# Tagged Questions

**3**

votes

**1**answer

173 views

### Projectives and Injectives in Functor Categories

Would it be possible to enlighten me (or even better give a reference) about enough projectives (injectives) in functor categories?
Here is a precise question. Let $C$ be a small category, whose ...

**2**

votes

**1**answer

143 views

### Showing a functorial isomorphism

I'm having trouble with this exercise from Elements of the Representation Theory of Associative Algebras I: Techniques of Representation Theory.
The exercise in question is from chapter IV.
So, let ...

**5**

votes

**1**answer

153 views

### In what generality does Eilenberg-Watts hold?

In homological algebra, the Eilenberg-Watts theorem states that if $F\colon\text{Mod}_R\to\text{Mod}_S$ is a right-exact coproduct preserving functor of modules, then $F\cong-\otimes_R F(R).$ The ...

**1**

vote

**0**answers

83 views

### Intuitive idea: What is the motivation for relative homological algebra

Basically I was wondering what is the purpous of choosing a smaller class of epis in a category? Does it allow for the existence of "more" "projective-like" objects?
What are some applications? For ...

**5**

votes

**0**answers

193 views

### Sign problem for the shift functor on DG modules

Let $\mathcal{A}$ be a Differential graded $k$ category, where $k$ is a commutative ring. I want to extend the shift functor $[1]$ on chain complexes to dg modules over $\mathcal{A}$. Now a right dg ...

**1**

vote

**0**answers

209 views

### Tensor product of d.g-algebras

I'd like to prove that the tensor product functor $- \otimes Y$, where $Y$ is a d.g-algebra over a field of characteristic 0, preserves finite products of d.g-algebras. This statement is in a paper by ...

**1**

vote

**1**answer

122 views

### arrows in the injective representations of quivers

Let $Q$ be a quiver (a directed graph) and $E$ be a representation of $Q$ by R-modules, that is, for each vertex $v$ in $Q$ there exists an R-module $E(v)$ and for each arrow $a:v\to w$ there exists a ...

**4**

votes

**1**answer

367 views

### Splitness of commutative diagrams

Consider the following commutative diagram in the category of $R$-modules where $R$ is an associative ring with identity and all modules are unital.
$
\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ ...

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**0**answers

127 views

### Is this a pure monomorphism?

Let $M$ be a representation of a quiver $Q=(V, E)$ by $R$-modules. By $M^{+}$ we mean a representation of $Q^{op}$ with $M^{+}(v)=\mathrm{Hom}(M(v), \frac{Q}{Z})$. One can easily see that there is ...

**1**

vote

**1**answer

201 views

### finitely presented representations

Let Q=(V,E) be a direct graph where V is the set of all its vertices and E denotes the set of all its arrows. $X$ is called a representation of Q by modules if it is a functor from Q to R-Mod. i.e. ...

**1**

vote

**1**answer

204 views

### Pure monomorphism of functors-

Suppose that $F, G: Q\rightarrow R{\rm -Mod}$ be two covariant functors where $Q$ is an abelain category and $R$ is a commuatative ring. Also let $\eta: F\rightarrow G$ be a natural transformation ...

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**1**answer

224 views

### Brutal truncation of indecomposable complexes

Let $P^{\bullet} = (P^i, d^i)_{i\leq 0}$ be an indecomposable object in category of complexes bounded above, where each $P_i$ is a finitely generated projective module over an finite dimensional ...

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203 views

### Extension to a right exact functor

Setup. Let $k$ be a field, $\mathcal{C}$ be a $k$-linear abelian category, $\mathcal{D}$ an arbitrary $k$-linear category. Let $\mathcal{C}'$ be a full subcategory of $\mathcal{C}$ with the following ...

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**0**answers

142 views

### Quasi-equivalences of DG categories

There are several definitions of a quasi-equivalence $\newcommand{\T}{\mathscr{T}}F : \T \to \T'$ of DG categories in the literature, e.g.
(i) the induced functor $H^0(F) : H^0(\T) ...

**4**

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**1**answer

193 views

### Is the functor $mod(Tw\mathcal{C})\rightarrow mod(\mathcal{C})$ cohomologically full and faithful?

Let $\mathcal{C}$ be a $c$-unital $A_\infty$-category. If $\mathcal{A}$ is a $c$-unital and triangulated $A_\infty$-category, then there is a $c$-unital $A_\infty$- functor
$$Tw: ...

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196 views

### Does abstract nonsense of model categories determine the “nonlinear” morphisms of $L_\infty$ algebras?

Recall that a Lie algebra is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a ...

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**2**answers

437 views

### Properties of quotient categories.

I asked this on math.stackexchange.com, but didn't get any answer.
Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre/thick/dense ...

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**0**answers

140 views

### Model structure on the category of chain complexes in an abelian category gives rise to the derived category

Let $\mathcal A$ be an abelian category with enough projectives. Consider a morphism $f \colon X^\bullet \longrightarrow Y^\bullet$ in the category $C(\cal A)$ of chain complexes in $\mathcal A$.
Is ...

