All Questions
Tagged with colimits or limits-and-colimits
105 questions with no upvoted or accepted answers
15
votes
0
answers
332
views
Which limits distribute over which colimits in $Set$? How about in $Spaces$?
I've never really thought much about distributivity of limits and colimits -- I tend to think more about commutativity of limits and colimits. This question makes me want to change that.
The question ...
- 63.9k
14
votes
0
answers
919
views
Grothendieck construction and coends
In category theory, both the Grothendieck construction and coends are represented by a sort of "integral sign", respectively:
$$
\int F
$$
for a functor $F:C\to\mathbf{Cat}$,
and:
$$
\int^x G(x,x)
$$
...
- 2,129
13
votes
1
answer
224
views
Large V-categories admitting the construction of V-presheaves
By a result of Foltz, and Freyd and Street, a category $C$ is essentially small (i.e. equivalent to a small category) if and only if both $C$ and $[C^{\text{op}}, \mathrm{Set}]$ are locally small. I ...
- 10.6k
11
votes
0
answers
442
views
A right adjoint preserves Phi-colimits if and only if the left adjoint does what?
Let $\Phi$ be a class of categories (e.g. filtered categories), and consider an adjunction $L : \mathbf C \rightleftarrows \mathbf D : R$. A $\Phi$-colimit is a colimit whose diagram is in $\Phi$. We ...
- 10.6k
10
votes
0
answers
228
views
Natural cotransformations and "dual" co/limits
$\DeclareMathOperator{\id}{\mathrm{id}}\DeclareMathOperator{\Hom}{\mathrm{Hom}}\DeclareMathOperator{\UnCoNat}{\mathrm{UnCoNat}}\DeclareMathOperator{\UnNat}{\mathrm{UnNat}}\DeclareMathOperator{\CoNat}{\...
- 11.8k
10
votes
0
answers
145
views
Do pseudo 2-limits commute?
It is a well-known fact that if $F:\mathcal{C}_1\times\mathcal{C}_2\rightarrow \mathcal{D}$ is a functor (between 1-categories), then $F$ has a limit if and only if $F:\mathcal{C}_1\rightarrow Fun(\...
- 1,001
10
votes
0
answers
446
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 ...
- 1,225
10
votes
0
answers
650
views
(Co-)Limits and fibrations of DG-Categories?
First of all, let me see if I got the 1-categorical version right:
Let $\mathcal F:C\to Cat $ be a
(pseudo-) functor. The 2-colimit
$\mathrm{colim}_C\mathcal F$ is then
given by the Grothendieck
...
- 3,249
9
votes
0
answers
103
views
Adjoining a morphism to a finitely complete category
Let $\mathscr C$ be a finitely complete category. Let $x, y$ be objects of $\mathscr C$. We can describe the universal property of freely adjoining a morphism $x \to y$ to $\mathscr C$: it comprises a ...
- 10.6k
9
votes
0
answers
129
views
Is totality a (large) cocompleteness condition?
A locally small category $A$ is called total if its Yoneda embedding $A \to [A^\circ, \mathbf{Set}]$ has a left adjoint. Such categories are necessarily small-cocomplete (since the presheaf category ...
- 10.6k
9
votes
0
answers
103
views
Cocompleteness of enriched categories of algebras
A useful result due to Linton is that for a cocomplete category $C$ and monad $T$ on $C$, if the category of algebras $C^T$ admits reflexive coequalisers, then it is cocomplete (see here for a sketch ...
- 10.6k
9
votes
0
answers
393
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
8
votes
1
answer
258
views
Sequential colimit of iterated quotients of Cauchy sequences
We work in constructive mathematics.
The sets and functions in the foundations form a Grothendieck topos, which means that all colimits exist, and in particular, that all sequential colimits exist. ...
- 2,624
8
votes
0
answers
165
views
Original reference for the Fam construction
For a category $\mathbf C$, the category of families of $\mathbf C$, denoted $\mathrm{Fam}(\mathbf C)$ is the free coproduct completion of $\mathbf C$. Explicitly, the objects of $\mathbf C$ are given ...
- 10.6k
8
votes
0
answers
106
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 ...
- 4,482
8
votes
0
answers
291
views
Loop space functor and sequential colimits of inclusions
The question is about a fact that is mentioned as "evident" everywhere in the literature, so my guess is that some small detail is passing over my head. Here it is:
Let $X_0\hookrightarrow X_1 \...
- 91
8
votes
0
answers
191
views
Yoneda embedding and Horn sentences
The following is taken from Borceux and Bourn's Mal'cev, Protomodular, Homological, and Semi-Abelian Categories.
