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Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

7 votes
0 answers
84 views

Can cyclic and simplicial objects be related in a similar way to how the species of linear o...

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 …
2 votes
0 answers
207 views

A map that names itself

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 …
1 vote
1 answer
91 views

Algebras for general transfors

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 …
3 votes
0 answers
149 views

Displaying displayed categories

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 …
5 votes
1 answer
185 views

Is there a monoidal category that coclassifies enriched category structures for a given set?

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 …
3 votes
0 answers
134 views

Fundamental monoid pertaining to adjunctions

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 …
2 votes
1 answer
292 views

Can tangent ($\infty$,1)-categories be described in terms of the higher Grothendieck constru...

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 …
3 votes
0 answers
133 views

Is there a construction capturing indexed families of adjunctions?

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 …
0 votes
1 answer
351 views

Writing categories as slice categories

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 $ …