Questions tagged [limits-and-colimits]
For questions on limits and colimts in the sense of category theory, and related notions.
99
questions with no upvoted or accepted answers
14
votes
0
answers
289
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 ...
14
votes
0
answers
842
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:
$$
\int^x G(x,x)
$$
...
12
votes
1
answer
178
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 ...
11
votes
0
answers
334
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
votes
0
answers
138
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(\...
9
votes
0
answers
122
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 ...
9
votes
0
answers
99
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 ...
9
votes
0
answers
195
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}{\...
9
votes
0
answers
401
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 ...
9
votes
0
answers
351
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 $X_\xi$ ...
8
votes
1
answer
186
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. ...
8
votes
0
answers
94
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 ...
8
votes
0
answers
143
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 ...
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 ...
8
votes
0
answers
279
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 \...
8
votes
0
answers
190
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 $\varphi\implies \psi$, ...
8
votes
0
answers
629
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
...
7
votes
0
answers
255
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
$...
7
votes
0
answers
212
views
Pushout of Nisnevich sheaves
Let us consider the projective line $\mathbb{P}^1$ over a field $k$ and take the following open embeddings
$$j_{\epsilon}\colon \mathbb{P}^1\setminus\{0,\infty\} \to \mathbb{P}^1\setminus\{\epsilon\}$$...
7
votes
0
answers
530
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 ...
7
votes
0
answers
251
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 ...
7
votes
0
answers
628
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'...
6
votes
0
answers
134
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
$$ \operatorname*{colim}_{i \in I} \int_{x \in C} A_i (x,x) \to \int_{...
6
votes
0
answers
136
views
Characterisation of essentially algebraic theories with a fixed set of sorts
It is well known (e.g. Palmgren–Vickers's Partial Horn logic and cartesian categories) that many-sorted essentially algebraic theories (equivalently partial Horn theories / quasi-equational theories / ...
6
votes
0
answers
464
views
When is the dual of a limit the same as the colimit of the duals?
We all know that the dual of the colimit of a diagram in the category of chain complexes (and similar categories) is the limit of the duals diagram. This follows immediately from the general fact that ...
6
votes
0
answers
283
views
When is every element of a coend of abelian groups contained in one of the summands?
Let $I$ be a small category and let $D : I^{\mathrm{op}} \times I \to \mathsf{Ab}$ be a functor. The coend
$$\int^{i \in I} D(i,i)$$
can be constructed as the direct sum $\bigoplus_{i \in I} D(i,i)$ ...
6
votes
0
answers
377
views
When do Kan extensions preserve colimits?
Assume that we have a pair of functors $Y:A \to B$ and $F:A \to C$ where $A$ is an essentially small category, $B,C$ are cocomplete categories and $Y,F$ preserve colimits. Assume also that for some ...
6
votes
0
answers
753
views
Limit of metric spaces
Let $\{X_n\}_{n\in \mathbb{N}}$ be a collection of T2 topological spaces, with maps $f_n\colon X_n \to X_{n+1}$. These maps are continuous and open. Let $X$ be the direct limit of this system.
Assume ...
5
votes
0
answers
202
views
Duality and compactness for pro vector spaces
I have a somewhat basic question which I haven't been able to piece together from the literature.
Background. We work over a field $\bf{k}$. Consider the category, $\bf{Pro}_{k}$, of pro- vector ...
5
votes
0
answers
301
views
$V$-cat and $V$-graph: coequalizers in the category of enriched functors
This question is regarding the 1974 JPAA paper $V$-cat and $V$-graph by Harvey Wolff.
To be precise, I don't understand a certain step in the proof of Corollary 2.9, which (the corollary) is crucial ...
5
votes
0
answers
143
views
Are weighted limits terminal in a category of cones?
Consider a Benabou-cosmos $(\mathcal{V},\otimes,J)$, $\mathcal{V}$-categories $\mathcal{I},\mathcal{C}$ and $\mathcal{V}$-functors $\mathcal{W}:\mathcal{I} \rightarrow \mathcal{V}$ and $\mathcal{D}:\...
5
votes
0
answers
122
views
Applications of $FP_\infty$ groups preserving direct systems
In [1], the author proves that given a group $G$ and a directed system $(M_\lambda)_\lambda$ of $G$-modules, the induced maps $$\varinjlim H^k(G,M_\lambda) \to H^k(G,\varinjlim M_\lambda)$$ are ...
5
votes
0
answers
201
views
Pushout of $C^*$-algebras using generalised morphisms
There is a known construction of pushout of $C^*$-algebras, or rather, the amalgamated free product, which is universal for commutative squares of $*$-homomorphisms. Jensen and Thomsen in their book ...
5
votes
0
answers
122
views
Is the module of Kähler differentials a coend?
Let $\phi\colon R\to S$ be a ring map. The module of Kähler differentials $\Omega_{S/R}$ of $\phi$ can be constructed as the following coequaliser:
$$\left(\bigoplus_{(a, b)\in S^2} S[(a, b)]\right) \...
5
votes
0
answers
66
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 ...
5
votes
0
answers
78
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 ...
5
votes
0
answers
204
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 ...
5
votes
0
answers
397
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,...
5
votes
0
answers
401
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 ...
5
votes
0
answers
149
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 $\sigma = (x_0 \geq x_1 \geq \cdots \geq x_k \geq p)$ ...
5
votes
0
answers
584
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
$$\...
4
votes
0
answers
128
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 ...
4
votes
0
answers
85
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 ...
4
votes
0
answers
241
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 \...
4
votes
0
answers
311
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(\...
4
votes
0
answers
191
views
When does the canonical $t$-structure restrict to perfect complexes?
I am interested in non-Noetherian(!) rings such that the canonical $t$-structure on $D(R)$ (the derived category of left $R$-modules) restricts to perfect complexes i.e. to the subcategory of ...
3
votes
0
answers
144
views
Weighted limit calculus
In coend calculus by Fosco Loregian, it is mentioned that Lawvere conjectured that co/ends constitute a categorification of logical calculus. In his own words:
The somewhat far-fetched conjecture ...
3
votes
0
answers
81
views
When is Tw(C) {ω}-filtered?
I am interested in categories $\mathsf{C}$ for which coends commute with $\omega$-chain limits. That is, given a chain of profunctors $P_n \colon \mathsf{C}^{op} \times \mathsf{C} \to \mathsf{Set}$ ...
3
votes
0
answers
177
views
For which categories $D$ is a $D^{\vartriangleleft\vartriangleright}$-shaped diagram in a stable $\infty$-category a limit iff it is a colimit?
Throughout, I'll omit the "$\infty$" from the term "$\infty$-category".
It is well-known (and sometimes even included in the definition, although not by Lurie) that pushouts and ...
3
votes
0
answers
108
views
Density with respect to a family of diagrams, versus a class of weights
In Theorem 5.19 of Kelly's Basic Concepts of Enriched Category Theory, it is proven that a fully faithful functor $K \colon \mathcal A \to \mathcal C$ is dense if and only if $\mathcal C$ is the ...