How to compute (co)limits of enriched categories?

I’ve asked this question on math.stackexchange some time ago ( and I received no complete answers, so I’m posting it here. Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories enriched over $\mathscr{V}$. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$. This leads to some questions. 1)When … Read more

Pushforward of an internal category along a functor

Let $F:C\to D$ be a “nice” functor (for example, $H_*(-;\mathbb{Z}):\mathbf{Top}\to \mathbf{Ab}^{\mathbb{Z}}$). Now assume that we have a category $O$ internal to $C$. Is there a canonical way to construct a category $F_*O$ which is internal to $D$? In the case where $C$ and $D$ are monoidal and $O$ is not internal, but only enriched, the … Read more

Kan extension of conservative functors

Suppose the right Kan extension RanFG of a conservative functor F along a conservative functor G exists (with the category domF=domG not necessarily small). Is it conservative itself? Answer No, take A to be any cocomplete category. For any category I the right Kan extension of Id:A→A along the embedding ι:A→Fun(I,A)(object a goes to the … Read more

A Reference on Multicategories with “Internal Hom”

The multicategory of Waldhausen categories is “enriched over itself”: the Hom-set of k-exact functors can be given a Waldhausen category structure by letting the morphisms be natural transformations, and cofibrations/weak equivalences be levelwise. With this construction, k-fold composition is a k-exact functor Hom(Ek−1,Ek)×⋯×Hom(E0,E1)⟶Hom(E0,Ek). (It also works for general composition, not just on the 1-level, but … Read more

Why is the category of all small S\mathbf{S}-enriched categories locally presentable?

In Lurie’s Higher Topos Theory Proposition A.3.2.4, the author used Proposition A.2.6.15 to prove that for any combinatorial monoidal model category S with all objects cofibrant and weak equivalences stable under filtered colimits, the category CatS is a left proper combinatorial model ctegory, where he implicitly used the statement: CatS is locally presentable. Why is … Read more

Is there a nice way to define discrete enriched categories?

In the setting of classical category theory, one defines the discrete category associated to a set $X$ as the category $X_\mathsf{disc}$ having the elements of $X$ as its objects and only identities as morphisms. This extends to a functor $\mathsf{disc}\colon\mathsf{Sets}\longrightarrow\mathsf{Cats}$, sitting in a quadruple adjunction $$ \pi_0\dashv\mathsf{disc}\dashv\mathrm{Obj}\dashv\mathsf{indisc}\colon\mathsf{Cats}\overset{\rightleftarrows}{\rightleftarrows}\mathsf{Sets}. $$ Passing now to $\mathcal{V}$-categories, is there a … Read more

Are weighted limits terminal in a category of cones?

Consider a Benabou-cosmos (V,⊗,J), V-categories I,C and V-functors W:I→V and D:I→C. The usual definition of a W-weighted limit of D is as a representing object of the functor [I,V](W,C(−,D)):Cop⟶V While this certainly works fine and one can do amazing things with it, it completely hides the limit-cones I know and love, what made it hard … Read more

$V$-cat and $V$-graph: coequalizers in the category of enriched functors

This question is regarding the 1974 JPAA paper $V$-cat and $V$-graph by Harvey Wolff. To be precise, I don’t understand a certain step in the proof of Corollary 2.9, which (the corollary) is crucial for the main theorem of the paper. Let $V$ be a closed symmetric monoidal category with coequalizers, $A$ an $V$-enriched category … Read more

Enriched Cauchy completions and underlying categories

The ordinary Cauchy completion ¯C of a small category C satisfies a number of conditions: Every idempotent in ¯C splits, there’s an equivalence of categories [Cop,Set]≃[¯Cop,Set], etc… There’s also a notion of Cauchy completion for enriched categories, my questions are about it: 1 – Let X be a V-enriched category (where V is a closed … Read more

Weighted Co/ends?

Recall: Limits Recall that the limit of a functor $D\colon\mathcal{I}\to\mathcal{C}$ is, if it exists, the pair $(\mathrm{lim}(D),\pi)$ with $\lim(D)$ an object of $\mathcal{C}$, and $\pi\colon\Delta_{\lim(D)}\Rightarrow D$ a cone of $\lim(D)$ over $D$ such that the natural transformation $$\pi_*\colon h_{\lim(D)}\Rightarrow\mathrm{Cones}_{(-)}(D),$$ is a natural isomorphism, where $\mathrm{Cones}_{(-)}(D)\overset{\mathrm{def}}{=}\mathrm{Nat}(\Delta_{(-)},D)$, and The component at $X\in\mathrm{Obj}(\mathcal{C})$ of $\pi_*$ is the map … Read more