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I can elaborate on the second example a bit more: Let $T$ be a Lawvere theory. Then $T$ is commutative iff $T \cong Lawv(\text{hom}_T(x,-))$, where $x$ is the generic object in $T$. A Lawvere theory …

A monoid is the same as a category with a single object. A monoidal category is the same as a bi-category with a single object. A commutative monoid is the same as a bi-category with a single object …

I am trying to understand Nikolai Durov's "New Approach to Arakelov Geometry" right now and it got me thinking about a particular thing. Let $R$ be a commutative, associative ring with unit. We can …

The axiom AB6 holds in any category of sheaves on a site that can be constructed using only finite coverings. The reason is the following: If checking for finite covers suffices, then filtered colimit …

If $(C,J)$ is a site, what is a natural condition on the Grothendieck topology $J$ to ensure that the category $Sh(C,J)$ is compactly assembled? I am both interested in the 1-categorical as well as th …

This is in some sense a follow up to the question asked here Properties of the category of compact Hausdorff spaces To clarify: The category $\text{Prof}$ of profinite sets sits inside the category $\ …

Efimov has in his recent preprint K-theory and localizing invariants of large categories, Appendix F, a long table of analogies between the categories $\text{Cat}^\text{dual}_\text{st}$ and $\text{Com …

It is the classifying category for the left action of $C$ on its product $C \times C$. Let me elaborate on this a bit further. Let $C$ be an $E_\infty$-monoid, for example represented by a symmetric m …

Consider an $E_n$-monoid X. We can deloop $X$ to an $\infty$-category $\mathbf{B}X$. There's a natural functor $X^\circlearrowleft : \mathbf{B}X \rightarrow \text{Spc}$ given by the left action of $X$ …