13
votes
Accepted
Homotopy groups of categories of elements as higher colimits
To answer these questions, the best is to note that $|\int_C D|$, the geometric realization of this total category, is equivalently the colimit of $D$, viewed as a functor with values in the $\infty$-...
12
votes
Accepted
Why isn't pullback-stability defined for individual colimits but for colimits with the same shape?
Pullback-stability is sometimes considered for individual colimits, or at least, smaller classes than “all colimits of shape $D$”. However, it’s most often used in settings where it holds for large ...
7
votes
Accepted
How do these definitions of factorization algebra compare?
I believe that Definition 2 and Definition 3 are equivalent. This involves that Definition 2 implies that F is multiplicative ("for each pairwise disjoint open sets
$U_{1},\dots,U_{k}\subset M$, ...
6
votes
Accepted
Inclusion of $1$-presheaves into $\infty$-presheaves preserves pushouts?
The inclusion from discrete presheaves to spaces doesn't preserve all pushouts, even when $\mathcal R$ is a point.
But it does preserves some pushouts. The usual condition is to check that at least ...
6
votes
Compact objects in slice categories of finitely presentable categories
This is true. It is for example easy to see that the full subcategory of objects of this form is closed under all finite colimits*, dense and that these objects are all finitely presentable. I ...
5
votes
Directed colimit of fully faithful functors
Both objects and morphisms in the colimit are given by the same colimit in the category of sets, which is to say that an object is an equivalence class of objects $\varphi=[c]$ with $c$ in some $\...
5
votes
Reference for homotopy colimit = total complex
I wrote a note for referential purposes. I hope this helps.
Arakawa, K. (2023). Homotopy Limits and Homotopy Colimits of Chain Complexes. arxiv.2310.00201
5
votes
Has the "Lambek embedding" into the category of (co)product-preserving presheaves been studied very much?
This construction is well known in enriched category theory, and has been studied in the increased generality of a class of weights (which recovers the finite discrete weights, describing finite (co)...
5
votes
Accepted
Does the 2-category of double categories and vertical transformations have flexible limits?
This 2-category is $T\text{-Alg}$ for a 2-monad $T$ on the complete 2-category ${\rm Cat}^{\rightrightarrows}$. Thus, by results in section 2 of Blackwell-Kelly-Power's "Two-dimensional monad ...
5
votes
Can every weighted colimit in a $\mathbf{Pos}$-enriched category be rephrased as a conical colimit?
Consider the terminal object $1\in\mathbf{Pos}$.
Then the closure of $1$ in $\mathbf{Pos}$ under all weighted colimits is $\mathbf{Pos}$ itself: If $X\in\mathbf{Pos}$, then $X\cong X\cdot 1$ is the ...
4
votes
When does base-change in topological spaces preserve quotient maps?
Thanks to the comments, I found the paper Exponentiability for maps means fibrewise core-compactness by G. Richter, which gives a more concrete (albeit still complicated) characterization of ...
4
votes
Has the "Lambek embedding" into the category of (co)product-preserving presheaves been studied very much?
It's not quite the same thing, but the full subcategory of presheaves preserving finite products (say call it $C^{f\times}$) is studied by Lurie in Higher Topos Theory section 5.5.8.
While $C\to\...
4
votes
What does it mean for a category to be generated under (some) colimits?
It seems that dense generation does not imply 1-naïve generation in general. For a counterexample it is enough to consider the class $\Phi$ for small (or just finite) coproducts.
Indeed, given any non ...
4
votes
Accepted
Co/limits and 2-co/limits of categories in the $\infty$- and $(\infty,2)$-category of $\infty$-categories
The case of limits is very distinct from the case of colimits. Let me start with the case of colimits.
In general, none of these agree. There is a slight "refinement" of question I which ...
3
votes
How do these definitions of factorization algebra compare?
OK, here is the full story, which confirms what is written in Daniel Bruegmann's answer. In what follows, I will work with a fixed
prefactorization algebra $F$ and an $n$-manifold $M$. I will make
use ...
3
votes
Does coproduct preserve cohomology in differential graded algebra category
The coproduct in the category of non-unital dg algebras is maybe easier to think about. Indeed note that the two relations in Jardine's note only have to do with the units in the two algebras. The non-...
3
votes
Categories that admit all finite products but not all finite coproducts
This thread was a bit awkward because the OP asked a question, got answers, then edited it to match the current question, got more answers, and then the edits were rolled back and the OP asked the ...
Community wiki
3
votes
Accepted
Does the forgetful functor from a pointed $\left(\infty, 1\right)$-category only create weakly contractible colimits?
A short answer is that if $C$ is the terminal category, then $1/C \to C$ is an equivalence, and hence creates all colimits.
Less flippantly, we can take $C$ to be the ordinary topos of sets. Then the ...
3
votes
Accepted
Does forgetting colimits preserve colimits?
Here is an example showing that $U_{\tau, \kappa}$ does not preserve pushouts for any pair of regular cardinals $\tau > \kappa$. Let $C = \tau$, where here $\tau$ is thought of as an ordinal (i.e. ...
3
votes
Comparing stabilization of stable category modulo injectives and a Verdier localization
This follows by applying Theorem 3.8 of Beligiannis' 2000 paper to the opposite
categories.
$\mathcal{I}$ is a full additive subcategory of $\mathcal{A}$, closed under
direct summands. It is ...
2
votes
Accepted
Vanishing of self-hom in Spanier–Whitehead stabilization category
What you want is statement 4.2 in Kropholler's paper, "On groups of type $(FP)_\infty$" (1993). This states that your colimit is zero if and only if the module $M$ has finite projective ...
2
votes
Homotopy colimit commutes with homotopy groups
Giving details along the hint given, note that given a (discrete) commutative ring $A$ and an element $x\in A$, the colimit of the sequence $$
A \xrightarrow{x} A \xrightarrow{x} A \xrightarrow{x} A \...
2
votes
Algebras for products or limits of monads
For finitary commutative monads on $\mathbf{Set}$, this has been studied in Faro–Kelly's On the canonical algebraic structure of a category. I have reworded Proposition 11 ibid. below in terms of ...
2
votes
Accepted
Weighted limits and Kan extension in Dist
I'm not sure I've followed your question exactly, so let me rephrase it to how I've understood it, and you can tell me if I've misunderstood. I shall use different letters to make sure I'm not ...
2
votes
Kernels and cokernels in a quotient of an abelian category
The kernel and cokernel can be defined in any pointed category $\mathcal C$ with finite limits and colimits. Recall that a category is pointed it has an object that is both initial and terminal, and ...
1
vote
Directed colimit of fully faithful functors
See Colimits of accessible categories by Paré and Rosicky.
1
vote
Day convolution and sheafification
$\require{AMScd}$I would say that the fact that $\bf a$ commutes with products is what allows you to apply it to all factors of the coend, but a more precise way to use its properties would be that
$${...
1
vote
Commuting homotopy colimits and arbitrary products in spaces
I will answer my own question, in hope that it is helpful to someone.
Given a functor $X:D \rightarrow Spc$ of $\infty$-categories, we can take the unstraightening of $X$ (the appropriate ...
1
vote
Pointwise Kan extensions VS weighted limits
I think the equivalence becomes trivial if you add the hidden assumption, that is the hypothesis of representability of the extension in $Dist$
This is akin to Kelly's remark about limit in enriched ...
1
vote
Does forgetting colimits preserve colimits?
Here's an example to play around with, with $\kappa = \omega$ and $\tau = \omega_1$ and $Vect = Vect_k$ for $k$ a finite field, look at the pushout of $Vect \leftarrow Set \to Vect$, where both ...
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