Definition df-termc 49963

Description: Definition of the proper class (termcnex 50066) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 49976). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 50005). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 50010). See also dftermc3 50021 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 50021, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 49981.

Followed directly from the definition, terminal categories are thin (termcthin 49967). The opposite category of a terminal category is "almost" itself (oppctermco 49995). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 50030).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 49991) and terminal objects (dftermo4 49992). The universal properties provide an alternate proof of initoeu1 17969, termoeu1 17976, initoeu2 17974, and termoeu2 49728. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 50006).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17645, 0thincg 49948, func0g 49579, 0funcg 49575, and initc 49581.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 49962	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1546	. . . . . 6 class 𝑐
4		cbs 17170	. . . . . 6 class Base
5	3, 4	cfv 6485	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1546	. . . . . 6 class 𝑥
8	7	csn 4555	. . . . 5 class {𝑥}
9	5, 8	wceq 1547	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1786	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 49907	. . 3 class ThinCat
12	10, 2, 11	crab 3391	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1547	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is referenced by:  istermc  49964