New answers tagged ct.category-theory
3
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Accepted
What is the group completion of finite sets with respect to cartesian product?
As already addressed in the comments:
Group completing the groupoid of finite pointed sets under the smash product gives a contractible space.
The groupoid of finite sets under the cartesian product ...
3
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Accepted
Are there any interesting classes of limits containing finite limits?
An example of a category with all finite colimits and $\aleph_1$-filetered colimits but not all sequential colimits is the $\aleph_1$-filtered $Ind$-completion $Ind_{\omega_1}(\mathcal C)$ of any ...
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Strictifying strong monoidal functors
Here is another example that shows the answer to the second part (where both $C_i$ are strict) is negative.
Let $C_1$ be the discrete strict monoidal category whose monoid of objects is the Booleans. ...
5
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Are there any interesting classes of limits containing finite limits?
There's one class which I know and which feels natural enough to mention: L-finite diagrams.
Robert Parè proved in his paper Simply connected limits that L-finite limits are precisely the "...
3
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Accepted
Proof that the inclusion $\Delta \to \mathbf{Pos}$ preserves colimits
I’ll show that $\Delta \to \mathbf{TOrd}$ preserves colimits. Let $F: J \to \Delta$ be a diagram that admits colimit $L$ with morphisms $\phi_j: F(j) \to L$ for each $j \in J$. I claim that $(L, \{\...
3
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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-...
8
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Is there a better name for the "Mayer-Vietoris Octahedral axiom" and has it been studied?
In May's paper The additivity of traces in triangulated categories he has the following things to say about this property. (At least, I think it is an equivalent property -- his notation is different ...
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Homotopical Combinatorics
Back when this question was asked, several people suggested they were not sure what was meant by "homotopical combinatorics." Well, now there's a subfield known as homotopical combinatorics. ...
2
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Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces
I certainly respect going back to primary sources. But, in this case, it's helpful to remember that a LOT has been written about bisimplicial sets since Quillen's 1973 paper. For example, a reference ...
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What are the 2-categorical mono/epimorphisms in the 2-category of relations?
$\newcommand{\procirc}{\mathbin{\diamond}}\newcommand{\rightproarrow}{\mathrel{\rightarrow\mkern-17mu|\mkern7mu}}$I just realised one can apply pretty much the same argument for showing that 1-...
5
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Is there a way to study the relationship between the category of finite groups and their conjugacy classes categorically?
This question will likely be closed, but I'll try to give the OP some references before it is. First, let me answer the OP's question in the comments, regarding why this is vague and why the question ...
5
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$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$
If $A$ is a braided ∞-group, the delooping $\def\B{{\sf B}}\B A$ is an ∞-group.
Consider the ∞-category of spaces equipped with an action of the ∞-group $\B A$.
Since $\B Ω G≃G$, this ∞-category is ...
5
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Variation on definition of logical functors avoiding power objects
I don't know of a notion of "logical functor" between predicative topoi that specializes automatically to the standard notion if they happen to be elementary topoi. But I do know a ...
2
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Accepted
Quotients in categories of metric spaces
$\newcommand{cr}{\operatorname{cr}}$ Start by letting $X_T$ be any second-countable, functionally Hausdorff space which is not regular. Edit: We'll need to add an additional assumption here, the ...
2
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Why does mathematics seem to have a polarity bias?
$\newcommand\Set{\mathrm{Set}}$Rather than what we use more or less than what, I've wondered at why while category theory is a completely symmetrical theory, many dual things feel very different from ...
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Directed colimit of fully faithful functors
See Colimits of accessible categories by Paré and Rosicky.
5
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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 $\...
3
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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
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Morphisms of hammocks in the simplicial localization
The link in the OP leads to a thread with an answer by Charles Rezk, who wrote
For each "shape" of zig-zag, there is a "hammock category" for it... whose objects are functors $f\...
2
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Accepted
Minus sign in rotated triangles in triangulated categories
Let $(\mathcal{D},T)$ be an additive category with an autoequivalence $T:\mathcal{D}\to\mathcal{D}$. One could theoretically modify the (TR2) axiom to instead require that $X\xrightarrow{u}Y\...
2
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Accepted
Factorization systems for vector bundles
Normally, we consider vector bundles over some base space $X$. The simplest case is if $X$ is a point. Then the category of vector bundles over $X$ is just the category of vector spaces (over whatever ...
2
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Reference on Operads
Most textbooks have a portion of the front-matter devoted to telling the reader what mathematical prerequisites are expected. Tom Leinster's excellent book has already been mentioned in an answer (and ...
4
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English Reference for the Bénabou-Roubaud theorem
I cooked up a detailed proof in this file, even if it is not original, because I wanted to understand everything.
