arXiv subjects
Publications and source records attributed to Angélica M. Osorno.
Transfer systems on finite posets have recently been gaining traction as a key ingredient in equivariant homotopy theory. Additionally, they also naturally occur in the data of a model structure. We give a complete characterization of all model category structures on a finite lattice, using transfer systems as our main tool, resulting in new connections between abstract homotopy theory and equivariant methods.
We show that various categories of trees can be modeled by Grothendieck constructions on categories of trees with a fixed set of leaves. We prove this result for the dendroidal category $Ω$, the category $Ω^G$ of trees with a $G$-action for a finite group $G$, and finally for the category of genuine equivariant trees $Ω_G$ that has played an important role in recent work on genuine equivariant operads.
We investigate how the notions of pairings of operads of May and compatible pairs of indexing systems of Blumberg--Hill relate via the correspondence between indexing systems and $N_{\infty}$-operads. We show that a pairing of operads induces a pairing on the associated indexing systems. Conversely, we show that in many cases, compatible pairs of indexing systems can be realized by a pairing of $N_{\infty}$-operads.
Given a finite commutative monoid $M$, we show that submonoids of $M\times [n]$ - where $[n] = \{0,1,\ldots,n\}$ is equipped with the max operation $\vee$ - may be enumerated via the transfer matrix method. When $M$ is also idempotent, we show that there are finitely many integers $λ$ and rational numbers $b_λ$ (only depending on $M$) such that the number of submonoids of $M\times [n]$ is $\sum_λb_λλ^n$. This answers a question of Knuth regarding ternary (and higher order) max-closed relations, and has applications to the enumeration of saturated transfer systems in equivariant infinite loop space theory.
We rework and generalize equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. There is a classical version which gives classical $Ω$-$G$-spectra for any topological group $G$, but our focus is on the construction of genuine $Ω$-$G$-spectra when $G$ is finite. We also show what is and is not true when $G$ is a compact Lie group. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for classical $G$-spectra for general $G$ but fails for genuine $G$-spectra. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving an illuminating direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving a number of corrections to results and proofs in the literature.
We define a tensor product for permutative categories and prove a number of key properties. We show that this product makes the 2-category of permutative categories closed symmetric monoidal as a bicategory.
We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics. In the case of a finite total order $[n]$, we enumerate all model structures, exhibiting a rich combinatorial structure encoded by Shapiro's Catalan triangle. This is an application of previous work of the authors on the theory of $N_\infty$-operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of $[n]$.
In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers. This is a version of Quillen's Theorem B amenable to applications. Second, we compute the $E^2$ page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of $S^{-1}S$. We show that if $S$ is a symmetric monoidal 2-groupoid with faithful translations then $S^{-1}S$ models the group completion of $S$.
We prove a multiplicative version of the equivariant Barratt-Priddy-Quillen theorem, starting from the additive version proven in arXiv:1207.3459. The proof uses a multiplicative elaboration of an additive equivariant infinite loop space machine that manufactures orthogonal $G$-spectra from symmetric monoidal $G$-categories. The new machine produces highly structured associative ring and module $G$-spectra from appropriate multiplicative input. It relies on new operadic multicategories that are of considerable independent interest and are defined in a general, not necessarily equivariant or topological, context. Most of our work is focused on constructing and comparing them. We construct a multifunctor from the multicategory of symmetric monoidal $G$-categories to the multicategory of orthogonal $G$-spectra. With this machinery in place, we prove that the equivariant BPQ theorem can be lifted to a multiplicative equivalence. That is the heart of what is needed for the presheaf reconstruction of the category of $G$-spectra in arXiv:1110.3571.
In previous work, we develop a generalized Waldhausen $S_{\bullet}$-construction whose input is an augmented stable double Segal space and whose output is a unital 2-Segal space. Here, we prove that this construction recovers the previously known $S_{\bullet}$-constructions for exact categories and for stable and exact $(\infty,1)$-categories, as well as the relative $S_{\bullet}$-construction for exact functors.
For a finite group G, we introduce the complete suboperad $Q_G$ of the categorical G-Barratt-Eccles operad $P_G$. We prove that $P_G$ is not finitely generated, but $Q_G$ is finitely generated and is a genuine $E_\infty$ G-operad (i.e., it is $N_\infty$ and includes all norms). For G cyclic of order 2 or 3, we determine presentations of the object operad of $Q_G$ and conclude with a discussion of algebras over $Q_G$, which we call biased permutative equivariant categories.
We give an operadic definition of a genuine symmetric monoidal G-category, and we prove that its classifying space is a genuine E_\infty G-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power, and Lack, to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal G-categories to genuine permutative G-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When G is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal G-categories as input to an equivariant infinite loop space machine that gives genuine G-spectra as output.
We prove that the homotopy theory of Picard 2-categories is equivalent to that of stable 2-types.
In a previous paper, we showed that a discrete version of the $S_\bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, by showing that there is a Quillen equivalence between a model category for unital 2-Segal objects and a model category for augmented stable double Segal objects which is given by an $S_\bullet$-construction. We show that this equivalence fits together with the result in the discrete case and briefly discuss how it encompasses other known $S_\bullet$-constructions.
In [MMO] (arXiv:1704.03413), we reworked and generalized equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. In this paper, we construct a new variant of the equivariant Segal machine that starts from the category $\scr{F}$ of finite sets rather than from the category ${\scr{F}}_G$ of finite $G$-sets and which is equivalent to the machine studied by Shimakawa and in [MMO]. In contrast to the machine in [MMO], the new machine gives a lax symmetric monoidal functor from the symmetric monoidal category of $\scr{F}$-$G$-spaces to the symmetric monoidal category of orthogonal $G$-spectra. We relate it multiplicatively to suspension $G$-spectra and to Eilenberg-MacLane $G$-spectra via lax symmetric monoidal functors from based $G$-spaces and from abelian groups to $\scr{F}$-$G$-spaces. Even non-equivariantly, this gives an appealing new variant of the Segal machine. This new variant makes the equivariant generalization of the theory essentially formal, hence is likely to be applicable in other contexts.
We show that the edgewise subdivision of a $2$-Segal object is always a Segal object, and furthermore that this property characterizes $2$-Segal objects.
We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of $Γ$-objects in 2-categories. In the course of the proof we establish strictfication results of independent interest for symmetric monoidal bicategories and for diagrams of 2-categories.
Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can transport the weak equivalences from one category to another with the same objects and a broader class of maps. Under mild hypotheses this process produces an equivalence of homotopy theories. We describe examples including algebras over an operad, such as symmetric monoidal categories and $n$-fold monoidal categories; and diagram categories, such as $Γ$-categories.