We are a learning seminar about \(\infty\)-nonsense. We will follow sections from Higher Algebra according to the interests of the participants.
We meet in SC 232 at 6-7pm on Thursdays. All talk titles are estimates, and subject to change.
Familiarity with \((\infty,1)\)-categories on the level of HTT §1-3 will be helpful, as well as some familiarity with model categories and stable homotopy theory; the fundamentals of higher category theory will be quickly reviewed.
This seminar is organized by Natalie Stewart and Taeuk Nam.
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Organizational meeting
Natalie and Taeuk
We will discuss logistics of the seminar; our main goal will be to gauge the interests of participants and get a chart outline of the curriculum and speakers for the first portion of the semester. -
HTT crash course I: \((\infty,1)\)-categories
Taeuk Nam
First, we will define infinity categories as simplicial sets that satisfy a certain lifting property. Then, we will compare our approach with other approaches to infinity categories. Finally, we will discuss how to extend ordinary category theoretic constructions to the setting of infinity categories.References:
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HTT crash course II: model structures on \(\mathbf{Set}_\Delta\)
Natalie Stewart
We define several classes of maps in \(\mathbf{Set}_\Delta\) and sketch the definition of (co)Cartesian fibrations.
We review the Quillen model structure on \(\mathbf{Set}_\Delta\), whose fibrations are Kan fibrations.
We go over the (unmarked) straightening and unstraightening constructions, and construct the covariant model structure on \((\mathbf{Set}_\Delta)_{/S}\), whose fibrations correspond with left fibrations over \(S\).
We go on to construct the Joyal model structure on \(\mathbf{Set}_\Delta\), which has fibrations the isofibrations (hence fibrant objects the quasicategories) and is Quillen equivalent to the Joyal model structure on \(\mathbf{Set}_\Delta-\mathbf{Cat}\), with left adjoint given by the simplicial nerve functor. In general, we emphasize a collection of quillen adjunctions relating the model structures mentioned above.
References:
- Lurie, Higher Topos Theory §2.1-2.3 for a full treatment, or
- Kerodon §4 for a readable summary of fibrations other than (co)Cartesian fibrations.
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HTT crash course III: the Cartesian model structure and the \((\infty,1)\)-category of \((\infty,1)\)-categories
Grant Barkley
We will discuss the notion of a Cartesian fibration of simplicial sets, generalizing the notion of a Grothendieck fibration of 1-categories. As in the 1-categorical setting, a Cartesian fibration of quasicategories should be thought as a family of quasicategories varying (contra)functorially over the base. By generalizing this perspective, we arrive at the construction of straightening and unstraightening functors. To place these constructions in the context of model structures, we will need to enhance the category of simplicial sets with some extra data, a marking of edges. The marked simplicial sets lying over a given simplicial set \(S\) have a model structure presenting the category of \((\infty,1)\)-presheaves on S. In particular, we get a new presentation of \(\mathbf{Cat}_\infty\) by taking \(S = *\). If we have time, we'll talk about how this perspective helps us compute limits in \(\mathbf{Cat}_\infty\).References:
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Stable \(\infty\)-Categories
Eunice Sukarto
We will define stability in an infinity-categorical setting. Roughly speaking, this means that fiber and cofiber sequences coincide. As in the 1-categorical setting, the homotopy category of a stable infinity category is triangulated. This means that we have a shift functor which is an equivalence, together with a collection of “distinguished triangles” (think: exact sequences). We will see that a pointed infinity category is stable iff it has all finite limits and colimits and pushout and pullback squares coincide. The correct notion of structure preserving morphisms between stable infinity categories are exact functors. This gives a subcategory \(\textbf{Cat}_{\infty}^{\mbox{Ex}} \subseteq \textbf{Cat}_{\infty}\) consisting of stable infinity categories and exact functors. In fact, the infinity category \(\textbf{Cat}_{\infty}^{\mbox{Ex}}\) has all small limits and small filtered colimits, which are preserved by the inclusion \(\textbf{Cat}_{\infty}^{\mbox{Ex}} \subseteq \textbf{Cat}_{\infty}\).References:
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Filtered objects, spectral sequences, and the Dold-Kan correspondence
Jit Wu Yap
In this talk, we will define the notion of t-structures for a triangulated category, and extend it to the setting of stable \(\infty\)-categories. We will then define the notion of a filtered object in a stable \(\infty\)-category, and construct a spectral sequence that generalizes the spectral sequence of a filtered complex in an abelian category. Finally we will review the classical Dold-Kan correspondence, and then introduce the \(\infty\)-categorical version of it.References:
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Special talk: monadicity and descent
Taeuk Nam
We will introduce the concept of monads and algebras over monads. Then, we will discuss the relationship between monads and adjunctions, which leads us to Beck’s monadicity theorem. Finally, we will go over applications to descent.References:
- These notes by Brantner for a general \(\infty\)-categorical sketch of monadicity.
