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Paula Verdugo

Publications and source records attributed to Paula Verdugo.

11 recordsLinked to original sources
arXiv2026

A model structure for cartesian 2-fibrations

Cartesian 2-fibrations provide a way to understand indexed categories, but their classical ``straightening'' construction requires several layers of weak coherence data. This paper develops a homotopical framework that replaces much of this bookkeeping with a fully strict model. By using marked 2-categories to record the cartesian morphisms and 2-cells, we construct a model structure whose fibrant objects are precisely the cartesian 2-fibrations over a fixed 2-category $\mathcal{C}$. We then show that the marked Grothendieck construction identifies these 2-fibrations, up to weak equivalence, with strict 2-functors from $\mathcal{C}$ into $2\mathrm{Cat}$. As an additional contribution, we construct localizations of 2-categories that simultaneously invert selected morphisms and 2-cells.

math.CT↗
arXiv2025

Realizing compatible pairs of transfer systems by combinatorial $N_\infty$-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.

math.AT↗
arXiv2025

Double categorical equivalences

We present an efficient and user-friendly method for constructing any cofibrantly generated model structure on the category of double categories whose trivial fibrations are the "canonical" ones: the double functors which are surjective on objects, full on both horizontal and vertical morphisms, and fully faithful on squares. We show that all of these model structures are left proper and that they are localizations of the gregarious model structure introduced by Campbell. As a notable consequence, this identifies the gregarious weak equivalences as the "canonical" equivalences of double categories, an elusive notion thus far. Moreover, the nature of our method gives an explicit description of the fibrant objects in terms of lifting conditions. We use this to recover several known model structures, as well as construct several new examples whose homotopy theories encode a range of $2$-dimensional structures.

math.AT↗
arXiv2025

On the equivalence invariance of formal category theory

We prove a result of equivalence invariance of formal category theory for statements that can be expressed within an equipment. To do this, we exploit Henry and Bardomiano Martínez's link between Makkai's FOLDS (first order logic with dependent sorts) and abstract homotopy theory. In the process, we construct a model structure on the category DblCat of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal and vertical morphisms, and fully faithful on squares, and whose fibrant objects are the equipments.

math.CT↗
arXiv2023

Internal Grothendieck construction for enriched categories

Given a cartesian closed category $\mathcal{V}$, we introduce an internal category of elements $\int_\mathcal{C} F$ associated to a $\mathcal{V}$-functor $F\colon \mathcal{C}^{\mathrm{op}}\to \mathcal{V}$. When $\mathcal{V}$ is extensive, we show that this internal Grothendieck construction gives an equivalence of categories between $\mathcal{V}$-functors $\mathcal{C}^{\mathrm{op}}\to \mathcal{V}$ and internal discrete fibrations over $\mathcal{C}$, which can be promoted to an equivalence of $\mathcal{V}$-categories. Using this construction, we prove a representation theorem for $\mathcal{V}$-categories, stating that a $\mathcal{V}$-functor $F\colon \mathcal{C}^{\mathrm{op}}\to \mathcal{V}$ is $\mathcal{V}$-representable if and only if its internal category of elements $\int_\mathcal{C} F$ has an internal terminal object. We further obtain a characterization formulated completely in terms of $\mathcal{V}$-categories using shifted $\mathcal{V}$-categories of elements. Moreover, in the presence of $\mathcal{V}$-tensors, we show that it is enough to consider $\mathcal{V}$-terminal objects in the underlying $\mathcal{V}$-category $\mathrm{Und}\int_\mathcal{C} F$ to test the representability of a $\mathcal{V}$-functor $F$. We apply these results to the study of weighted $\mathcal{V}$-limits, and also obtain a novel result describing weighted $\mathcal{V}$-limits as certain conical internal limits.

math.CT↗
arXiv2023

Fibrantly-transferred model structures

We develop new techniques for constructing model structures from a given class of cofibrations, together with a class of fibrant objects and a choice of weak equivalences between them. As a special case, we obtain a more flexible version of the classical right-transfer theorem in the presence of an adjunction. Namely, instead of lifting the classes of fibrations and weak equivalences through the right adjoint, we now only do so between fibrant objects, which allows for a wider class of applications.