Metatheorem 0.1.3. Let $\mathcal P$ be a statement of the form $\varphi\implies \psi$, ...
- 10.5k
7
votes
0
answers
266
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
$...
- 446
7
votes
0
answers
417
views
When do Kan extensions preserve colimits?
Assume that we have a pair of functors $Y:A \to B$ and $F:A \to C$ where $A$ is an essentially small category, $B,C$ are cocomplete categories and $Y,F$ preserve colimits. Assume also that for some ...
- 718
7
votes
0
answers
219
views
Pushout of Nisnevich sheaves
Let us consider the projective line $\mathbb{P}^1$ over a field $k$ and take the following open embeddings
$$j_{\epsilon}\colon \mathbb{P}^1\setminus\{0,\infty\} \to \mathbb{P}^1\setminus\{\epsilon\}$$...
- 303
7
votes
0
answers
555
views
maximal tensor product commutes with inductive limits
Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras and let $B$ be an arbitary $C^*$ algebra.
I want to prove $(\varinjlim A_n)\otimes_{max} B \cong \varinjlim (A_n \otimes_{max} B)$. This ...
- 1,188
7
votes
0
answers
260
views
Topological localization of (infinite) inverse limits
The classical localization of topological spaces at a given set of primes $\mathcal{P}$ is a functor $\mathcal{T}\xrightarrow{(-)_{(\mathcal{P)}}}\mathcal{T}$ from a suitable category of topological ...
- 5,596
7
votes
0
answers
639
views
Constructing pointwise Kan extensions as adjoints to some functor
Background
I'm working on formalizing some category theory in Coq at https://bitbucket.org/JasonGross/catdb. Currently, I'm in the process of formalizing pointwise Kan extensions. Partly because I'...
- 695
6
votes
0
answers
152
views
Covering categories with posets
Let $C$ be a small (1-)category.
There is always a poset $D$ and a functor $p : D \to C$ such that:
$p$ is surjective on objects, i.e. for every $c$ in $C$ there is a $d$ in $D$ such that $p (d) = c$,...
- 15.9k
6
votes
0
answers
142
views
Which Ends preserve filtered colimits?
Can we characterize entirely for which categories $C$ the end on $C$ preserves filtered colimits, in a sense that the natural map
$$ \operatorname*{colim}_{i \in I} \int_{x \in C} A_i (x,x) \to \int_{...
- 42.4k
6
votes
0
answers
139
views
Characterisation of essentially algebraic theories with a fixed set of sorts
It is well known (e.g. Palmgren–Vickers's Partial Horn logic and cartesian categories) that many-sorted essentially algebraic theories (equivalently partial Horn theories / quasi-equational theories / ...
- 10.6k
6
votes
0
answers
555
views
When is the dual of a limit the same as the colimit of the duals?
We all know that the dual of the colimit of a diagram in the category of chain complexes (and similar categories) is the limit of the duals diagram. This follows immediately from the general fact that ...
6
votes
0
answers
291
views
When is every element of a coend of abelian groups contained in one of the summands?
Let $I$ be a small category and let $D : I^{\mathrm{op}} \times I \to \mathsf{Ab}$ be a functor. The coend
$$\int^{i \in I} D(i,i)$$
can be constructed as the direct sum $\bigoplus_{i \in I} D(i,i)$ ...
- 63.1k
6
votes
0
answers
812
views
Limit of metric spaces
Let $\{X_n\}_{n\in \mathbb{N}}$ be a collection of T2 topological spaces, with maps $f_n\colon X_n \to X_{n+1}$. These maps are continuous and open. Let $X$ be the direct limit of this system.
Assume ...
- 2,384
6
votes
0
answers
629
views
Inverse limit of graded rings
Let $(I,\le)$ be a directed set and let $(\rho^{\beta\alpha}: R^\beta \to R^\alpha)_{\alpha \le \beta}$ be an $I$-directed system of $\mathbb{Z}$-graded rings whose multiplication is denoted by
$$\...
- 16.2k
5
votes
1
answer
156
views
The inverse limit of a sequence of ring surjections commutes with taking difference subsets of the respective units & gluing in some primes?
Define $R_n := \Bbb{Z}/p_n\#$ the ring of integers modulo primorial $p_n\# = p_n p_{n-1} \cdots p_1$. Let $U_n$ denote the group of units modulo $p_n\#$ in these rings.
Then if $f_{n,n+1}: \Bbb{Z}/p_{...