- Lurie, Higher Algebra §4.7 for a full higher categorical overview, including some on an \(\infty\)-version of descent.
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The derived \((\infty,1)\)-category of an abelian category
Wyatt Reeves
We will talk about how the formalism of derived infinity categories can be used to understand vanishing cycles.References:
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The \((\infty,1)\)-category of spectra
Keeley Hoek
We'll discuss a few settings in which spectra naturally arise, before doing a bit of work and proving Lurie's \(\infty\)-version of Brown's representability theorem. Then we'll see how to build a stable infinite category from any infinity category with finite limits, a construction called taking spectrum objects. If we get time, after proving some basic properties of this construction---in particular that the resulting infinity category is always stable---we'll specialize to the infinity category of spaces and consider honest spectra.References:
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Foundations of \(\infty\)-operads.
Rushil Mallarapu
We will begin section 2 of HA with a discussion of the challenge of finding the right definition of “symmetric monoidal \(\infty\)-category” and why the formalism of operads saves the day. We will then define multityped operads, \(\infty\)-operads, morphisms of \(\infty\)-operads, algebra objects, and present some propositions characterizing how to build \(\infty\)-operads out of classical ones and recognizing coCartesian fibrations of operads. Finally, we will give some preliminary examples of \(\infty\)-operads, and if time permits, attempt to convince you that \(\infty\)-operads are really topological operads (up to some Quillen equivalence).References:
- Lecture notes here
- Lurie, Higher Algebra §2.1
- Lurie, Higher Algebra §6.3 for comparison with (co)monoids in symmetric sequences.
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Constructions of \(\infty\)-Operads
Natalie Stewart
We describe several methods of constructing \(\infty\)-operads and \(\mathcal{O}\)-monoidal \(\infty\)-categories from other ones. We first show that a \(\otimes\)-closed full subcategory of an \(\mathcal{O}\)-monoidal \(\infty\)-category is canonically \(\mathcal{O}\)-monoidal. We then state that a slice category of an \(\mathcal{O}\)-monoidal \(\infty\)-category over a \(\mathcal{O}\)-algebra is canonically \(\mathcal{O}\)-monoidal.We go on to construct coproducts in \(\operatorname{Op}_\infty\). We then construct the Boardman-Vogt tensor product for operads in \(\mathbf{Set}\) and for preoperads, and hencewe present the symmetric monoidal \(\infty\)-category \(\operatorname{Op}_\infty^\otimes\). We sketch the Eckmann Hilton argument in \(\mathbf{Set}\), and its homotopy-coherent generalization, called Dunn additivity. Time permitting, we describe the constructions of the monoidal envelope and Day convolution.
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Disintegration and assembly
Grant Barkley
We introduce the notion of unital \(\infty\)-operad, which generalizes the 1-operads with a unique nullary operation. We also introduce generalized \(\infty\)-operads, which generalize \(\infty\)-operads by allowing the space of nullary operations to be nontrivial. These can be thought of as families of \(\infty\)-operads over a base \infty-category. The usual \(\infty\)-operads are a reflective sub-\(\infty\)-category of generalized \(\infty\)-operads, and the reflection functor is called assembly. We call a family of \(\infty\)-operads reduced if each fiber \(\infty\)-operad has a contractible underlying category (i.e. as a colored operad it has one color). It turns out that assembly restricts to an equivalence between reduced families of \(\infty\)-operads over \(\infty\)-groupoids and those unital \(\infty\)-operads whose underlying category is an \(\infty\)-groupoid. The inverse of this equivalence is called disintegration.References:
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Products and coproducts
Michael Kural
In this talk, we investigate Cartesian and coCartesian symmetric monoidal structures on an \(\infty\)-category \(\mathcal{C}\). These structures are induced by taking finite products and finite coproducts, respectively (if they exist). We discuss the existence and uniqueness of such structures and analyze their properties using the \(\infty\)-operadic formalisms previously introduced. In particular, studying the \(\mathcal{O}\)-algebra objects of a Cartesian symmetric monoidal structure leads us to the notion of a \(\mathcal{O}\)-monoid, which specializes to a \(\mathcal{O}\)-monoidal \(\infty\)-category when applied to \(\operatorname{Cat}_{\infty}\). Our constructions also allow us to understand which parts of a (co)Cartesian structure still remain even without the assumption of the existence of finite (co)products on the underlying \(\infty\)-category.References:
We would like to acknowledge that this seminar takes place on the traditional, ancestral, and unceded territory of the Massachusett, the Pawtucket, the Wampanoag, and the Nipmuc peoples. They have stewarded for hundreds of generations the land that is now called Cambridge and the areas surrounding it.