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I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.4 Examples of Categories ...
I need some further help in order to fully understand some further aspects of Section 1.4 Example 13 ...
Section 1.4 Example 13 reads as follows:View attachment 8347
View attachment 8348
In the above text by Awodey we read the following:
" ... ...In detail, a homomorphism from a monoid $$M$$ to a monoid $$N$$ is a function $$h : M \to N$$ such that for all $$m,n \in M$$,$$ h( m \bullet_M n ) = h(m) \bullet_N h(N) $$
and
$$h( u_M ) = u_N $$
Observe that a monoid homomorphism from $$M$$ to $$N$$ is the same thing as a functor from $$M$$ regarded as a category to $$N$$ regarded as a category. ... ... "I cannot see how (exactly and rigorously) the monoid homomorphism specified above fits the 3 conditions (a), (b) and (c) that Awodey lays down for a functor (see text below) ... ...
Can someone please demonstrate explicitly and rigorously how (exactly) the monoid homomorphism specified above fits the 3 conditions (a), (b) and (c) that Awodey lays down for a functor (see text below) ... ...Help will be appreciated ... ...
Peter=======================================================================================
*** NOTE ***
In order to help and answer the question posed in the above post, MHB readers of the post need access to Awodey's definition of a functor ... so I am providing access to the same ... as follows:View attachment 8345
https://www.physicsforums.com/attachments/8346Hope that helps ...
Peter
I need some further help in order to fully understand some further aspects of Section 1.4 Example 13 ...
Section 1.4 Example 13 reads as follows:View attachment 8347
View attachment 8348
In the above text by Awodey we read the following:
" ... ...In detail, a homomorphism from a monoid $$M$$ to a monoid $$N$$ is a function $$h : M \to N$$ such that for all $$m,n \in M$$,$$ h( m \bullet_M n ) = h(m) \bullet_N h(N) $$
and
$$h( u_M ) = u_N $$
Observe that a monoid homomorphism from $$M$$ to $$N$$ is the same thing as a functor from $$M$$ regarded as a category to $$N$$ regarded as a category. ... ... "I cannot see how (exactly and rigorously) the monoid homomorphism specified above fits the 3 conditions (a), (b) and (c) that Awodey lays down for a functor (see text below) ... ...
Can someone please demonstrate explicitly and rigorously how (exactly) the monoid homomorphism specified above fits the 3 conditions (a), (b) and (c) that Awodey lays down for a functor (see text below) ... ...Help will be appreciated ... ...
Peter=======================================================================================
*** NOTE ***
In order to help and answer the question posed in the above post, MHB readers of the post need access to Awodey's definition of a functor ... so I am providing access to the same ... as follows:View attachment 8345
https://www.physicsforums.com/attachments/8346Hope that helps ...
Peter