Questions tagged [limits-and-colimits]
For questions on limits and colimts in the sense of category theory, and related notions.
333
questions
9
votes
0
answers
404
views
Milnor's universal bundle as a colimit?
I have had Milnor's construction of the classifying space of a topological group explained to me on multiple occasions, and seen it described briefly in various places. But only now am I reading the ...
- 1,175
5
votes
1
answer
298
views
Is Cauchy completion the largest extension with the same free cocompletion?
EDIT Title has been edited.
Let $C$ be a category, and $$\hat{C} = [C^{op}, (Set)]$$ be its free cocompletion. Despite its name, the free cocompletion of free cocompletion is not equivalent to the ...
- 5,038
1
vote
0
answers
70
views
Gluing categorical limit over subgraphs
Let $C$ be a category, and $\Gamma$ a graph in $C$. Under good conditions it makes sense to talk about the limit $\lim \Gamma$ of $\Gamma$ in $C$.
Suppose $\Gamma$ is the union of two subgraphs $\...
- 5,038
3
votes
1
answer
385
views
Filtered colim of F-groups
A group G is said to have a property F if there exists a finite aspherical CW-complex of which it is the fundamental group (according to wikipedia).
question: is there a full characterization of ...
- 1,603
5
votes
1
answer
223
views
Colimits of short exact sequences of C*-algebras
Assume I have an inductive system of short exact sequences of $C^{\ast}$-algebras (i.e., short exact sequences $0 \to A_n \to B_n \to C_n \to 0$ together with transformations from the $n$-th to the $(...
- 2,926
1
vote
0
answers
81
views
Examples of spaces which have explicit expression as colimits in $\mathrm{Top}$
$\DeclareMathOperator\Ball{Ball}$Question: What "well-known" spaces can be explicitly written down in the form $\bigcup_k \phi_k C(K_n,\mathbb{R}^m)$; where $K_n$ is a non-empty compact ...
- 4,969
4
votes
0
answers
319
views
Continuity property for Čech cohomology
Suppose we have an inverse system of compact Hausdorff spaces $\lbrace X_i , \varphi_{ij} \rbrace_{i\in I}$ and that each space has a presheaf $\Gamma_i$ assigned to it in such a way that $\Gamma_i(\...
- 255
7
votes
2
answers
734
views
Do filtered colimits commute with finite limits in the category of pointed sets?
It seems to be the case that filtered colimits commute with finite limits in the category Set (for instance, this is shown in Why do filtered colimits commute with finite limits?), but does the same ...
- 199
5
votes
0
answers
202
views
Pushout of $C^*$-algebras using generalised morphisms
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 ...
- 33.9k
7
votes
0
answers
256
views
Relation between two limit presentations of Eilenberg--Moore objects
Let $\mathbb{T}=({\cal T}\colon C\to C,\mu,\eta)$ be a monad (in the
$2$-category $\mathsf{Cat}$), which we view as a $2$-functor
$\mathbb{T}\colon\mathsf{B}\Delta_{\mathrm{a}}\to\mathsf{Cat}$ (where
$...
- 406
9
votes
1
answer
306
views
2-monads for categories with a class of (co)limits
This question concerns the strictness of (co)completions, at various levels of generality.
In Blackwell–Kelly–Power's Two-dimensional monad theory, the authors state
For instance, the 2-category $\...
- 8,755
9
votes
3
answers
774
views
Decomposing a (co)limit by decomposing the indexing diagram
Let $D: I \to \mathcal C$ be a diagram, and suppose we have a colimit decomposition $I = \varinjlim_{j \in J} I_j$ in $Cat$. Then under certain conditions, we can decompose the colimit of $D$ as $\...
- 61.6k
3
votes
0
answers
149
views
Classifying spaces of amalgamated topological monoids
Let $\mathsf{Top}_*$ be the category of well-based spaces and $\mathsf{TopMon}$ the category of topological monoids. Recall the James construction $\mathcal{J}:\mathsf{Top}_*\to \mathsf{TopMon}$ which ...