math.AT↗
arXiv2021

A 2Cat-inspired model structure for double categories

We construct a model structure on the category $\mathrm{DblCat}$ of double categories and double functors. Unlike previous model structures for double categories, it recovers the homotopy theory of 2-categories through the horizontal embedding $\mathbb{H}\colon2\mathrm{Cat}\to\mathrm{DblCat}$, which is both left and right Quillen, and homotopically fully faithful. Furthermore, we show that Lack's model structure on $2\mathrm{Cat}$ is both left- and right-induced along $\mathbb{H}$ from our model structure on $\mathrm{DblCat}$. In addition, we obtain a $2\mathrm{Cat}$-enrichment of our model structure on $\mathrm{DblCat}$, by using a variant of the Gray tensor product. Under certain conditions, we prove a Whitehead theorem, characterizing our weak equivalences as the double functors which admit an inverse pseudo double functor up to horizontal pseudo natural equivalence. This retrieves the Whitehead theorem for 2-categories. Analogous statements hold for the category $\mathrm{wkDblCat}_s$ of weak double categories and strict double functors, whose homotopy theory recovers that of bicategories. Moreover, we show that the full embedding $\mathrm{DblCat}\to\mathrm{wkDblCat}_s$ is a Quillen equivalence.

math.AT↗
arXiv2021

A model structure for weakly horizontally invariant double categories

We construct a model structure on the category $\mathrm{DblCat}$ of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal and vertical morphisms, and fully faithful on squares; and whose fibrant objects are the weakly horizontally invariant double categories. We show that the functor $\mathbb H^{\simeq}\colon \mathrm{2Cat}\to \mathrm{DblCat}$, a more homotopical version of the usual horizontal embedding $\mathbb H$, is right Quillen and homotopically fully faithful when considering Lack's model structure on $\mathrm{2Cat}$. In particular, $\mathbb H^{\simeq}$ exhibits a levelwise fibrant replacement of $\mathbb H$. Moreover, Lack's model structure on $\mathrm{2Cat}$ is right-induced along $\mathbb H^{\simeq}$ from the model structure for weakly horizontally invariant double categories. We also show that this model structure is monoidal with respect to Böhm's Gray tensor product. Finally, we prove a Whitehead Theorem characterizing the weak equivalences with fibrant source as the double functors which admit a pseudo inverse up to horizontal pseudo natural equivalence.

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arXiv2020

Stable homotopy hypothesis in the Tamsamani model

We prove that symmetric monoidal weak n-groupoids in the Tamsamani model provide a model for stable n-types. Moreover, we recover the classical statement that Picard categories model stable 1-types.

math.AT↗
arXiv2019

Real K-theory for Waldhausen infinity categories with genuine duality

We develop a new framework to study real $K$-theory in the context of $\infty$-categories. For this, we introduce Waldhausen $\infty$-categories with genuine duality, which will be the input for such $K$-theory. These are Waldhausen $\infty$-categories in the sense of Barwick equipped with a compatible duality and a refinement of their (lax) hermitian objects generalizing the concept of Poincar\'e $\infty$-categories of Lurie. They may also be thought of as a version of complete Segal spaces enriched in genuine $C_2$-spaces whose underlying $\infty$-category carries a compatible Waldhausen structure, since we show that their respective $\infty$-categories are equivalent. We define the real $K$-theory genuine $C_2$-spaces by means of an enriched version of the $S_\bullet$-construction, defined for Waldhausen $\infty$-categories with genuine duality. Moreover, we prove an Additivity Theorem for this $S_\bullet$-construction which leads to an Additivity Theorem for real $K$-theory. Furthermore, such real $K$-theory satisfy a universal property -- analogous to that proved by Barwick for algebraic $K$-theory of Waldhausen $\infty$-categories --: We prove that every theory can be universally turned into an additive theory and identify our real K-theory with the universal additive theory associated to the functor that associates to a Waldhausen $\infty$-category with genuine duality its maximal subspace. Finally, we promote the real $K$-theory genuine $C_2$-spaces to genuine $C_2$-spectra.

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