotent types and fracture squares in homotopy type theory

AbstractWe develop the basic theory of nilpotent types and their localizations away from sets of numbers in Homotopy Type Theory. For this, general results about the classifying spaces of fibrations with fiber an Eilenberg–Mac Lane space are proven. We also construct fracture squares for localizations away from sets of numbers. All of our proofs are constructive.

Mac Lane\u2019s Comparison Theorem for the Kleisli Construction Formalized in Coq

(co)Monads are used to encapsulate impure operations of a computation. A (co)monad is determined by an adjunction and further determines a specific type of adjunction called the (co)Kleisli adjunction. Mac Lane introduced the comparison theorem which allows comparing these adjunctions bridged by a (co)monad through a unique comparison functor. In this paper we specify the foundations of category theory in Coq and show that the chosen representations are useful by certifying Mac Lane’s comparison theorem and its basic consequences. We also show that the foundations we use are equivalent to the foundations by Timany. The formalization makes use of Coq classes to implement categorical objects and the axiom uniqueness of identity proofs to close the gap between the contextual equality of objects in a categorical setting and the judgmental Leibniz equality of Coq. The theorem is used by Duval and Jacobs in their categorical settings to interpret the state effect in impure programming languages.