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**0**answers

105 views

### Why are the left exact functors from an abelian category to abelian groups cocomplete and have a injective generator?

Let $\mathcal{C}$ be an abelian category, $\mathcal{Ab}$ the category of abelian groups and $Lex(\mathcal{C}, \mathcal{B})$ the category of left exact functors between abelian categories.
What is the ...

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**2**answers

127 views

### Why every complex of injectives is homotopically injective (provided that, the injective dimension is finite)?

Let $\scr A$ be an abelian category with exact products and a cogenerator (e.g. $\scr A$ is a category of modules). Let ${\mathbf K}(\scr A)$ be the homotopy category of cochain complexes over $\scr ...

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**1**answer

262 views

### Drect limit of sequences

Let $\mathcal{C}$ is a grothendiect category and consider all of what follows in $\mathcal{C}$.
Let $$\{\varepsilon_i: 0\to A_i \to B_i \to C_i\to 0\ ,\ \phi_i^j\}$$ be a direct system of
short ...

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**0**answers

187 views

### Homology of the dg-nerve vs Hochschild homology of the dg-category

Lurie in Higher Algebra, section 1.3 associates a quasi-category to a dg-category A via the so called dg-nerve construction, extending the classical nerve. I have a feeling the homology of the ...

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**2**answers

284 views

### Why is $Lex(\mathcal{A},\mathcal{Ab})$ abelian? Does $Lex(\mathcal{A},\mathcal{Ab})\rightarrow Func(\mathcal{A},\mathcal{Ab})$ admit a left-adjoint?

What is the best way to show, that $Lex(\mathcal{A},\mathcal{Ab})$ is abelian, where $\mathcal{A}$ is an abelian category and $\mathcal{Ab}$ is the category of abelian groups from scratch?
There is ...

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**2**answers

320 views

### Is this square a push-out square?

Consider the following diagram which lives in the category of $R$-modules.
$$
\begin{array}{ccccccccc}
0 & \xrightarrow{i} & A & \xrightarrow{f} & B & \xrightarrow{q} & C ...

**3**

votes

**1**answer

217 views

### Resolutions chain homotopic to projective ones

Motivation. In my research I have a situation where a monoid $M$ is acting by nice cellular maps on a contractible cell complex and so the augmented chain complex is a resolution of the trivial module ...

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vote

**1**answer

232 views

### Coequalizer in category of dg-algebras

It is known that there is a model structure on category of dg algebras (non-commutative over arbitrary commutative ring). In particular it is complete and co-complete category. My question is how to ...

**3**

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**0**answers

126 views

### Repeated Homotopy Category of Chain Complexes

Consider an additive category $\mathcal{C}$. It is known that the category $Ch(\mathcal{C})$ of chain complexes in $\mathcal{C}$ is again an additive category and hence one can consider the category ...

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396 views

### Examples of applications of the Freyd-Mitchell embedding theorem.

The Freyd-Mitchell embedding theorem states the following:
Let $\mathcal{A}$ be a small abelian category. There exists a unital ring $R$ and a full, faithful and exact functor ...

**5**

votes

**2**answers

365 views

### Are subfunctors of left exact functors also left exact?

Consider the direct image functor $f_*: Sh(X) \rightarrow Sh(Y)$, let $X$ and $Y$ be topological spaces, let $f: X \rightarrow Y$ be a continuous map, let $G \in Sh(X)$ be a sheaf. I was reading this ...

**13**

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**1**answer

520 views

### How much of a variety can be reconstructed from codimension-zero data?

This is an improved version of this question. Maybe it should be an edit -- I'm not sure what the MO convention is.
I'm curious, more or less, how much information one can get out of the derived ...

**1**

vote

**1**answer

294 views

### How much of a variety can be reconstructed from codimension-zero data?

Suppose $X$ is a nice (finite type over $\mathbb{C}$, smooth and proper if necessary) variety. Suppose we're given the $\mathbb{C}$-points of $X$ as a set and for any finite subset $S\subset X$ we ...

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**2**answers

328 views

### Let $M$ be a $R$-Bimodule that happens to be projective, is its associated left $R \otimes R^{op}$-module projective too?

Let $k$ be a commutative ring, a $R$-Bimodule $M$ over a $k$-algebra $R$ is a $k$-module with two actions of $R$ on $M$, on the left and on the right, the classical example of this being $R$ itself ...

**5**

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226 views

### Exactness is often an open condition. How often?

Let $\mathcal C$ denote an abelian category. Consider a formal $\mathbb Z$-graded object $V_\bullet$ in $\mathcal C$, by which I simply mean a $\mathbb Z$-indexed list $n\mapsto V_n$ of objects. A ...

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votes

**1**answer

490 views

### Universality of Ext functor using Yoneda extensions

Theses are simple and natural questions, but I could not find anything about it. If anyone has an answer or a reference this would be very much appreciated.
Let $\mathcal{C}$ be an abelian category ...

