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Let $F: C \to D$ be a functor between two categories without a right adjoint. What are some natural ways to create a right adjoint for $F$? Of course, this does not make sense on the nose. One needs t …

I am learning A-infinity category with Fukaya category in mind, and would like to understand the meaning of A-infinity relations. In particular, as $N=1$, it means $dd=0$. As $N=2$, it means that $d$ …

If I understand correctly, Dijkgraaf-Witten TQFT in dimension 2 is the following. Fix any finite group $G$, we define a field over a closed 2-manifold to be a principle $G$ bundle (it's automatically …

Space-level homotopy induces (co)chain homotopy, but I've never heard of the converse. I am not sure if it is simply not true? However, for good enough spaces (finite type nilpotent), Mandell proved …

Let $F$ be a finite field, $Sq^i$ be the $i$-th Steenrod operation $$ H^*(-;F) \to H^{*+i}(-;F).$$ By Yoneda lemma, such operation is a map $\phi_i: B^{*}F \to B^{*+i} F$, where $B$ denotes the delo …

This question is an extension of my previous question last year (see [2020]) in which I asked about the (consensus of a) definition of a weak $n$-category. Here are some background: while strict $n$-c …

In the nPOV, cohomology is realized as the connected component of the derived hom space [1]. Namely, $$H^n(X,Y) = \pi_0 \mathbb{H}(X,B^nY),$$ where $\mathbb{H}$ is an $(\infty,1)$-topos and $B$ is the …

1. Algebra from action groupoids Let $G$ be a finite group acting on a finite set $X$ from the right (denoted in element as $x^{g}$). We have an algebra (of the action groupoid) over $\mathbb{C}$: the …

Two possibly noncommutative rings are called Morita equivalent if their left-module categories are equivalent. In the commutative case, Morita equivalence is nothing more than ring isomorphism. Otherw …

This question is motivated by Why do combinatorial abstractions of geometric objects behave so well? The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials Kazhdan-Lusztig-Stanley polynomial …

Let $G$ be a finite group. It can be think of as a $1$-category with one object and $|G|$ many morphisms. If $A$ happens to be abelian, then one can think of it to an $n$-category. Conversly, this doe …

Let $C$ be a category, and $\Gamma$ a graph in $C$. Under good conditions it makes sense to talk about the limit $\lim \Gamma$ of $\Gamma$ in $C$. Suppose $\Gamma$ is the union of two subgraphs $\Gamm …

TL;DR. Many cohomologies can be unified using comonads. Question: which cohomologies cannot be? For each algebraic theory, there is an adjunction, and therefore a (co)monad (or called a (co)triple). …

Classically, two rings $R$ and $S$ are Morita equivalent if and only if any of the following is true ($R$-Mod) $\simeq$ ($S$-Mod). $S \simeq Hom_R(M,M)$, where $M$ is a finitely generated projective …

What's the standard definition, if any, of an $n$-category as of 2020? The literature I can tap into is quite limited, but I'll try my best to list what I had so far. In [Lei2001], Leinster demonstrat …