Algebraic Theories and (Infinity,1)-Categories

We adapt the classical framework of algebraic theories to work in the setting of (infinity,1)-categories developed by Joyal and Lurie. This gives a suitable approach for describing highly structured objects from homotopy theory. A central example, treated at length, is the theory of E_infinity spaces: this has a tidy combinatorial description in terms of span diagrams of finite sets. We introduce a theory of distributive laws, allowing us to describe objects with two distributing E_infinity stuctures. From this we produce a theory of E_infinity ring spaces. We also study grouplike objects, and produce theories modelling infinite loop spaces (or connective spectra), and infinite loop spaces with coherent multiplicative structure (or connective ring spectra). We use this to construct the units of a grouplike E_infinity ring space in a natural manner. Lastly we provide a speculative pleasant description of the K-theory of monoidal quasicategories and quasicategories with ring-like structures.

💡 Research Summary

The paper undertakes a systematic extension of classical algebraic theories into the realm of (∞,1)-categories as developed by Joyal and Lurie, thereby providing a flexible language for describing highly structured homotopical objects. After a brief recollection of Lawvere theories in ordinary categories, the author introduces the notion of an “∞‑theory” as a limit‑preserving functor from a small ∞‑category of finite‑set‑based operations into a target (∞,1)-category. The central construction is a combinatorial model for the theory of E∞‑spaces: it is presented as the free ∞‑theory generated by span diagrams of finite sets, where a span encodes both product and coproduct information simultaneously. This span‑theory is shown to be equivalent, via a straightening‑unstraightening argument, to the usual model of E∞‑algebras in spaces, and it enjoys a transparent description of composition that makes higher coherence automatic.

Building on this, the author defines a “theory of distributive laws” by taking two copies of the span‑theory and adjoining 2‑dimensional cells that enforce a homotopy‑coherent distributivity condition. The resulting double‑span theory encodes precisely the data of an E∞‑ring space: a space equipped with two commuting E∞‑structures (addition and multiplication) together with a coherent distributive homotopy. The paper proves that algebras over this double‑span theory are equivalent to the classical notion of E∞‑ring spaces, but the proof avoids any strictification or rectification steps, relying entirely on the ∞‑categorical machinery.

The next major theme is the treatment of grouplike objects. By imposing a grouplike condition (π0‑level invertibility) on the span‑theory, the author obtains a theory whose algebras are infinite loop spaces. The associated “spectrification” functor is constructed directly inside the ∞‑theoretic framework, yielding connective spectra as models of the grouplike E∞‑theory. Extending the distributive‑law construction to the grouplike setting produces a theory of connective ring spectra, and the paper shows how the unit (the space of invertible elements) of a grouplike E∞‑ring space arises naturally as a sub‑algebra of the double‑span theory. This construction bypasses the traditional bar‑construction or operadic group‑completion techniques, offering a more conceptual description of units.

Finally, the author ventures into speculative territory by proposing a “pleasant” description of the K‑theory of monoidal quasicategories and of quasicategories equipped with ring‑like structures. The idea is to replace Waldhausen’s S‑construction with a span‑based K‑theory construction that respects the higher‑coherent algebraic operations encoded by the earlier theories. Although full proofs are deferred, the outline suggests that the span‑framework could lead to a more computationally tractable and conceptually unified approach to algebraic K‑theory in the ∞‑categorical setting.

Overall, the paper delivers a coherent, combinatorial, and highly flexible approach to algebraic structures in homotopy theory. By grounding E∞‑spaces, E∞‑ring spaces, infinite loop spaces, and connective ring spectra in a single span‑based ∞‑theory, it unifies several classical constructions under one homotopy‑coherent umbrella and opens new avenues for studying K‑theory and other higher‑categorical invariants.