Abstracts for Abstract Homotopy Theory Seminar

Commutative rings are commutative algebra objects in abelian groups, but they can also be viewed as models of a Lawvere theory. If we don't insist on having inverses for addition, this admits a nice description: commutative semirings are product-preserving functors to sets from a category of "bispans" of finite sets. In this talk I will explain an infinity-categorical version of this comparison; in particular, using results of Gepner-Groth-Nikolaus, we can describe connective commutative ring spectra in terms of bispans of finite sets.

Gabriel-Ulmer duality is a contravariant biequivalence between small finite-limit categories and locally finitely presentable categories, which in the spirit of Lawvere's functorial semantics can be viewed as a *theory-model duality*: small finite-limit categories C are viewed as theories, and the lfp category FL(C, Set) of finite-limit preserving Set-valued functors is viewed as category of models of of C. The opposite of C can be reconstructed up to equivalence from Lex(C,Set) as full subcategory of *compact* objects. 

In my talk I will discuss a variant of lax colimit and lax limit for diagrams of weak (infinity, infinity)-categories thought of as directed homotopy types. I will demonstrate that this notion is appropriate to perform directed analogues of classical constructions of homotopy theory like suspensions, loop spaces and homotopy fibers, and therefore may be thought of as a directed version of homotopy colimits and homotopy limits. This is joint work with David Gepner.