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

The k-ification of the compact-open topology for weak Hausdorff compactly generated spaces

Let CGWH be the category of weak Hausdorff compactly generated spaces; see e.g. N.P. Strickland. THE CATEGORY OF CGWH SPACES: Preprint available from 2009 or Tammo tom Dieck. Algebraic topology. Zurich: European Mathematical Society (EMS), 2008. (7.9) CGWH is well known to be a cartesian closed category (references above prove this). However, there appears … Read more

Is simplicial localization part of a Quillen equivalence between relative categories and simplicial categories?

There are many models for ∞-categories. One of them is relative categories – AKA categories with weak equivalences – which have a model structure due to Barwick–Kan. Another one is simplicial categories with Bergner’s structure. It is otherwise known that these two are Quillen equivalent, but the equivalence goes through complete Segal spaces and quasi-categories … Read more

When is a right lifting property closed under pushouts?

A class of morphisms defined by a right Quillen lifting property (weak orthogonality) is always closed under pullbacks (limits); under what assumptions will it be closed under pushouts (colimits)? In a model category it makes sense to use fibrant replacement and ask when will it be closed under taking fibrant replacement of pushouts or colimits. … Read more

Projective objects in the category of simplicial objects in an abelian category

Let $S\mathcal{A}$ be the category of simplicial objects in an abelian category $\mathcal{A}$. In exercise 8.4.5 in Weibel’s An Introduction to Homological algebra, it is said that $P \in S\mathcal{A}$ is projective if each $P_i$ is projective and $P$ is null-homotopic. By Dold-Kan correspondence we can pass to the category of chain complex $C\mathcal{A}_{\ge0}$. Consider … Read more

K theory of a simplicial monoidal category, Cofinality theorem

Let X=(d↦Xd) be a simplicial symmetric monoidal category. We define the K-theory space of X to be K(X)=|d↦K(Xd)|, the geometric realisation of the simplicial space d↦K(Xd). Classically (i.e. for non-simplicial categories) we have the cofinality theorem that states that a full and cofinal functor Y→X between symmetric monoidal categories induces an isomorphism on K-theory in … Read more

Non-enriched Bousfield localizations

We know that whenever we have a Bousfield localization between two simplicial model categories, this gives rise to a reflective subcategory in ∞-categories (or coreflective, depending on the direction of the Bousfield localization). I’m interested in what happens if the model categories at issues are not simplicial, or even in the intermediate case when the … Read more

Kan complexes and semigroups

Given a simplicial commutative semigroup: (1) is it true that its underlying simplicial set is a Kan complex if and only if the simplicial semigroup was a simplicial group? (2) is the constant simplicial set on a set, Kan fibrant? A positive answer to (2) would give a negative answer to (1), since the constant … Read more

Homotopy coherent space maps induces homotopy coherent chain complex morphisms

It is an elementary fact that a map f:X→Y between spaces induces a chain complex morphism f∗:C∗(X)→C∗(Y). This allows one to transfer 1-category theoretic arguments from spaces to associated chain complexes. When dealing with ∞ stuff, sometimes a bit more is needed. Let Sing(Top) be the simplicially enriched category of topological spaces with MapTop(X,Y)n=HomTop(X×Δn,Y) And … Read more

Filtered colimit of fibrations

In a model category $\mathcal{C}$, is the filtered colimit of fibrations, resp. trivial fibrations, a fibration, resp. trivial fibration? Thm. in Toen-Vezzosi’s “Homotopical algebraic geometry, II” ( seems to give a criterion, but it points to the wrong reference, as nothing of the sort is in Hovey’s book “Model Categories”. Answer In Lemma 7.4.1 … Read more