ct.category theory – When is a locally presentable category (locally) cartesian-closed?

Let $kappa$ be a regular cardinal. A category $mathscr C$ is locally $kappa$-presentable iff it is the free completion of a small $kappa$-cocomplete category under $kappa$-filtered colimits. Is there a known characterisation of the categories $mathscr C$ that are:

  1. locally $kappa$-presentable and cartesian-closed;
  2. locally $kappa$-presentable and locally cartesian-closed;

in terms of being the free cocompletion of a small $kappa$-cocomplete category with particular structure under $kappa$-filtered colimits?

ct.category theory – In a rigid monoidal category, why is $W^*otimes V^*$ a left dual of $V otimes W$?

My approach: We want to produce a coevaluation map $c: mathbf{1} rightarrow (V otimes W) otimes (W^* otimes V^*)$ and an evaluation map $e : (W^* otimes V^*) otimes (V otimes W) rightarrow mathbf{1} $ such that the maps

$$r_{Votimes W} circ (1_{Votimes W} otimes e) circ a_{Votimes W, W^* otimes V^*, Votimes W} circ cotimes 1_{Votimes W} circ l_{Votimes W}^{-1}$$

$$l_{W^*otimes V^*} circ (eotimes 1_{W^*otimes V^*} ) circ a_{W^*otimes V^*, V otimes W, W^*otimes V^*}^{-1} circ 1_{W^*otimes V^*}otimes c circ r_{W^*otimes V^*}^{-1}$$

are the identity maps on $V otimes W$ and $W^* otimes V^*$, respectively. Naturally one would want to use the maps $e_V, c_V, e_W, c_W$, so I suggest the following definition of $e$, suppressing the associativity constraint:

$$ e = W^* otimes V^* otimes V otimes W xrightarrow{1_W^* otimes e_V otimes 1_V^*} W^*otimes mathbf{1} otimes W cong W^* otimes W xrightarrow{ev_W} mathbf{1}$$

and similarly

$$ c = mathbf{1} xrightarrow{c_V} V^*otimes V cong (V^* otimes 1) otimes V xrightarrow{1_{V^*} otimes e_W otimes 1_V } V otimes W otimes W^* otimes V^*$$

It can be checked that these give $W^* otimes V^*$ the structure of a left dual by verifying the definition. Then, since all left duals are unique up to unique isomorphism, it follows that $(V otimes W)^* cong W^*otimes V^*$.

ct.category theory – In an abelian category, how to see surjective and injective imply isomorphism?

Suppose $f : X rightarrow Y$ is a map in an abelian category such that $ker f = 0$ and $mathrm{im} f = Y$ (up to isomorphism). How does one show $f$ is an isomorphism? If we had elements to work with, it’s easy to define the inverse map, but I am not sure how to categorify the idea of an inverse. My idea is to consider both the pullback $(Z, i_X, i_Y)$ of $f: X rightarrow Y$ and $1_Y : Y rightarrow Y$ and the pushout $(P, pi_X, pi_Y)$ of $f: X rightarrow Y$ and $1_X : X rightarrow X$, but I am not able to construct a map $g$ from $Y$ to $X$ such that $f circ g = 1_Y$ and $g circ f = 1_X$.

ct.category theory – Trees in chain complexes

$DeclareMathOperator{Ch}{mathit{Ch}}$Let $Ch_mathbb{Q}$ denotes the model category of chain complexes over rational numbers. Let $T_ast$ be a tree in $Ch_{mathbb{Q}}$ with $n$ vertices.

How to classify trees with respect to weak equivalences i.e., chain homotopies? Is it true that the classification can be recovered from the $mathit{ho}(Ch_{mathbb{Q}})$?

I think the key factor is that any chain complex $C_astcong oplus_n Vlangle nrangle_ast$, here $Vlangle nrangle_ast$ is the chain complex concentrated in degree $n$ and $Vlangle nrangle_n= H_n(C_ast)$

For example if we take a path with length 2, $f_ast : C_ast to C_ast’$ then it is equivalent to maps $H_n(f_ast): H_n(C_ast) to H_n(C_ast’)$, that is maps between vector spaces. We know that any map between vector spaces is completely describe by the dim(Ker). In this case, any path of length 2 is fully describe by $dim(mathrm{ker}(H_n(f_ast)))_n$.

I really appreciate it if someone could say something for trees.

ct.category theory – Pushout of generalised morphisms $C^*$-algebras

There is a known construction of pushout of $C^*$-algebras, or rather, the amalgamated free product, which is universal for commutative squares of $*$-homomorphisms. Jensen and Thomsen in their book Elements of KK-theory give, in an appendix, a detailed treatment for the free product (i.e the coproduct), for instance. However, $*$-homomorphisms are not the only morphisms between $C^*$-algebras that are reasonable to consider. For instance, a non-proper map $Xto Y$ between locally compact Hausdorff spaces induces a $*$-homomorphism $C_0(Y) to M(C_0(X)) simeq C_b(X)$, which is continuous for the strict topology on $C_0(Y)$. This motivates somewhat the more general type of morphism, where we can take a map $Arightsquigarrow B$ between $C^*$-algebras to be a strictly continuous $*$-homomorphism $Ato M(B)$, where $M(B)$ is the multiplier algebra of $B$. Any $*$-homomorphism $Ato B$ gives one of these more general morphisms, namely the composite $Ato B hookrightarrow M(B)$. It is a fun fact that as a set $M(B)$ is the completion in the strict topology of $B$, though we regard it as being equipped with both the strict topology and its (Banach) $C^*$-topology. These generalised morphisms compose by using the universal property of the strict completion, and so we have a category.

