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I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ...
I need some further help in order to fully understand some aspects of the definition of the product of two categories as it applies to the category Groups ... ...
The definition of the product of two categories ... reads as follows:
For the category Groups of groups and group homomorphisms, the product category of two categories ##C## and ##D##, namely ##C \times D##, has objects of the form ##(G,H)## where ##G## and ##H## are groups and where ##G \in C## and ##H \in D## ...
Arrows would be of the form
##(f,g) : (G,H) \to (G',H')##
for ##f: G \to G'## and ##g: H \to H'##
... BUT ...
... now ... you would expect ... indirectly at least! ... that the definition of the category and its rules would specify the product ...
##(g_1, h_1) \star (g_2, h_2) = (g_1 \bullet_1 g_2, h_1 \bullet_2 h_2)## ... ...
... BUT! ...
how does the product category definition imply this in the case of groups ...
Hope someone can help ...
Peter
I need some further help in order to fully understand some aspects of the definition of the product of two categories as it applies to the category Groups ... ...
The definition of the product of two categories ... reads as follows:
For the category Groups of groups and group homomorphisms, the product category of two categories ##C## and ##D##, namely ##C \times D##, has objects of the form ##(G,H)## where ##G## and ##H## are groups and where ##G \in C## and ##H \in D## ...
Arrows would be of the form
##(f,g) : (G,H) \to (G',H')##
for ##f: G \to G'## and ##g: H \to H'##
... BUT ...
... now ... you would expect ... indirectly at least! ... that the definition of the category and its rules would specify the product ...
##(g_1, h_1) \star (g_2, h_2) = (g_1 \bullet_1 g_2, h_1 \bullet_2 h_2)## ... ...
... BUT! ...
how does the product category definition imply this in the case of groups ...
Hope someone can help ...
Peter
Attachments

Awodey  1  Product of Two Categories, Section 1.6 ... ... PART 1 .png27.9 KB · Views: 376

Awodey  2  Product of Two Categories, Section 1.6 ... ... PART 2.png24.1 KB · Views: 372

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