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Model categories, pro-categories and functors
Stockholm University, Faculty of Science, Department of Mathematics.
2023 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of five papers. The first three are concerned with various model structures on ind- and pro-categories, while the last two are concerned with the homotopy theory of functors.

In Paper I, a general method for constructing simplicial model structures on ind- and pro-categories is described and its basic properties are studied. This method is particularly useful for constructing "profinite" analogues of known model categories. It recovers various known model structures and also constructs many interesting new model structures. 

In Paper II, it is shown 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 of the latter model category is inspired by the method described in Paper I, but there are a few subtle differences that make its construction more involved.

In Paper III, the general method from Paper I is used to give an alternative proof of a result by Arone, Barnea and Schlank. This result states that the stabilization of the category of noncommutative CW-complexes can be modelled as the category of spectral presheaves on a certain category M. The advantage of this alternative proof is that it mainly relies on well-known results on (stable) model categories.

In Paper IV, the question of whether an ordinary functor between enriched categories is equivalent to an enriched functor is addressed. This is done for several types of enrichments: namely when the base of enrichment is (pointed) topological spaces, (pointed) simplicial sets or orthogonal spectra. Simple criteria are obtained under which this question has a positive answer.

In Paper V, the Goodwillie calculus of functors between categories of enriched diagram spaces is described. It is shown that the layers of the Goodwillie tower are classified by certain types of diagrams in spectra, directly generalizing Goodwillie's original classification. Using this classification, an operad structure on the derivatives of the identity functor is constructed that generalizes an operad structure originally constructed by Ching.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2023. , p. 43
Keywords [en]
Homotopy theory, Quillen model categories, Pro-categories, Enriched categories, Goodwillie calculus
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-215391ISBN: 978-91-8014-232-8 (print)ISBN: 978-91-8014-233-5 (electronic)OAI: oai:DiVA.org:su-215391DiVA, id: diva2:1742833
Public defence
2023-05-25, lärosal 4, hus 1, Albano, Albanovägen 28, Stockholm, 13:15 (English)
Opponent
Supervisors
Available from: 2023-05-02 Created: 2023-03-12 Last updated: 2023-03-27Bibliographically approved
List of papers
1. Simplicial model structures on pro-categories
Open this publication in new window or tab >>Simplicial model structures on pro-categories
(English)Manuscript (preprint) (Other academic)
Abstract [en]

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 ∞-categorical approach to ind- and pro-categories in an appendix.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-197652 (URN)
Available from: 2021-10-12 Created: 2021-10-12 Last updated: 2023-03-12Bibliographically approved
2. Profinite ∞-operads
Open this publication in new window or tab >>Profinite ∞-operads
2022 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 408, article id 108601Article in journal (Refereed) Published
Abstract [en]

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 ∞-operads to a certain model category of profinite ∞-operads. The construction is based on a notion of lean ∞-operad, and we characterize those ∞-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) ∞-operads and of ∞-categories. 

Keywords
Dendroidal sets, Infinity-operads, Lean infinity-operads, Profinite completion, Quillen model categories
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-211817 (URN)10.1016/j.aim.2022.108601 (DOI)000860763500003 ()2-s2.0-85135502790 (Scopus ID)
Available from: 2022-11-28 Created: 2022-11-28 Last updated: 2023-03-12Bibliographically approved
3. A note on noncommutative CW-spectra
Open this publication in new window or tab >>A note on noncommutative CW-spectra
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We use the machinery of Paper I to give an alternative proof of one of the main results of [ABS21]. 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.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-197659 (URN)
Available from: 2021-10-12 Created: 2021-10-12 Last updated: 2023-03-12Bibliographically approved
4. Replacing functors with enriched ones
Open this publication in new window or tab >>Replacing functors with enriched ones
(English)Manuscript (preprint) (Other academic)
Abstract [en]

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 result on simplicial Dwyer-Kan localizations that may be of independent interest.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-215388 (URN)
Available from: 2023-03-12 Created: 2023-03-12 Last updated: 2023-03-12
5. Goodwillie calculus for diagram categories
Open this publication in new window or tab >>Goodwillie calculus for diagram categories
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We describe the Goodwillie calculus of functors between categories of enriched diagrams and generalize Goodwillie’s classification of n-homogeneous functors to this setting. Using this classification, we construct an operad structure on the derivatives of the identity functor from TopC to itself that directly generalizes Ching’s operad structure on the derivatives of the identity functor from Top to itself.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-215389 (URN)
Available from: 2023-03-12 Created: 2023-03-12 Last updated: 2023-03-12

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Blom, Thomas

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