- 1,623
5
votes
2
answers
3k
views
Does a fully faithful functor always preserve limits and colimits?
I read on this n-lab page that a fully faithful functor $F: C\to D$ reflects all limits and colimits by the universal property.
On the other hand, I think a fully faithful functor does not always ...
- 8,707
2
votes
0
answers
109
views
Cofinality for natural transformations
Given a diagram $D\colon\mathcal{C}\longrightarrow\mathcal{D}$, we say that a functor $J\colon\mathcal{I}\longrightarrow{C}$ is cofinal if we have a natural isomorphism
$$
\mathrm{colim}\left(\mathcal{...
- 10.7k
6
votes
1
answer
348
views
Ferrand pushouts for algebraic stacks
Given algebraic spaces $X$, $Y$, $Z$ with a finite morphism $Y \rightarrow X$ and a closed immersion $Y \hookrightarrow Z$, the pushout $P \cong X \amalg_Y Z$ exists as an algebraic space (cf. Temkin ...
- 63
6
votes
1
answer
319
views
Weibel's H-book, Milnor's exact sequence for spectral sequence of filtered complex, Theorem 5.5.5
This is a question which I asked on StackExchange first, but might be more suited here.
I got stuck on the proof of Theorem 5.5.5 in Weibel's book. Not only that, but I also could not even find the ...
- 1,021
0
votes
1
answer
75
views
Ultrabornological representation for the space of uniformly continuous functions?
Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space
$$
C_{\omega_i}(\mathbb{R}^n,\...
- 4,969
9
votes
0
answers
356
views
When is an increasing union a colimit?
Let's consider a diagram $\Phi: \lambda \to \mathcal{T}_*$
$$
X_0 \to X_1 \to \cdots \to X_\xi \to X_{\xi+1} \to \cdots
$$
of pointed spaces,
indexed by some ordinal $\lambda$, in which each $X_\xi$ ...
- 12.5k
2
votes
0
answers
238
views
Every spectrum is the homotopy colimit of shifted suspension spectra
Let $X$ be a spectrum. In various places, I have encountered the statement that
$$
X \simeq \text{hocolim}_n \Sigma^{\infty-n}X_n.
$$
I was wondering how this homotopy colimit is defined, and why we ...
- 173
3
votes
1
answer
104
views
Is $C(\mathbb{R}^n)$ is a DF-Space?
I recently have begun reading about DF-spaces and its clear to me that $C(K)$ is a DF-space for any compact subset (non-empty) $K$ of some $\mathbb{R}^D$ for finite D, since $C(K)$ is Banach. However,...
- 4,969
11
votes
2
answers
945
views
How to understand adjoint functors?
I asked this same question on MathUnderflow two weeks ago but didn't receive any answer. Now that I am thinking more, it feels like the most suitable place for this question is here.
I have a good ...
- 1,019
5
votes
1
answer
321
views
Weighted Co/ends?
Recall: Limits
Recall that the limit of a functor $D\colon\mathcal{I}\to\mathcal{C}$ is, if it exists, the pair $(\mathrm{lim}(D),\pi)$ with
$\lim(D)$ an object of $\mathcal{C}$, and
$\pi\colon\...
- 10.7k
7
votes
1
answer
191
views
Algebras for products or limits of monads
If a category $C$ has limits of a certain type, then the category of monads on $C$ has the same type of limits, and these limits are computed "levelwise" (i.e. are preserved by the forgetful ...
- 40.2k
2
votes
0
answers
226
views
Ultralimit of metric spaces vs. inductive limits of underlying topological spaces
Let $\{(X_n,d_n)\}_{n =1}^{\infty}$ be a sequence of bounded metric spaces such that:
$X_n \subseteq X_{n+1}$ is a metric subspace of $X_n$.
Let $\omega$ denote a non-principal ultrafilter (i.e.: ...
