arXiv subjects

Thomas Blom

Publications and source records attributed to Thomas Blom.

9 recordsLinked to original sources

arXiv2026

The product rule in Goodwillie calculus

In this paper, we prove a product rule for Goodwillie derivatives: given a differentiable $\infty$-category $\mathcal{C}$ whose stabilization is equivalent to the $\infty$-category $\mathrm{Sp}$ of spectra, we show that the derivatives functor $\partial_* \colon \mathrm{Fun}^\omega(\mathcal{C}, \mathrm{Sp}) \to \mathrm{RMod}_{\partial_*\mathrm{id}_\mathcal{C}}(\mathrm{SSeq}(\mathrm{Sp}))$ is strong symmetric monoidal, where the source is equipped with the pointwise tensor product and the target with Day convolution. Since the Koszul dual of $\partial_*\mathrm{id}_\mathcal{C}$ can be recovered as a coendomorphism operad from Day convolution, this product rule is useful for calculating the operad $\partial_*\mathrm{id}_\mathcal{C}$ in examples. We derive the product rule as a consequence of the more general statement that taking derivatives preserves cartesian products on the $(\infty, 2)$-categorical level. In fact, the main theme of this paper is that the extraction of Goodwillie derivatives preserves a lot of structure when regarded as a functor of $(\infty, 2)$-categories: apart from products, it also preserves cotensors and certain pullbacks. We illustrate how our results can be used to calculate Goodwillie derivatives by determining the operad structure on the derivatives of the identity functor in pointed spaces, algebras over an operad and sheaves on a site.

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arXiv2026

Day convolution for algebraic patterns

We characterize the exponentiable objects for a wide range of structures prevalent in $\infty$-categorical algebra, extending the construction of Day convolution to more general structures than $\infty$-operads. More precisely, we give a criterion that is both necessary and sufficient for many of these structures encountered in practice, such as (equivariant) $\infty$-operads and virtual double $\infty$-categories. We work within the framework of algebraic patterns of Chu-Haugseng that describe these structures in terms of weak Segal fibrations. As part of the proof, we give a new description of weak Segal fibrations in terms of generalized Segal spaces on certain "tree" categories. We also define the "underlying graph" of a weak Segal fibration, extending the notion of the underlying $\infty$-category for $\infty$-operads, and explicitly describe the underlying graph of exponential objects in weak Segal fibrations.

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arXiv2024

On the chain rule in Goodwillie calculus

We prove a generalization of the Arone-Ching chain rule for Goodwillie derivatives by showing that for any pair of reduced finitary functors $F \colon \mathcal{D} \to \mathcal{E}$ and $G \colon \mathcal{C} \to \mathcal{D}$ between differentiable $\infty$-categories, there is an equivalence $\partial_*(FG) \simeq \partial_*F \circ_{\partial_*{\mathrm{id}_{\mathcal{D}}}} \partial_*G$. This confirms a conjecture of Lurie. The proof of this theorem consists of two parts, which are of independent interest. We first show that the Goodwillie derivatives can be refined to a lax functor $\partial_* \colon \mathrm{Diff} \to \mathrm{Pr}^{\mathrm{Sym}}_{\mathrm{St}}$ from the $(\infty, 2)$-category of differentiable $\infty$-categories and reduced finitary functors to a certain $(\infty, 2)$-category of generalized symmetric sequences. Such a lax structure on the Goodwillie derivatives was long believed to exist, but has not been constructed prior to this work. We then finish the proof by studying the interaction of this lax functor with Koszul duality. In order to do so, we establish a new universal property of the bar-cobar adjunction.

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arXiv2024

On the straightening of every functor

We show that any functor between $\infty$-categories can be straightened. More precisely, we show that for any $\infty$-category $\mathcal{C}$, there is an equivalence between the $\infty$-category $(\mathrm{Cat}_{\infty})_{/\mathcal{C}}$ of $\infty$-categories over $\mathcal{C}$ and the $\infty$-category of unital lax functors from $\mathcal{C}$ to the double $\infty$-category $\mathrm{Corr}$ of correspondences. The proof relies on a certain universal property of the Morita category which is of independent interest.

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arXiv2023

Profinite completions of topological operads

We show that the particular profinite completion used by Boavida-Horel-Robertson in their study of the Grothendieck-Teichm\"uller group fits in the framework of profinite completion as a left Quillen functor. More precisely, we construct a model category of profinite up-to-homotopy operads based on dendroidal objects in Quick's model category of profinite spaces and show that the construction of Boavida-Horel-Robertson extends to a left Quillen functor into this model category. We also characterize the underlying $\infty$-category of this model category and obtain a Dwyer-Kan style characterization of the weak equivalences between such profinite up-to-homotopy operads. Since this model category of profinite up-to-homotopy operads is Quillen equivalent to the one considered in our earlier paper "Profinite $\infty$-operads", we obtain analogous results in that setting.

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arXiv2022

Replacing functors with enriched ones

We describe simple criteria under which a given functor is naturally equivalent to an enriched one. We do this for several bases of enrichment, namely (pointed) simplicial sets, (pointed) topological spaces and orthogonal spectra. We also describe a few corollaries, such as a new proof of a result of Lurie on Dwyer-Kan localizations.

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arXiv2022

A note on noncommutative CW-spectra

We use the machinery of arXiv:2009.07539 to give an alternative proof of one of the main results of arXiv:2101.09775. This result states that the category of noncommutative CW-spectra can be modelled as the category of spectral presheaves on a certain category M, whose objects can be thought of as "suspension spectra of matrix algebras". The advantage of our proof is that it mainly relies on well-known results on (stable) model categories.

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arXiv2021

Profinite $\infty$-operads

We show that a profinite completion functor for (simplicial or topological) operads with good homotopical properties can be constructed as a left Quillen functor from an appropriate model category of infinity-operads to a certain model category of profinite infinity-operads. The construction is based on a notion of lean infinity-operad, and we characterize those infinity-operads weakly equivalent to lean ones in terms of homotopical finiteness properties. Several variants of the construction are also discussed, such as the cases of unital (or closed) infinity-operads and of infinity-categories.

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arXiv2020

Simplicial model structures on pro-categories

We describe a method for constructing simplicial model structures on ind- and pro-categories. Our method is particularly useful for constructing "profinite" analogues of known model categories. Our construction quickly recovers Morel's model structure for pro-p spaces and Quick's model structure for profinite spaces, but we will show that it can also be applied to construct many interesting new model structures. In addition, we study some general properties of our method, such as its functorial behaviour and its relation to Bousfield localization. We compare our construction to the infinity-categorical approach to ind- and pro-categories in an appendix.

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