Intereting Posts

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At what point does exponential growth dominate polynomial growth?
Spectrum of finite $k$-algebras
isoperimetric inequality using Fourier analysis
Quadratic sieve algorithm
IMO 2016 P3, number theory with the area of a polygon
Complexity classes and number of problems
Can any two disjoint nonempty convex sets in a vector space be separated by a hyperplane?
The asymptotic behaviour of $\sum_{k=1}^{n} k \log k$.
Show that $\inf(\frac{1}{n})=0$.
What are the solutions to $z^4+1=0$?

Suppose I have a directed system $C_i$, $i\in\mathbb{N}$ of chain complexes over free abelian groups (bounded below degree $0$)

$$C_i=0\rightarrow C^{0}_{(i)}\rightarrow C^{1}_{(i)}\rightarrow\cdots\rightarrow C^{n-1}_{(i)}\rightarrow C^n_{(i)}\rightarrow \cdots$$

with chain maps $f_i\colon C_i\rightarrow C_{i+1}$. Can I say that

$$H_*\left(\lim_{\rightarrow}(C_i,f_i)\right)\cong\lim_{\rightarrow}\left(H_*(C_i),(f_i)_*\right),$$

where $\displaystyle\lim_{\rightarrow}$ is the direct limit (colimit) in the respective category, $H_*(C)$ is the *th homology of the the chain complex $C$, and $f_*$ is the induced homomorphism in homology of the chain map $f$?

I imagine the answer will involve some categorical property of the functor $H_*$.

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- Foundations book using category theory for student embarking on PhD in mathematical biology?
- Grothendieck Topology on the Category of Elements

- why is this composition well defined?
- Why is every category not isomorphic to its opposite?
- Advice on self study of category theory
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- $\operatorname{Ann}_RA+\operatorname{Ann}_RB\subseteq\operatorname{Ann}_R\operatorname{Ext}^n_R(A,B)$?
- Additive functor over a short split exact sequence.
- Is it possible to define a ring as a category?

Any exact functor between abelian categories will preserve homology, and colimits indexed by filtered or directed diagrams are exact in $\mathbf{Ab}$.

The first claim is straightforward, because homology is computed using kernels and cokernels. Indeed, given a chain complex $C_{\bullet}$, we form the object of cycles as a kernel,

$$0 \longrightarrow Z_n \longrightarrow C_n \longrightarrow C_{n-1}$$

and we form the object of boundaries as a cokernel,

$$0 \longrightarrow Z_{n+1} \longrightarrow C_{n+1} \longrightarrow B_n \longrightarrow 0$$

and then the homology object is also a cokernel:

$$0 \longrightarrow B_n \longrightarrow Z_n \longrightarrow H_n \longrightarrow 0$$

The second claim is best checked by hand using the concrete description of filtered/directed colimits in $\mathbf{Ab}$. By general nonsense, colimits are additive and preserve cokernels, so it is enough to check that kernels are preserved by filtered/directed colimits.

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