- 4,969
5
votes
1
answer
450
views
Computation on homotopy colimit cocomplete triangulated categories
I have a couple of questions about dealing with homotopy (co)limits cocomplete triangulated categories.
Question I:The first one concerns a comment by Peter Arndt in this discussion about derived ...
- 6,000
1
vote
1
answer
294
views
Creating an inverse system which "stratifies density"
Setting:
Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying
$$
\bigcup_{n ...
- 4,969
4
votes
2
answers
1k
views
Sheaf cohomology commutes with colimits of sheaves
Let $X$ be a Noetherian scheme over a Noetherian ring $R$ and $(F_{\alpha})_{\alpha \in I}$ a direct system of $O_X$-module sheaves on $X$. I'm looking for source literature where I can find a proof ...
- 6,000
6
votes
0
answers
472
views
When is the dual of a limit the same as the colimit of the duals?
We all know that the dual of the colimit of a diagram in the category of chain complexes (and similar categories) is the limit of the duals diagram. This follows immediately from the general fact that ...
2
votes
0
answers
109
views
Primitive ideals of inductive limits of $C^*$-algebras
I am trying to understand ideals of direct limits in the category of $C^{\ast}$-algebras.
Let $(A_n,f_n)$ be a direct sequence of $C^{\ast}$-algebras and let $I$ be a primitive (modular) ideal of ...
- 1,085
8
votes
2
answers
830
views
Reference for homotopy colimit = total complex
I'm looking for a reference for the following fact:
take a simplicial chain complex $ X:\Delta^{op}\to Ch_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough ...
- 13.7k
5
votes
0
answers
123
views
Is the module of Kähler differentials a coend?
Let $\phi\colon R\to S$ be a ring map. The module of Kähler differentials $\Omega_{S/R}$ of $\phi$ can be constructed as the following coequaliser:
$$\left(\bigoplus_{(a, b)\in S^2} S[(a, b)]\right) \...
- 10.7k
0
votes
3
answers
1k
views
Injective maps and direct limits [closed]
I have the following question. Assume you have a category $\mathcal{C}$ such that direct limits exists. Let $(C_n)_{n\in\mathbb{N}}$ be a sequence of objects in $\mathcal{C}$ and consider the ...
8
votes
0
answers
103
views
Reference for limits of schemes with non-affine transitions?
Inverse systems of projective schemes appear in several contexts, for example:
in constructing the Zariski-Riemann space of a projective variety,
in studying subvarieties of a projective variety ...
- 4,382
12
votes
2
answers
510
views
Does forgetting colimits preserve colimits?
For each regular cardinal $\kappa$ let $\operatorname{Cat}_{\kappa}$ be the $(2,1)$-category of small categories with $\kappa$-small colimits, and functors that preserve those colimits. For each pair ...
- 698
1
vote
1
answer
396
views
Convergence in $C_c$ but not in $C$
Let $C_c(\mathbb{R})$ be the set of compactly-supported continuous functions on $\mathbb{R}$. We can view this with a number of different topologies but I have my eye on two in particular. Let $X$ ...
- 4,969
1
vote
0
answers
67
views
Comparison of inductive limit topology with very-rapidly decaying non-convex $L^{!/2}$-space topology
This is related to these posts and here.
Let $L^1([n,n+1])$ denote the subspace of $L^p$-functions on $[0,\infty)$ essentially supported on $[-n,n]$. Denote the accelerated $\ell^1$-direct sum ...
- 4,969
0
votes
1
answer
151
views
Convergence in LB-spaces
Edit:
Let $X$ be a strict LB-space described by $\lim X_n$ and suppose that $\{x_n\}_{n \in \mathbb{N}}$ converges in $X$. I'm looking for a reference showing that $x_n$ must converge in some $X_N$.
- 4,969
5
votes
0
answers
68
views
Relative completeness of a relative cocompletion of a subcategory
I'm going to use the language from Lack and Rosicky's Notions of Lawvere theory, but I won't be touching on actual enriched category theory.