**12**

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**1**answer

419 views

### Using the Yoneda embedding to talk about exactness in an additive category

Suppose I have an additive category $\mathcal{C}$ and a pair of composable arrows:
$$A \longrightarrow B \longrightarrow C.$$
It makes no sense to ask if this sequence is exact at $B$ since the ...

**2**

votes

**0**answers

840 views

### Homotopy Equivalence of Mapping Cones

Suppose we have an additive category $\mathcal{A}$ and we consider the homotopy category of chain complexes in $\mathcal{A}$, denoted by $\mathcal{K}(\mathcal{A})$. If we have $X_1, X_1', X_2, X_2' ...

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votes

**3**answers

328 views

### Example: a pair of nonisomorphic parallel morphisms with isomorphic cones

First of all, let me fix some notation.
Let $\mathcal D$ be a triangulated category in the sense of Verdier-Grothendieck (for example, the homotopy category $\mathbf{K}(k)$ of cochain complexes ...

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votes

**3**answers

773 views

### Characterising categories of vector spaces

Consider the category $FdVect_k$ of finite dimensional $k$-vector spaces, for some given field. It is abelian, semisimple, in that each object is a finite sum of simple objects (of which there is only ...

**7**

votes

**1**answer

360 views

### Is there an algebraic “derived mapping space” construction that encompasses both Hochschild homology and loop spaces of non-simply-connected spaces?

I'm looking for directions to the literature that might contain fairly explicit constructions that might be called (the algebra of functions on) the "derived mapping space" from a simplicial set to a ...

**0**

votes

**1**answer

117 views

### Inductive vs projective limit of sequence of split surjections

Let
$$
A_1\twoheadrightarrow
A_2\twoheadrightarrow
A_3\twoheadrightarrow
A_4\twoheadrightarrow
\cdots
$$
be an inductive sequence of countable abelian groups, the connecting homomorphisms of which are ...

**6**

votes

**1**answer

265 views

### $A_\infty$-categories and their equivalent dg-categories: the case of $\mathcal{RH}om(\mathcal A,\mathcal B)$

Hello everyone,
I'm currently working on dg-categories, in particular I'm looking for some convenient characterization of the dg-category $\mathcal{RH}om(\mathcal A,\mathcal B)$, for two given ...

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votes

**3**answers

331 views

### Free Objects in Functor Categories

As a bit of background, consider the category of all covariant, additive functors from a small Abelian category $C$ to Abelian groups, which I'll denote $[C,Ab]$. First, $[C,Ab]$ is an Abelian ...

**5**

votes

**2**answers

336 views

### Resolutions of unbounded complexes and homotopy (co)limits.

I want to understand once and for all what the resolution of an unbounded complex is. I've been trying to read 'Homotopy limits in triangulated categories' by Marcel Bokstedt and Amnon Neeman and ...

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vote

**2**answers

426 views

### Theorem on composition of derived functors, question about proof

I got a question about a proof I found in Gelfand-Manin's "Methods of homological algebra" (Page 200):
Theorem 1. Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be three abelian categories, $F: ...

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votes

**0**answers

174 views

### Reference Request: The Categorification of $\mathbb{Z}$ as cochain complexes of vector spaces

Just as the fact that a categorification of $\mathbb{N}$ is the category of finite dimensional vector spaces, a categorification of $\mathbb{Z}$ (in my mind) is the category of bounded cochain ...

**5**

votes

**1**answer

311 views

### Does there exist a wide but not full abelian subcategory of an abelian category?

Does there exist an example of an abelian category A and abelian subcategory B, where B is wide but not full (as a subcategory of A)?

**4**

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492 views

### $Ext$ preserves products

Let $\mathcal{A}$ be an $Ab4$ category. Define $\mathrm{Ext}^{n}(-,-):\mathcal{A}^{op}\times\mathcal{A}\rightarrow Ab$ bifunctor using n-extensions. Consider $A$ an object of $\mathcal{A}$. Does ...

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vote

**0**answers

157 views

### Two-point desuspension for augmented chain complexes?

Let $X$ be a chain complex augmented over $\mathbf{Z}$ with augmentation $\varepsilon_X:X_0 \to \mathbf{Z}$. We define $[1](X)$ to be the $\mathbf{Z}$-augmented chain complex such that $[1](X)_0 = ...

**3**

votes

**2**answers

468 views

### left derived functors commute with filtered colimits

Let $\mathcal{A}$ be an $\mathbf{AB5}$ category with enough projectives and let $F:\mathcal{A}\rightarrow\mathcal{B}$ be a right exact functor into abelian category that commutes with filtered ...

**11**

votes

**5**answers

1k views

### Cov. right-exact additive functors that don't commute with direct sums?

Background
Recently, I have been writing up some notes on derived functors and I came across the Eilenberg-Watts theorems [1], which essentially explain why it is hard to find derived functors ...