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There are a few ways to go about this. First, let us observe that the surjections in $\mathbf{Set}$ are precisely the maps that have the left lifting property with respect to the inclusion $\emptyse …

Your understanding of the definition is incorrect, but that is probably because the cited nLab page is misleading. Let me spell it out a little bit more accurately: Let $\mathfrak{K}$ be a 2-categ …

Here is the correct statement: Let $\mathfrak{Cat}$ be the 2-category of locally small categories (and all functors) and let $\mathfrak{Cocomp}$ be the 2-category of locally small cocomplete cate …

Since you mentioned locally presentable categories, I'll give one sufficient condition involving that. Let $\mathcal{A}$ be a locally finitely presentable additive category. Then the canonical morphi …

As Daniel Schäppi pointed out, this is the same as asking whether the composite of two monadic functors is monadic. The answer, unfortunately, is no. Consider locally presentable categories. By the …

When $\mathcal{A}$ is a finite category, the finitely presentable objects in $[\mathcal{A}, \mathcal{C}]$ are precisely the diagrams that are componentwise finitely presentable: see e.g. Proposition 2 …

Since $Z$ is right orthogonal to every component of the adjunction unit, it is in particular right orthogonal to $\eta_Z : Z \to R Z$. Thus $\eta_Z : Z \to R Z$ admits a retraction, say $r : R Z \to Z …

Let's start with the cartesian monoidal case, since it is the easiest one to understand. Recall that the Yoneda embedding is left exact, so if we have a monoid $M$ in a category $\mathcal{C}$, then $\ …

In principle $\textrm{End}(X)$ could have two elements, $\textrm{id}_X$ and $0_X$. If it has only one element, then $\textrm{id}_X = 0_X$, so for all $f : X \to Y$, $f = f \circ \textrm{id}_X = f \cir …

I previously indicated how to construct the tensor product for $\textbf{Ab}(\mathcal{V})$ when $\mathcal{V}$ is a sufficiently nice cartesian closed category. (The comments are probably more helpful t …

Given an orthogonal (resp. weak) factorisation system $(\mathcal{E}, \mathcal{M})$, we can straightforwardly define when a sink $( U_i \to V : i \in I )$ (where the morphisms $U_i \to V$ may be repeat …

Here's a simple observation: the category of commutative monoids has an object that is both initial and terminal (just like the category of groups or abelian groups), so it cannot be cartesian closed. …

Here is a tentative definition based on finitary unbiased monoidal categories – I make no claims of usefulness! An infinitary unbiased monoidal category consists of the following data: An ordinary …

In short: always. Indeed, given a functor $F : \mathcal{C} \to \mathcal{D}$ left adjoint to $G : \mathcal{D} \to \mathcal{C}$, the triangle identities say that the composites of the canonical morphis …

There is no difference for $\kappa = \aleph_0$. The point is that you can build colimits for filtered diagrams using just colimits for chains. Every filtered category $\mathcal{J}$ admits a cofinal …