Suppose I have a category $\mathbb{C}$ with a class of ...
- 1,253
2
votes
3
answers
451
views
Cocomplete and finitely complete category with nice pullbacks that is not locally presentable
I have a result that holds for cocomplete and finitely complete categories such that pullbacks preserve directed colimits, by which I mean $A \times_B (\operatorname{colim}_{i \in I} C_i) = colim_{i \...
- 758
1
vote
1
answer
272
views
Comparison of product topology and colimit topology in sequence spaces
In Munkres Theorem 20.4 it is shown that the (relative) uniform topology induced by:
$$
d(x,y)\triangleq \sup_{n \in \mathbb{N}} d(x_n,y_n)
$$
is strictly finer than the product topology on $\prod_{n \...
- 4,969
8
votes
2
answers
422
views
Classification of absolute 2-limits?
Let $\mathcal V$ be a good enriching category. Recall that an enriched limit weight $\phi: D \to \mathcal V$ is called absolute if $\phi$-weighted limits are preserved by any $\mathcal V$-enriched ...
- 61.6k
2
votes
1
answer
98
views
Sobolev topology on essentially compactly supported Sobolev-"functions"
The locally convex space of essentially compactly-supported $p$-integrable "functions" $\operatorname{L}_{\mathrm{comp}}^p(\mathbb{R}^d,\mathbb{R})$ is defined as the set
$$
\bigcup_{n \in \mathbb{N}} ...
- 4,969
5
votes
1
answer
161
views
Projective module which splits off sequence of submodules, but not the sum
Does there exist an example of a module $X$ over some ring $R$ together with submodules $T_i$ such that:
$X$ is projective,
$X$ splits as an internal direct sum $X\cong T_1\oplus T_2\oplus \ldots \...
- 858
16
votes
3
answers
926
views
Cofinality for coends?
Recall that a functor $I \xrightarrow u J$ is cofinal if it has the property that for any functor $J \xrightarrow F C$, we have that $\varinjlim F \cong \varinjlim Fu$ via the canonical map, either ...
- 61.6k
6
votes
1
answer
375
views
Is there such a thing as a weighted Kan extension?
The title pretty much sums it up.
More in detail. Let $C$, $D$ and $E$ be categories, let $F:C\to D$ and $G:C\to E$ be functors, and let $P:C^{op}\to \mathrm{Set}$ be a presheaf. The colimit of $F$ ...
- 2,129
4
votes
1
answer
276
views
Limit of split short exact sequences
Let $X$ be a module over some ring which splits as $$X\cong M_1\oplus S_1\cong M_1\oplus M_2 \oplus S_2 \cong M_1\oplus M_2 \oplus M_3\oplus S_3\cong \ldots$$
where the isomorphisms come from ...
- 858
21
votes
4
answers
2k
views
Conceptual reason that monadic functors create limits?
Let $U: Alg_T \to C$ be the forgetful functor from the category of algebras of $T: C \to C$ ($T$ could be a monad; I'm happy to think about the simpler case where $T$ is just an endofunctor or pointed ...
- 61.6k
1
vote
1
answer
192
views
Continuous function on colimit
Let $X$ be a Banach space and $f:X\rightarrow \mathbb{R}$ be continuous. Suppose that $\{X_n\}_{n \in \mathbb{N}}$ is a strictly nested sequence of sub-Banach spaces, for which $\cup_{n \in \mathbb{N}...
- 4,969
6
votes
0
answers
285
views
When is every element of a coend of abelian groups contained in one of the summands?
Let $I$ be a small category and let $D : I^{\mathrm{op}} \times I \to \mathsf{Ab}$ be a functor. The coend
$$\int^{i \in I} D(i,i)$$
can be constructed as the direct sum $\bigoplus_{i \in I} D(i,i)$ ...
- 61.5k