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The derivative of a combinatorial species $S: core(FinSet) \to core(FinSet)$ is given by $S^\prime [N] = S[N\sqcup 1]$. Intuitively, an $S^\prime$-structure is built by introducing a "hole" to the set …

Call the walking arrow $\Delta_{1}$, containing exactly one nontrivial 1-cell $[0<1] : 0 \to 1$. I am interested in a map $\Phi : \Delta_{1} \to \mathrm{Type}$, such that $\Phi [0<1] = \Phi$ (and ther …

Algebras for endofunctors bridge the gap between functors acting on a category and structures defined in it. An algebra for an endofunctor $F$ is instantiated by some morphism $Fa \to a$, and more cru …

Displayed categories provide a natural categorification from classifying functions to the world of functors. The spirit of the idea is to encode a functor $ F: D \to C $ using a suitable 2-functor (la …

Let $S$ be a set. Is there a monoidal category $TS$ that we can construct from $S$ such that monoidal functors $F: TS \to M$ (up to monoidal natural isomorphism) correspond to $M$-enriched categories …

Marco Grandis has been working to collect and formalize the ideas of directed homotopy theory (his main work on the subject has been listed in the references at the nLab page on the subject: directed …

Given a locally presentable ($\infty$,1)-category $C$, can the fibrewise stabilization of it's codomain fibration, also called its tangent category $TC$, be given in terms of the Grothendieck construc …

I'm sorry in advance if this question does not belong on this site. I am curious as to what is "really" going on when you have a family of functors indexed by elements in a base category, all of which …

Is there a known method to write any category $ C $ as being equivalent to a slice category bundle $ \bar{C}_{/c}\to\bar{C} $, where $ C\simeq \bar{C}_{/c} $? It seems one can try to find a category $ …