I’m interested in the pushout, in this category, of an arbitrary generalised morphism and a generalised morphism arising from an injective $*$-homomorphism, in particular the inclusion of a full corner.

Conjecture: A full corner inclusion pushes out to also give (the generalised morphism arising from) a full corner inclusion.

ct.category theory – Reference on internal categories and externalization

I’m looking for a reference on internal categories and externalization of internally defined notions.

The nlab has a stub on externalization (more details are available under small fibration) and the page on internal categories gives enough of an introduction that I can sketch most internal notions, but I could really use a concise introduction to internal categories and externalization, and if possible the relationship between internalization and externalization. Are they adjoint in some sense?

I’m fine assuming a background of $2$-category theory, so talking about the $2$-category of internal categories in a category with pullbacks etc. would make sense, but ideally the reference would assume no familiarity with internal category theory or externalization. Any assistance is appreciated.

ct.category theory – Filled natural transformations

According to the nlab, a commutative square


can be viewed as a ‘lifting problem’ between $f$ and $g$, and a solution to this lifting problem is a morphism $ell:Bto C$ filling the above square, so


commutes. Taking this view, a natural transformation $alpha:FRightarrow G$ is a collection of lifting problems between the images of $F$ and $G$ on Hom-sets.

Is the notion of a ‘solved’ or ‘filled’ natural transformation, such that for each arrow $f:Xto Y$ in the domain category we have a filled square, useful or studied anywhere?

By a filled natural transformation square for $f$ I mean a commutative square as below


This seems superficially related to the notion of a pseudonatural transformation, where $ell_f$ would be replaced by an isomorphic $2$-cell satisfying coherence diagrams, but I’m not sure there’s any actual connection. I would also be interested in the notion of an ‘orthogonal transformation’, where each $ell_f$ is unique so $F(f)$ and $G(f)$ are orthogonal for all $f$.

ct.category theory – tensor triangulated categories

Let $(mathcal{T},otimes)$ be a monoidal (not necessarily symmetric and possibly without unit object) triangulated category where $-otimes- $ is exact on both variables. Let $S$ be a set of objects in $mathcal{T}$ such that for any $s_1in S$ and $s_2in S$ we have that $s_1otimes s_{2}in S$.

I was wondering if the thick subcategory generated by $S$ is automatically monoidal?

ct.category theory – Lie monoids as monoids internal to the category of smooth manifolds?

This question can be thought as a complement to this one.

Lie groups can be defined as groups internal to the category of smooth manifolds. Lie monoids, however, as a particular case of Lie semigroups, seem to deserve a much more complicated definition (see, for instance, ‘Lie semigroups and their applications’, by Hilgert and Neeb, section 1.4).

Briefly, these are thought as closed subsemigroups of Lie groups, satisfying an extra property. This property, on its turn, is related to the infinitesimal counterpart of the notion of Lie semigroup (in the above reference, the notion of ‘Lie wedge’, whose definition, consequently, must precede that of a Lie semigroup).

What kind of difficulties appear if one tries to define a Lie monoid simply as a monoid internal to the category of smooth manifolds (or some related category)?


Lie groupoids, on their turn, can be defined as groupoids internal to the category of smooth manifolds. Is there an analogous notion of ‘Lie category’, in which morphisms are allowed not to be isomorphisms? Of course, the same question holds for its infinitesimal counterpart.

I tried to find some reference dealing with such a notion, but couldn’t. Though, it seems to be a reasonable one to consider even within the realm of Lie groupoid theory. For example, if one wants to allow distinct objects to have distinct automorphism groups, but still be connected by morphisms, this notion seems to be a necessary step.

In particular, that’s the case if one wants to allow morphisms between distinct objects to be not only isomorphisms between their automorphism groups, but also covering maps between them. I can’t think right now of a concrete example coming, say, from Physics, but it sounds possible that the ‘internal symmetries’ of a system might ‘collapse’ in this particular way.

Besides that, exactly as Lie groupoids can be considered as natural generalizations of Lie groups (even if this shouldn’t be considered the most appropriate point of view, for many reasons…), the ‘Lie categories’ would be natural generalizations of Lie monoids. Indeed, a ‘Lie category’ with one object would amount precisely to a Lie monoid.

Any references will be appreciated.

ct.category theory – Grothendieck derivator vs Infinite category

I have some questions on derivator and infinite category,

I would be grateful if someone could help me.

Is there some problems that infinite category/derivator can resolve but derivator/infinite category cannot resolve?

Why so much people prefer infinite category than Grothendieck derivator ?

is there a good place to learn about infinite category and Grothendieck derivator but with a historical and comparing point of view.

Thank you in advance!!