Calculate (co)limits as (co)equalisers (two examples)

There is a general formulation for constructing limits as equalisers: see Theorem 1 in Section V.2, Maclane. For the dual version, see Theorem A.2.1 in Appendix A written by me.

The constructions look like these (see the links above for details):

limitscolimits

But in practice, these diagrams may not be helpful to see what the equalisers should be. Now I give proofs for the (co)equalisers in two examples: the connected component of a simplicial set and the sheaf condition.

The connected components [Background]

For definitions and other backgrounds, see Subsection 00G5. For the record, see [P12, DLOR07] for the cosimplicial identities and Tag 000G for simplicial identities. (These identities are used in my proofs.)

Two examples of (co)limits as (co)equalisers

Pdf here: Two examples of (co)limits as (co)equalisers

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[Short Notes] Non-compactness of the closed unit ball in an infinite-dimensional Banach space

This is about an exercise in [Bass]:

Exercise 19.5. Prove that if H is infinte-dimensional, that is, it has no finite basis, then the closed unit ball in H is not compact.

Proof. Choose an orthonormal basis \{x_i\}, then ||x_i-x_j||^2=||x_i||^2+||x_j||^2=2. This means the sequence is not Cauchy hence has no convergent subsequence.

For a Banach space, by Riesz’s lemma to find a non-Cauchy sequence.

 

[Bass] Bass, R. F. (2013). Real analysis for graduate students. Createspace Ind Pub.