- 260
5
votes
0
answers
361
views
On a simple alternative correction to Roos' theorem on $\varprojlim^1$
Here is a discussion about an incorrect theorem of Roos, later corrected, some counterexamples and so on. Reading over this, I was a bit shocked because it contradicted something from Weibel's ...
- 1,021
5
votes
0
answers
89
views
Free cocompletion of a 2-category under pseudo colimits, lax colimits, and colax colimits
Let $\mathscr K$ be a small 2-category. It follows from $\mathrm{Cat}$-enriched category theory that the free cocompletion of $\mathscr K$ under strict 2-colimits of 2-functors is given by the 2-...
- 10.6k
5
votes
0
answers
154
views
Compact objects in categories of categories
I am interested in the compact objects of various categories of categories.
For example, $Cat^{small}$ is presentable and has compact objects that are retracts of finite colimits of $\Delta^n$, the $n$...
- 521
5
votes
0
answers
245
views
Duality and compactness for pro vector spaces
I have a somewhat basic question which I haven't been able to piece together from the literature.
Background. We work over a field $\bf{k}$. Consider the category, $\bf{Pro}_{k}$, of pro- vector ...
- 51
5
votes
0
answers
343
views
$V$-cat and $V$-graph: coequalizers in the category of enriched functors
This question is regarding the 1974 JPAA paper $V$-cat and $V$-graph by Harvey Wolff.
To be precise, I don't understand a certain step in the proof of Corollary 2.9, which (the corollary) is crucial ...
- 1,115
5
votes
0
answers
157
views
Are weighted limits terminal in a category of cones?
Consider a Benabou-cosmos $(\mathcal{V},\otimes,J)$, $\mathcal{V}$-categories $\mathcal{I},\mathcal{C}$ and $\mathcal{V}$-functors $\mathcal{W}:\mathcal{I} \rightarrow \mathcal{V}$ and $\mathcal{D}:\...
- 225
5
votes
0
answers
146
views
Applications of $FP_\infty$ groups preserving direct systems
In [1], the author proves that given a group $G$ and a directed system $(M_\lambda)_\lambda$ of $G$-modules, the induced maps $$\varinjlim H^k(G,M_\lambda) \to H^k(G,\varinjlim M_\lambda)$$ are ...
5
votes
0
answers
368
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(\...
- 275
5
votes
0
answers
220
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 ...
- 35.4k
5
votes
0
answers
131
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) \...
- 11.8k
5
votes
0
answers
72
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
5
votes
0
answers
80
views
Is there an analogue of final functors for genuine 2-categorical limits
A functor $I\to J$ of $1$-categories is called final, if each undercategory $(j, I)$ is connected.
More generally, for $(\infty,1)$-categories there is an analogous notion where one requires the ...
- 725
5
votes
0
answers
211
views
A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff
We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...
- 1,270
5
votes
0
answers
448
views
Examples of nonstable ∞-categories in which sifted colimits commute with finite limits
What are some natural examples (if any) of nonstable ∞-categories in which finite limits commute with sifted colimits (or rather just colimits over Δ^op)?
Stable ∞-categories do satisfy this property,...
- 37.8k
5
votes
0
answers
433
views
Is it possible to assume that an étale neighborhood is connected?
I am new to étale topology (though I've seen Grothendieck's sites before).
Let $S:=\mathcal{O}^\textrm{sh}_{X,x}$ be the strict local ring of a point $x$ of a scheme $X=\operatorname{Spec}R$ (over a ...
- 51
5
votes
0
answers
156
views
Contractibility of a poset-indexed colimit
Let $(X,\leq)$ be a poset with distinguished element $p$, and let $P'$ be the poset of "finite chains which weakly descend to $p$" given by all $\sigma = (x_0 \geq x_1 \geq \cdots \geq x_k \geq p)$ ...
- 15.5k
4
votes
0
answers
131
views
Weakening of terminal object in a category
I’ve come across a category $\mathcal{C}$ recently with an object $T$ such that any other object $X$ has a map $f:X\rightarrow T$, and for any two maps $f,g:X\rightarrow T$, there exists a (not ...
- 1,949
4
votes
0
answers
88
views
Homotopy colimits in subcategories of combinatorial model categories
We know that, in a combinatorial simplicial model category $\mathbf{M}$, we can find a regular cardinal $\lambda$ large enough that all $\lambda$-filtered homotopy colimits can be computed as strict ...
- 1,071
4
votes
0
answers
244
views
Free vector space on a filtered limit
$\DeclareMathOperator\Set{Set}\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Coalg{Coalg}\DeclareMathOperator\ProVect{ProVect}\DeclareMathOperator\prolim{prolim}
$Let $K$ be a field and $F: \Set \...
- 1,361