About Zoran Škoda's blog: it is factually false. Fibred categories appear right at the ...
0
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Is there a notion of a complex/analytic diffeological space?
Perhaps the general solution to your question can be found in "A theory of plots" by Atsushi Yamaguchi here? Slides from his talk at the last conference on diffeology and differential ...
2
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Categories that admit all products but not all coproducts
Opposites of Kleisli categories.
If C is a category with coproducts, and T a monad on C, then it is easy to see that the Kleisli category Kl(T) will inherit the coproducts from C. On the other hand, ...
Community wiki
0
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Limits in category theory and analysis
A more abstract version of the answer https://mathoverflow.net/a/120183 :
Recall that a filter $F$ has limit $x$ if all open sets containing $x$ are in $F$, equivalently, if
$$U_x\subseteq F$$
where $...
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1-categorical universal properties for the smash product of pointed sets
Since I can't quite figure it out as I mentioned in the comments, let me already answer what I can - the answer to Question I is yes, and I'm enclined that it is so also for Question II but I'm ...
15
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Categories that admit all products but not all coproducts
Now that the question was changed to existence of finite (co)products, here is a well-known example: the category $\mathbf{Gp}_{\text{fin}}$ of finite groups has finite limits, but not finite ...
Community wiki
10
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Categories that admit all products but not all coproducts
Here is an example appearing 'in nature': the category $\mathbf{Sch}$ of schemes has arbitrary (small) coproducts (disjoint unions), but existence of (small) products is a subtle question (on the ...
Community wiki
9
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Categories that admit all products but not all coproducts
Jeremy Rickard gives a nice example of a (locally small) category with all small colimits but not all small limits in an answer to Cocomplete but not complete abelian category. His category is even ...
Community wiki
7
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Categories that admit all products but not all coproducts
If $X$ is a large set and $\mathsf{Set}$ is the category of small sets, then $\mathsf{Set}/X$ is small-cocomplete, but it fails to have a terminal object. Now dualize this.
Community wiki
5
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Categories that admit all products but not all coproducts
As the question is stated, there are actually no examples, at least assuming classical logic.
Classically, every small complete category is thin, i.e., a preorder with arbitrary meets. (https://...
Community wiki
6
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Categories that admit all products but not all coproducts
For the version of the question about finite products and coproducts:
Any meet-semilattice that does not have joins. For example, a linear order with no least element (which has no empty join).
Take ...
Community wiki
5
votes
Accepted
Presentability rank of tensor product of presentable categories
The answer is yes, you can bound it by the maximum of the two accessibility ranks.
The point is the following: let $\otimes_\kappa$ denote the Lurie tensor product on $Cat_\kappa$, the category of ...
4
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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 ...
0
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Large V-categories admitting the construction of V-presheaves
(this should be a comment rather than an answer but I'm new here so I can't comment yet)
I take it that by "The $\mathscr V$-enriched presheaf category $[C^{op},\mathscr V]$ exists", you ...
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Poisson and homotopy Poisson operads
A non-cofibrant dg operad whose homology is $\mathrm{Pois}$ and which looks like $\mathrm{Com}\circ\mathrm{Lie}_\infty$ appears in Section 4.1 of this paper of Anton Khoroshkin and Pedro Tamaroff, ...
4
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Accepted
Why can we take the colimit over the category of elements?
If I'm not mistaken, a cosequence of the arguments in Murre's proof is that the inclusion $J^{\text{op}}\colon\mathcal I^{\text{op}}\hookrightarrow \mathcal E ^{\text{op}}$ is a final functor: It's ...
13
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Accepted
Plus construction on Simplicial Sets?
The answer is yes. This is spelled out in the book The local structure of algebraic K-theory by Bjørn Ian Dundas, Thomas G. Goodwillie and Randy McCarthy. Check out Section 1.6.1 on page 26, where ...
6
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Accepted
Locating the typed version of Hoàng Xuân Sính's thesis on Gr-categories
The TeXed version can be found on
https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/SinhThesis.pdf
Cristian David Gonzalez Avilés is the author of the TeX version. For reasons unknown to ...
2
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Infinite radical ideal cubed equals zero for tame hereditary Artin algebras
Any nonzero map between indecomposable preprojective modules cannot lie in the infinite radical.
For, we may as well assume the source is indecomposable projective. If there are $n$ indecomposable ...
2
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Infinite radical ideal cubed equals zero for tame hereditary Artin algebras
I think that for tame hereditary algebras, the maps in the infinite radical go preprojective $\to$ regular $\to$ preinjective, so composing three of these gives zero.
A possible route to confirming ...
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